Finite Group Databasebeta

← #33137825

Subgroups — #33137825

Order 18688.

838 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988865C211yesmaximal, characteristic
2C2×C8×Dic146C211yesmaximal, characteristic, complemented
3#22614918C211yesmaximal, characteristic, complemented
4#22614851C211yesmaximal, characteristic
5#22611422C211yesmaximal, characteristic, complemented
6#22611421C211yesmaximal, characteristic, complemented
7#22611319C211yesmaximal, characteristic
8#22611317C211yesmaximal, characteristic
9#22610890C211yesmaximal, characteristic, complemented
10#22610688C211yesmaximal, characteristic, complemented
11#22610664C211yesmaximal, characteristic, complemented
12#22610663C211yesmaximal, characteristic, complemented
13#22610616C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#1934C211yesmaximal, characteristic
16C2×#14556590—21no—
17C2×#14556590C2211yescharacteristic
18C2×#14556568—21no—
19C2×C4×Dic146C2211yescharacteristic
20C2×D4×Dic73C2211yescharacteristic
21#17988865—24no—
22C8×Dic146—24no—
23C2×#14556525C2211yescharacteristic
24C2×#14556525C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556524C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556520C2211yescharacteristic
29C2×#14556520C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33C2×C8×Dic73C2211yescharacteristic
34#17988521—21no—
35#17988520—21no—
36#17988510—21no—
37#17988510—21no—
38#17988491—21no—
39#17988490C2211yescharacteristic
40#17988486C2211yescharacteristic
41#17988486C2211yescharacteristic
42#17988485C2211yescharacteristic
43#17988485C2211yescharacteristic
44#17988483C2211yescharacteristic
45#17988456C2211yescharacteristic
46#17988445—21no—
47C4⋊C4×Dic73C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57#390×C73C2211yescharacteristic
58#389×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×Dic146D411yescharacteristic
63C23×Dic73—21no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556590—22no—
67#14556590—22no—
68#14556590—44no—
69#14556568—44no—
70C4×Dic146—22no—
71C4×Dic146—22no—
72D4×Dic73—22no—
73D4×Dic73—22no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73D411yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×#11924007D411yescharacteristic
88C2×#11924007D411yescharacteristic
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C22⋊Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73C2311yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95#14556525—22no—
96#14556525—22no—
97#14556524—22no—
98#14556524—22no—
99C4⋊#10596831—22no—
100#14556520—22no—
101#14556520D414yes—
102#14556512—22no—
103#14556512—22no—
104C8×Dic73—22no—
105C8×Dic73—22no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112C22×D4×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#102×C73—22no—
123#102×C73Q814yescomplemented
124C4×C8×C73—22no—
125#95×C73—21no—
126C2×Dic146—22no—
127C2×Dic146—22no—
128C2×Dic146—42no—
129C2×Dic146—42no—
130C2×Dic146—42no—
131C2×Dic146D814yes—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—42no—
137C22×Dic73—42no—
138C22×Dic73—42no—
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831C2×D412yes—
149C4⋊Dic73—21no—
150C4⋊Dic73—22no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—42no—
158C4⋊Dic73—42no—
159C4⋊Dic73#4112yes—
160#11924007—22no—
161#11924007—22no—
162#11924007—42no—
163#11924007—42no—
164#11924007—42no—
165#11924007—42no—
166#11924007—42no—
167#11924007—81no—
168C22⋊Dic73—42no—
169C22⋊Dic73—42no—
170C22⋊Dic73—81no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—42no—
174C4×Dic73C2×D412yes—
175C4×Dic73#4112yes—
176C24×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73C2×Q812yes—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73C2×D412yes—
194C2×C8×C73C2×Q812yes—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73C2×Q812yes—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—81no—
204C42×C73—21no—
205C42×C73—21no—
206C23×C73—22no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73C22×D411yescharacteristic
214D4×C73—22no—
215D4×C73—22no—
216D4×C73—41no—
217D4×C73—41no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73C2×D812yes—
233C2×C4×C73C22×D411yescharacteristic
234C2×C4×C73C22×Q811yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C732−1+411yescharacteristic
238C2×C4×C732−1+411yescharacteristic
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—22no—
245C2×Dic73—21no—
246C2×Dic73—21no—
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—84no—
259C2×Dic73#12112yes—
260C2×Dic73#12112yes—
261C2×Dic73C2×D812yes—
262C2×Dic73C2×D812yes—
263C2×Dic73#13512yes—
264Dic146—28no—
265Dic146—84no—
266C8×C73—22no—
267C8×C73—22no—
268#10596831—22no—
269#10596831—41no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73C2×#12111yescharacteristic
278C22×C73#54811yescharacteristic
279C22×C73D4×Q811yescharacteristic
280C22×C73C22×D811yescharacteristic
281C22×C73C2×#13511yescharacteristic
282C22×C73#57711yescharacteristic
283C22×C73#57811yescharacteristic
284C4×C73—22no—
285C4×C73—42no—
286C4×C73—41no—
287C4×C73—81no—
288C4×C73—81no—
289C4×C73#53311yescharacteristic
290C4×C73#54111yescharacteristic
291C4×C73#54711yescharacteristic
292C4×C73D4×Q811yescharacteristic
293Dic73—22no—
294Dic73—22no—
295Dic73—42no—
296Dic73—41no—
297Dic73—81no—
298Dic73—81no—
299Dic73—81no—
300Dic73#46614yes—
301#25469—731nomaximal, Sylow
302C2×C73—82no—
303C2×C73#242911yescharacteristic
304C2×C73C2×#46611yescharacteristic
305C2×C73#306311yescharacteristic
306C2×C73#335811yescharacteristic
307C2×C73#336411yescharacteristic
308C2×C73Q8×D811yescharacteristic
309C2×C73#338111yescharacteristic
310C2×#474—731no—
311C2×C8×Q8—731no—
312#2546—731no—
313#2538—731no—
314#2086—731no—
315#2086—731no—
316#2067—731no—
317#2067—731no—
318#1990—731no—
319#1947—731no—
320#1942—731no—
321#1942—731no—
322#1934—731no—
323#1934D7311yescharacteristic
324C4×#102—731no—
325C73#2546911yescharacteristic, complemented, Sylow
326C2×C4⋊Q8—731no—
327C2×C4⋊Q8—1461no—
328C2×C22⋊Q8—1461no—
329C2×C4×Q8—731no—
330C2×C4×D4—731no—
331#474—1464no—
332C8×Q8—1464no—
333C2×#107—731no—
334C2×#107—731no—
335C2×#107—731no—
336C2×#107—731no—
337C2×C4⋊C8—731no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×#102—731no—
341C2×#102—731no—
342C2×#102C2×D7311yescharacteristic
343C2×#102C2×D7311yescharacteristic
344C2×C4×C8—731no—
345C2×C4×C8—731no—
346C2×C4×C8C2×D7311yescharacteristic
347#398—1461no—
348#398—1461no—
349#396—1461no—
350#396—1461no—
351#391—1461no—
352#390—731no—
353#390C2×D7311yescharacteristic
354#389—731no—
355#389—731no—
356#389—731no—
357#389—731no—
358#389—731no—
359#389C2×D7311yescharacteristic
360#384—731no—
361#380—1461no—
362C4×C4⋊C4—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340C2×D7311yescharacteristic
370#340C2×D7311yescharacteristic
371C22×Q8—731no—
372C22×Q8—1461no—
373C22×D4C22×D7311yescharacteristic
374C23×C4—1461no—
375C22×C8—731no—
376C22×C8—731no—
377C22×C8C22×D7311yescharacteristic
378C22×C8C22×D7311yescharacteristic
379C4⋊Q8—2924no—
380C4⋊Q8—1462no—
381C4⋊Q8—1462no—
382C22⋊Q8—2924no—
383C4×Q8—1462no—
384C4×Q8—1462no—
385C4×D4—1462no—
386C4×D4—1462no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1461no—
409C2×C22⋊C4—1461no—
410C2×C22⋊C4—1461no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—1461no—
414C2×C42C22×D7311yescharacteristic
415#107—1462no—
416#107—1462no—
417#107—1462no—
418#107—1462no—
419C4⋊C8—734no—
420C4⋊C8—1462no—
421C4⋊C8—1462no—
422#102—22no—
423#102—1462no—
424#102—1462no—
425#102Dic14614yescomplemented
426C4×C8—22no—
427C4×C8—1462no—
428C4×C8—1462no—
429#95—21no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436C24—21no—
437C2×Q8—2922no—
438C2×Q8—2922no—
439C2×Q8—2922no—
440C2×Q8—734no—
441C2×Q8—1462no—
442C2×Q8—1462no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×Dic14612yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—2922no—
453C22×C4—2922no—
454C22×C4—2922no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4D4×D7311yescharacteristic
464C22×C4D4×D7311yescharacteristic
465C22×C4C23×D7311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—732no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8D4×D7312yes—
477C2×C8C2×Dic14612yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—2922no—
485C4⋊C4—2922no—
486C4⋊C4—2922no—
487C4⋊C4—2922no—
488C4⋊C4—2922no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—5841no—
492C4⋊C4—732no—
493C4⋊C4—1462no—
494C4⋊C4—1462no—
495C4⋊C4—1462no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4C2×Dic14612yes—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—2922no—
507C22⋊C4—2922no—
508C22⋊C4—5841no—
509C42—21no—
510C42—21no—
511C42—2922no—
512C42—732no—
513C42—732no—
514C42—1461no—
515C42—1461no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C2×D4×D7311yescharacteristic
524Q8—5844no—
525Q8—1468no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
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—5844no—
556C2×C4—732no—
557C2×C4—732no—
558C2×C4—732no—
559C2×C4—732no—
560C2×C4—732no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1462no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4D8×D7312yes—
570C2×C4C2×D4×D7311yescharacteristic
571C2×C4C22×Dic14611yescharacteristic
572C2×C4#41×D7311yescharacteristic
573C2×C4#41×D7311yescharacteristic
574C2×C4#1455663811yescharacteristic
575C2×C4#1455663811yescharacteristic
576C8—22no—
577C8—22no—
578C8—2921no—
579C8—1462no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#121×D7311yescharacteristic
588C22#1798924511yescharacteristic
589C22D4×Dic14611yescharacteristic
590C22C2×D8×D7311yescharacteristic
591C22#135×D7311yescharacteristic
592C22#1798939311yescharacteristic
593C22#1798940111yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—2922no—
600C4—2921no—
601C4—5841no—
602C4—5841no—
603C4—5841no—
604C4—734no—
605C4—1462no—
606C4—1462no—
607C4#1798914011yescharacteristic
608C4#1798919411yescharacteristic
609C4#1798923511yescharacteristic
610C4D4×Dic14611yescharacteristic
611C2—82no—
612C2#2261352711yescharacteristic
613C2#466×D7311yescharacteristic
614C2#2261961511yescharacteristic
615C2#2262256311yescharacteristic
616C2#2262261311yescharacteristic
617C2D8×Dic14611yescharacteristic
618C2#2262280711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1934C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#1934D73
hypocenterC73#25469
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic146C2semidirect
3. #22614918C2semidirect
5. #22611422C2semidirect
6. #22611421C2semidirect
9. #22610890C2semidirect
10. #22610688C2semidirect
11. #22610664C2semidirect
12. #22610663C2semidirect
123. #102×C73Q8semidirect
325. C73#25469semidirect
425. #102Dic146semidirect

Maximal subgroups

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

subgroupindex
C2×#179888652
C2×C8×Dic1462
#226149182
#226148512
#226114222
#226114212
#226113192
#226113172
#226108902
#226106882
#226106642
#226106632
#226106162
#102×Dic732
C73×#19342
#2546973