Finite Group Databasebeta

← #33130322

Subgroups — #33130322

Order 18688.

854 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614239C211yesmaximal, characteristic
2D4×C4⋊Dic73C211yesmaximal, characteristic
3#22611490C211yesmaximal, characteristic, complemented
4#22611404C211yesmaximal, characteristic, complemented
5#22611354C211yesmaximal, characteristic, complemented
6#22611354C211yesmaximal, characteristic, complemented
7#22610503C211yesmaximal, characteristic, complemented
8#22610503C211yesmaximal, characteristic, complemented
9#22610121C211yesmaximal, characteristic, complemented
10#22610121C211yesmaximal, characteristic, complemented
11#22610018C211yesmaximal, characteristic, complemented
12#22610018C211yesmaximal, characteristic, complemented
13#22609960C211yesmaximal, characteristic, complemented
14C2×#17988350C211yesmaximal, characteristic, complemented
15C73×#1869C211yesmaximal, characteristic
16C2×#14556572—21no—
17C2×D4×Dic73—21no—
18C22×C4⋊Dic73C2211yescharacteristic
19C22×C4⋊Dic73C2211yescharacteristic
20C22×C4⋊Dic73C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556525C2211yescharacteristic
23C2×#14556525C2211yescharacteristic
24C2×#14556525C2211yescharacteristic
25#17988517—21no—
26#17988512—21no—
27#17988486—21no—
28#17988485C2211yescharacteristic
29#17988485C2211yescharacteristic
30#17988462—21no—
31#17988462—21no—
32#17988457C2211yescharacteristic
33#17988448—21no—
34#17988447C2211yescharacteristic
35#17988447C2211yescharacteristic
36#17988447C2211yescharacteristic
37#17988447C2211yescharacteristic
38#17988447C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40#17988350C2211yescharacteristic
41#17988350C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C23×C8×C73C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C23×Dic73—21no—
60C23×Dic73—21no—
61C23×Dic73—21no—
62#14556572—44no—
63D4×Dic73—44no—
64C2×C4⋊Dic73—22no—
65C2×C4⋊Dic73—22no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73—22no—
73C2×C4⋊Dic73—41no—
74C2×C4⋊Dic73—41no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C4×Dic73—21no—
91#14556525—22no—
92#14556525—22no—
93#14556525—41no—
94#14556525—41no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C22×D4×C73D411yescharacteristic
104C23×C4×C73C2311yescharacteristic
105C23×C4×C73C2311yescharacteristic
106C22×C8×C73—22no—
107C22×C8×C73—22no—
108C22×C8×C73—41no—
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C4×D4×C73—22no—
114C4×D4×C73—22no—
115C2×C4⋊C4×C73D411yescharacteristic
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C22⋊C8×C73—22no—
120C22⋊C8×C73—22no—
121C22⋊C8×C73—41no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—42no—
131C22×Dic73—42no—
132C22×Dic73—42no—
133C22×Dic73—42no—
134C22×Dic73—42no—
135C22×Dic73—42no—
136C22×Dic73—42no—
137C22×Dic73—81no—
138C22×Dic73—81no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—42no—
155C4⋊Dic73—42no—
156C4⋊Dic73—42no—
157C4⋊Dic73—42no—
158C4⋊Dic73—81no—
159C4⋊Dic73#4112yes—
160#11924007—42no—
161#11924007—42no—
162C22⋊Dic73—42no—
163C22⋊Dic73—42no—
164C22⋊Dic73—42no—
165C22⋊Dic73—42no—
166C22⋊Dic73—42no—
167C4×Dic73—42no—
168C24×C73C2×D411yescharacteristic
169C24×C73C2×D411yescharacteristic
170C2×D4×C73—22no—
171C2×D4×C73—22no—
172C2×D4×C73D814yes—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73#4112yes—
188C22×C4×C73#4112yes—
189C22×C4×C73C2411yescharacteristic
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—44no—
202C2×C8×C73#4112yes—
203C4⋊C4×C73—22no—
204C22⋊C4×C73—22no—
205C22⋊C4×C73—22no—
206C42×C73—21no—
207C42×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73#12312yes—
219C23×C73C2×D812yes—
220C23×C73C2×D812yes—
221C23×C73#13712yes—
222C23×C73C22×D411yescharacteristic
223D4×C73—28no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73#12312yes—
243C2×C4×C73C2×D812yes—
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73H(5,𝔽2)11yescharacteristic
249C2×C4×C732−1+411yescharacteristic
250C2×Dic73—22no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—84no—
275C2×Dic73—84no—
276C2×Dic73—84no—
277C8×C73—22no—
278C8×C73—41no—
279C8×C73—41no—
280C8×C73—41no—
281C22×C73—22no—
282C22×C73—22no—
283C22×C73—22no—
284C22×C73—22no—
285C22×C73—22no—
286C22×C73—22no—
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73#48014yes—
295C22×C73C2×#12311yescharacteristic
296C22×C73#54611yescharacteristic
297C22×C73#54711yescharacteristic
298C22×C73C22×D811yescharacteristic
299C22×C73C2×#13711yescharacteristic
300C22×C73#57711yescharacteristic
301C22×C73#57811yescharacteristic
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—41no—
305C4×C73—41no—
306C4×C73#52911yescharacteristic
307C4×C73#53911yescharacteristic
308C4×C73#55111yescharacteristic
309C4×C73#55211yescharacteristic
310Dic73—42no—
311Dic73—81no—
312Dic73—81no—
313Dic73—81no—
314Dic73—81no—
315Dic73—81no—
316Dic73—81no—
317Dic73—81no—
318#24933—731nomaximal, Sylow
319C2×C73—24no—
320C2×C73—42no—
321C2×C73#241911yescharacteristic
322C2×C73C2×#48011yescharacteristic
323C2×C73#309711yescharacteristic
324C2×C73#329411yescharacteristic
325C2×C73#331211yescharacteristic
326C2×C73#333411yescharacteristic
327C2×C73#334411yescharacteristic
328#2486—731no—
329D4×C4⋊C4—731no—
330#2099—731no—
331#2083—731no—
332#2075—731no—
333#2075—731no—
334#1918—731no—
335#1918—731no—
336#1869D7311yescharacteristic
337#1852—731no—
338#1852—731no—
339#1823—731no—
340#1823—731no—
341#1808—731no—
342C2×#340—731no—
343C73#2493311yescharacteristic, complemented, Sylow
344C23×C8C2×D7311yescharacteristic
345C2×#123—1461no—
346C2×C4×D4—1461no—
347C2×C4×D4C2×D7311yescharacteristic
348C22×C4⋊C4—731no—
349C22×C4⋊C4—731no—
350C22×C4⋊C4—731no—
351C2×#107—731no—
352C2×#107—731no—
353C2×#107—731no—
354C2×#107—731no—
355C2×C22⋊C8C2×D7311yescharacteristic
356C2×C22⋊C8C2×D7311yescharacteristic
357C2×C22⋊C8C2×D7311yescharacteristic
358#398—1461no—
359#397—1461no—
360#389—731no—
361#389—731no—
362#389—1461no—
363#385—1461no—
364#385—1461no—
365#384—731no—
366#381—1461no—
367#380—731no—
368#380—731no—
369#380—731no—
370#380—731no—
371#380—731no—
372C4×C4⋊C4—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#336C2×D7311yescharacteristic
386#336C2×D7311yescharacteristic
387C22×D4D29211yescharacteristic
388C23×C4—1461no—
389C23×C4—1461no—
390C23×C4—1461no—
391C23×C4C22×D7311yescharacteristic
392C23×C4C22×D7311yescharacteristic
393C22×C8—22no—
394C22×C8—22no—
395C22×C8—41no—
396C22×C8C22×D7311yescharacteristic
397C22×C8C22×D7311yescharacteristic
398C22×C8C22×D7311yescharacteristic
399C22×C8C22×D7311yescharacteristic
400#123—2924no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—2924no—
404C2×C4⋊C4—2921no—
405C2×C4⋊C4—2921no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4—1461no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1462no—
421C2×C4⋊C4—1462no—
422C2×C4⋊C4—1462no—
423C2×C4⋊C4—1462no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4D29211yescharacteristic
426C2×C22⋊C4—1461no—
427C2×C22⋊C4—1461no—
428C2×C22⋊C4—1461no—
429C2×C22⋊C4—1461no—
430C2×C22⋊C4—1461no—
431C2×C22⋊C4D29211yescharacteristic
432C2×C22⋊C4D29211yescharacteristic
433C2×C42—1461no—
434C2×C42C22×D7311yescharacteristic
435#107—2921no—
436#107—2921no—
437#107—1462no—
438#107—1462no—
439C22⋊C8—22no—
440C22⋊C8—22no—
441C22⋊C8—41no—
442#95—1461no—
443#95—1461no—
444#95—1461no—
445#95—1461no—
446#95—1461no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450C24C2×D29211yescharacteristic
451C24C2×D29211yescharacteristic
452C2×D4—22no—
453C2×D4—22no—
454C2×D4D58414yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—2922no—
466C22×C4—2922no—
467C22×C4—2922no—
468C22×C4—2922no—
469C22×C4—2922no—
470C22×C4—2922no—
471C22×C4—2922no—
472C22×C4—5841no—
473C22×C4—5841no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4C2×D29211yescharacteristic
483C22×C4C2×D29211yescharacteristic
484C22×C4C2×D29211yescharacteristic
485C22×C4C2×D29211yescharacteristic
486C22×C4#1192402712yes—
487C22×C4#1192402712yes—
488C22×C4C23×D7311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—44no—
501C2×C8#1192402712yes—
502C4⋊C4—22no—
503C4⋊C4—2922no—
504C4⋊C4—2922no—
505C4⋊C4—2922no—
506C4⋊C4—2922no—
507C4⋊C4—2922no—
508C4⋊C4—2922no—
509C4⋊C4—5841no—
510C4⋊C4—732no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4—1461no—
523C4⋊C4—1461no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—2922no—
529C22⋊C4—2922no—
530C22⋊C4—2922no—
531C22⋊C4—2922no—
532C22⋊C4—2922no—
533C42—21no—
534C42—21no—
535C42—2922no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23#1455657512yes—
547C23C2×D58412yes—
548C23C2×D58412yes—
549C23#1455662412yes—
550C23C22×D29211yescharacteristic
551D4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—5844no—
594C2×C4—5844no—
595C2×C4—5844no—
596C2×C4—1462no—
597C2×C4#1455657512yes—
598C2×C4C2×D58412yes—
599C2×C4#41×D7311yescharacteristic
600C2×C4C2×#1192402711yescharacteristic
601C2×C4C2×#1192402711yescharacteristic
602C2×C4C2×#1192402711yescharacteristic
603C2×C4#1455663611yescharacteristic
604C2×C4#1455663811yescharacteristic
605C8—22no—
606C8—41no—
607C8—41no—
608C8—41no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22#1798890114yes—
623C22C2×#1455657511yescharacteristic
624C22#1798923211yescharacteristic
625C22#1798924111yescharacteristic
626C22C22×D58411yescharacteristic
627C22C2×#1455662411yescharacteristic
628C22#1798939911yescharacteristic
629C22#1798940411yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—2922no—
635C4—5841no—
636C4—5841no—
637C4—5841no—
638C4—5841no—
639C4—5841no—
640C4—5841no—
641C4—5841no—
642C4#1798911811yescharacteristic
643C4#1798918211yescharacteristic
644C4#1798926811yescharacteristic
645C4#1798927411yescharacteristic
646C2—24no—
647C2—42no—
648C2#2261340611yescharacteristic
649C2C2×#1798890111yescharacteristic
650C2#2261990111yescharacteristic
651C2#2262174511yescharacteristic
652C2#2262198411yescharacteristic
653C2#2262227911yescharacteristic
654C2#2262238811yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1869C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenter#1869D73
hypocenterC73#24933
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22611490C2semidirect
4. #22611404C2semidirect
5. #22611354C2semidirect
6. #22611354C2semidirect
7. #22610503C2semidirect
8. #22610503C2semidirect
9. #22610121C2semidirect
10. #22610121C2semidirect
11. #22610018C2semidirect
12. #22610018C2semidirect
13. #22609960C2semidirect
14. C2×#17988350C2semidirect
343. C73#24933semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226142392
D4×C4⋊Dic732
#226114902
#226114042
#226113542
#226105032
#226101212
#226100182
#226099602
C2×#179883502
C73×#18692
#2493373