Finite Group Databasebeta

← #33127345

Subgroups — #33127345

Order 18688.

746 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988883C211yesmaximal, characteristic, complemented
2#22615752C211yesmaximal, characteristic
3#22612613C211yesmaximal, characteristic
4#22611359C211yesmaximal, characteristic
5#22611354C211yesmaximal, characteristic
6#22611135C211yesmaximal, characteristic
7#22611128C211yesmaximal, characteristic
8#22610882C211yesmaximal, characteristic, complemented
9#22610701C211yesmaximal, characteristic, complemented
10#22610674C211yesmaximal, characteristic, complemented
11#22610664C211yesmaximal, characteristic, complemented
12#22610621C211yesmaximal, characteristic, complemented
13#22610607C211yesmaximal, characteristic, complemented
14#22609947C211yesmaximal, characteristic, complemented
15C2×#433×C73C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19#17988883—24no—
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556525C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28#17988526—21no—
29#17988523—21no—
30#17988507—21no—
31#17988499—21no—
32#17988493C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988482—21no—
37#17988481—21no—
38#17988475—21no—
39#17988468—21no—
40#17988453C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988447C2211yescharacteristic
43#17988447C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×#117×C73C2211yescharacteristic
54#433×C73—24no—
55C2×C4⋊C8×C73C2211yescharacteristic
56C2×C4⋊C8×C73C2211yescharacteristic
57C2×C22⋊C8×C73C2211yescharacteristic
58C2×C22⋊C8×C73C2211yescharacteristic
59C2×#97×C73C2211yescharacteristic
60C2×C4×C8×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×Dic146C2311yescharacteristic
63#14556591—22no—
64#14556591—22no—
65#14556570—22no—
66#14556570—22no—
67C4×Dic146—22no—
68C4×Dic146—22no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88#14556525—22no—
89#14556525—22no—
90#14556525—22no—
91#14556523—22no—
92#14556516—22no—
93#14556516—22no—
94#14556516—22no—
95#14556516—22no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107C23×C4×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112#117×C73—22no—
113#117×C73—22no—
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73D411yescharacteristic
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C4⋊C8×C73—22no—
121C4⋊C8×C73D414yes—
122C22⋊C8×C73—22no—
123C22⋊C8×C73—22no—
124#97×C73—22no—
125C4×C8×C73—22no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—42no—
132C2×Dic146—81no—
133C2×Dic146—81no—
134C2×Dic146#4112yes—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×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—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73#4112yes—
158#11924007—24no—
159#11924007—42no—
160#11924007—81no—
161#11924007—81no—
162#11924007—81no—
163C22⋊Dic73—24no—
164C22⋊Dic73—81no—
165C22⋊Dic73—81no—
166C22⋊Dic73—81no—
167C4×Dic73—42no—
168C4×Dic73—81no—
169C24×C73C2×D411yescharacteristic
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2411yescharacteristic
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73C2×D412yes—
193C2×C8×C73C2×D412yes—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—22no—
196C22⋊C4×C73—22no—
197C22⋊C4×C73—22no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73C2×D412yes—
201C42×C73#4112yes—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73C22×D411yescharacteristic
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73#12112yes—
228C2×C4×C73#12112yes—
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C732−1+411yescharacteristic
234C2×C4×C732−1+411yescharacteristic
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259Dic146—22no—
260Dic146—22no—
261Dic146—41no—
262Dic146—41no—
263Dic146—84no—
264C8×C73—22no—
265C8×C73—22no—
266C8×C73—41no—
267C8×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73C2×#12111yescharacteristic
276C22×C73#52511yescharacteristic
277C22×C73#53611yescharacteristic
278C22×C73#57611yescharacteristic
279C22×C73#57711yescharacteristic
280C22×C73#57811yescharacteristic
281C22×C73#57811yescharacteristic
282C4×C73—22no—
283C4×C73—22no—
284C4×C73—41no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73#54711yescharacteristic
288C4×C73#54811yescharacteristic
289C4×C73#55211yescharacteristic
290C4×C73#55611yescharacteristic
291Dic73—42no—
292Dic73—42no—
293Dic73—81no—
294Dic73—81no—
295Dic73—81no—
296Dic73—81no—
297Dic73—81no—
298Dic73—81no—
299#24714—731nomaximal, Sylow
300C2×C73—81no—
301C2×C73#241511yescharacteristic
302C2×C73#305911yescharacteristic
303C2×C73#306211yescharacteristic
304C2×C73#312111yescharacteristic
305C2×C73#312411yescharacteristic
306C2×C73#324211yescharacteristic
307C2×C73#324311yescharacteristic
308C2×#477—731no—
309C2×#433D7311yescharacteristic
310#2628—731no—
311#2323—731no—
312#2076—731no—
313#2075—731no—
314#2036—731no—
315#2035—731no—
316#1988—731no—
317#1949—731no—
318#1944—731no—
319#1942—731no—
320#1935—731no—
321#1933—731no—
322#1805—731no—
323C73#2471411yescharacteristic, complemented, Sylow
324C2×C4⋊Q8—731no—
325C2×C22⋊Q8—731no—
326C2×C4×Q8—731no—
327C2×#117C2×D7311yescharacteristic
328#477—1464no—
329#433—24no—
330C2×#107—731no—
331C2×#107—731no—
332C2×#107—731no—
333C2×#106—731no—
334C2×C4⋊C8C2×D7311yescharacteristic
335C2×C4⋊C8C2×D7311yescharacteristic
336C2×#103—731no—
337C2×#103—731no—
338C2×#103—731no—
339C2×#103—731no—
340C2×C22⋊C8C2×D7311yescharacteristic
341C2×C22⋊C8C2×D7311yescharacteristic
342C2×#97C2×D7311yescharacteristic
343C2×C4×C8C2×D7311yescharacteristic
344#400—1461no—
345#399—1461no—
346#395—1461no—
347#393—1461no—
348#391—731no—
349#389—731no—
350#389—731no—
351#389—731no—
352#389—1461no—
353#388—1461no—
354#388—1461no—
355#387—1461no—
356#382—731no—
357#380—731no—
358#380—731no—
359#380—731no—
360C4×C4⋊C4—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369C22×Q8—731no—
370C22×Q8—1461no—
371C23×C4C22×D7311yescharacteristic
372C22×C8C22×D7311yescharacteristic
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375C22×C8C22×D7311yescharacteristic
376C4⋊Q8—1462no—
377C4⋊Q8—1462no—
378C22⋊Q8—1462no—
379C22⋊Q8—1462no—
380C4×Q8—1462no—
381C4×Q8—1462no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4D29211yescharacteristic
398C2×C4⋊C4D29211yescharacteristic
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4D29211yescharacteristic
404C2×C22⋊C4D29211yescharacteristic
405C2×C42—1461no—
406C2×C42—1461no—
407C2×C42C22×D7311yescharacteristic
408C2×C42C22×D7311yescharacteristic
409#107—1462no—
410#107—1462no—
411#107—1462no—
412#106—1462no—
413C4⋊C8—22no—
414C4⋊C8C22⋊D7314yes—
415#103—1462no—
416#103—1462no—
417#103—1462no—
418#103—1462no—
419C22⋊C8—22no—
420C22⋊C8—22no—
421#97—22no—
422C4×C8—22no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434C24C2×D29211yescharacteristic
435C2×Q8—2922no—
436C2×Q8—5841no—
437C2×Q8—5841no—
438C2×Q8—732no—
439C2×Q8—1461no—
440C2×Q8—1461no—
441C2×Q8—1461no—
442C2×Q8—1461no—
443C2×Q8—1461no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4C2×D29211yescharacteristic
459C22×C4C2×D29211yescharacteristic
460C22×C4C2×D29211yescharacteristic
461C22×C4C2×D29211yescharacteristic
462C22×C4C2×D29211yescharacteristic
463C22×C4C23×D7311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8C2×C22⋊D7312yes—
475C2×C8C2×C22⋊D7312yes—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—2922no—
479C4⋊C4—2922no—
480C4⋊C4—5841no—
481C4⋊C4—5841no—
482C4⋊C4—5841no—
483C4⋊C4—732no—
484C4⋊C4—1464no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—5841no—
501C22⋊C4—5841no—
502C22⋊C4—5841no—
503C22⋊C4—1464no—
504C42—21no—
505C42—21no—
506C42—2922no—
507C42—5841no—
508C42C2×C22⋊D7312yes—
509C42#1192402812yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C22×D29211yescharacteristic
518Q8—2921no—
519Q8—2921no—
520Q8—5844no—
521Q8—1462no—
522Q8—1462no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—1462no—
563C2×C4—1462no—
564C2×C4#1455656512yes—
565C2×C4#1455656512yes—
566C2×C4C22×C22⋊D7311yescharacteristic
567C2×C4#41×D7311yescharacteristic
568C2×C4#41×D7311yescharacteristic
569C2×C4C2×#1192402811yescharacteristic
570C2×C4#1455663811yescharacteristic
571C2×C4#1455663811yescharacteristic
572C8—22no—
573C8—22no—
574C8—41no—
575C8—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1455656511yescharacteristic
584C22#1798909911yescharacteristic
585C22#1798916311yescharacteristic
586C22#1798939211yescharacteristic
587C22#1798939911yescharacteristic
588C22#1798940411yescharacteristic
589C22#1798940411yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—2922no—
596C4—2922no—
597C4—5841no—
598C4—5841no—
599C4—5841no—
600C4—5841no—
601C4—5841no—
602C4—5841no—
603C4#1798923611yescharacteristic
604C4#1798924411yescharacteristic
605C4#1798927611yescharacteristic
606C4#1798930511yescharacteristic
607C2—81no—
608C2#2261335411yescharacteristic
609C2#2261958611yescharacteristic
610C2#2261961111yescharacteristic
611C2#2262016211yescharacteristic
612C2#2262019511yescharacteristic
613C2#2262120111yescharacteristic
614C2#2262121611yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#433×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenterC2×#433D73
hypocenterC73#24714
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#17988883C2semidirect
8. #22610882C2semidirect
9. #22610701C2semidirect
10. #22610674C2semidirect
11. #22610664C2semidirect
12. #22610621C2semidirect
13. #22610607C2semidirect
14. #22609947C2semidirect
323. C73#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#179888832
#226157522
#226126132
#226113592
#226113542
#226111352
#226111282
#226108822
#226107012
#226106742
#226106642
#226106212
#226106072
#226099472
C2×#433×C732
#2471473