Finite Group Databasebeta

← #33135550

Subgroups — #33135550

Order 18688.

800 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988806C211yesmaximal, characteristic
2C2×Q8×#10596831C211yesmaximal, characteristic, complemented
3#22615297C211yesmaximal, characteristic, complemented
4#22611319C211yesmaximal, characteristic
5#22611312C211yesmaximal, characteristic, complemented
6#22611253C211yesmaximal, characteristic, complemented
7#22611248C211yesmaximal, characteristic
8#22610791C211yesmaximal, characteristic, complemented
9#22610784C211yesmaximal, characteristic
10#22610743C211yesmaximal, characteristic
11#22610699C211yesmaximal, characteristic, complemented
12#22610617C211yesmaximal, characteristic
13#22610610C211yesmaximal, characteristic, complemented
14C4×#14556514C211yesmaximal, characteristic, complemented
15C73×#2592C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C22×#11924016C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19#17988806—24no—
20Q8×#10596831—24no—
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×C4⋊#10596831C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×#14556512C2211yescharacteristic
34C2×C4×#10596831C2211yescharacteristic
35C2×C4×#10596831C2211yescharacteristic
36C2×C4×#10596831C2211yescharacteristic
37#17988525—21no—
38#17988506—21no—
39#17988493C2211yescharacteristic
40#17988493C2211yescharacteristic
41#17988485C2211yescharacteristic
42#17988485C2211yescharacteristic
43#17988468—21no—
44#17988468—21no—
45#17988453C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×#124×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52#397×C73—21no—
53#393×C73—21no—
54#391×C73C2211yescharacteristic
55#391×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#390×C73C2211yescharacteristic
58#388×C73—21no—
59#388×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C2×#11924018—22no—
63C2×#11924018—22no—
64C2×#11924018—41no—
65C2×#11924016—22no—
66C2×#11924016—22no—
67C2×#11924016—41no—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C4×Dic146—22no—
73C4×Dic146—22no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×C4×Dic73—21no—
81#14556524—41no—
82#14556522—41no—
83C4⋊#10596831—22no—
84C4⋊#10596831—22no—
85C4⋊#10596831D414yes—
86#14556515—22no—
87#14556515—22no—
88#14556515—41no—
89#14556514—22no—
90#14556514—22no—
91#14556512—22no—
92#14556512—22no—
93#14556512—41no—
94C4×#10596831—22no—
95C4×#10596831—22no—
96C4×#10596831—22no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105C22×Q8×C73D411yescharacteristic
106C22×Q8×C73C2311yescharacteristic
107C22×D4×C73C2311yescharacteristic
108#124×C73—22no—
109#124×C73—22no—
110C4×Q8×C73—22no—
111C4×Q8×C73—22no—
112C2×C4⋊C4×C73D411yescharacteristic
113C2×C4⋊C4×C73D411yescharacteristic
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123C2×C42×C73—21no—
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C2×C42×C73C2311yescharacteristic
127#95×C73—21no—
128#95×C73—21no—
129#95×C73—21no—
130#95×C73—21no—
131#95×C73—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146#4112yes—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142#11924018—44no—
143#11924016—44no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831C2×D412yes—
154C2×#10596831C2×D412yes—
155C2×#10596831#4112yes—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162#11924007—42no—
163#11924007—42no—
164#11924007—42no—
165C4×Dic73—42no—
166C24×C73—21no—
167C2×Q8×C73—22no—
168C2×Q8×C73—22no—
169C2×Q8×C73—22no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2411yescharacteristic
192C4⋊C4×C73—22no—
193C4⋊C4×C73—22no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C22⋊C4×C73—42no—
200C22⋊C4×C73—42no—
201C22⋊C4×C73—42no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—81no—
204C22⋊C4×C73—81no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—81no—
210C42×C73C2×D412yes—
211C42×C73#4112yes—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73C22×D411yescharacteristic
220Q8×C73—28no—
221Q8×C73—41no—
222Q8×C73—41no—
223Q8×C73—41no—
224Q8×C73—41no—
225D4×C73—41no—
226D4×C73—41no—
227D4×C73—41no—
228D4×C73—41no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
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—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73#12112yes—
253C2×C4×C73#12112yes—
254C2×C4×C73#13512yes—
255C2×C4×C73#13512yes—
256C2×C4×C73C22×D411yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73H(5,𝔽2)11yescharacteristic
261C2×C4×C732−1+411yescharacteristic
262C2×Dic73—22no—
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—
274Dic146—22no—
275Dic146—22no—
276Dic146—41no—
277Dic146—41no—
278#10596831—22no—
279#10596831—22no—
280#10596831—22no—
281#10596831—41no—
282C22×C73—81no—
283C22×C73—81no—
284C22×C73—81no—
285C22×C73—81no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73C2×#12111yescharacteristic
290C22×C73#54811yescharacteristic
291C22×C73#55011yescharacteristic
292C22×C73C2×#13511yescharacteristic
293C22×C73C2×#13511yescharacteristic
294C22×C73#57711yescharacteristic
295C22×C73#57811yescharacteristic
296C4×C73—22no—
297C4×C73—22no—
298C4×C73—22no—
299C4×C73—22no—
300C4×C73—41no—
301C4×C73—41no—
302C4×C73—81no—
303C4×C73—81no—
304C4×C73—81no—
305C4×C73#52911yescharacteristic
306C4×C73#54211yescharacteristic
307C4×C73#54611yescharacteristic
308C4×C73#54811yescharacteristic
309Dic73—42no—
310Dic73—81no—
311Dic73—81no—
312Dic73—81no—
313#25297—731nomaximal, Sylow
314C2×C73—161no—
315C2×C73#242611yescharacteristic
316C2×C73#305011yescharacteristic
317C2×C73#306311yescharacteristic
318C2×C73#336411yescharacteristic
319C2×C73#336511yescharacteristic
320C2×C73#339311yescharacteristic
321C2×C73#339411yescharacteristic
322C2×#464—731no—
323C2×C8×Q8—731no—
324#2592D7311yescharacteristic
325#2581—731no—
326#2067—731no—
327#2066—731no—
328#2056—731no—
329#2055—731no—
330#1969—731no—
331#1968—731no—
332#1958—731no—
333#1949—731no—
334#1934—731no—
335#1933—731no—
336C4×#103—731no—
337C73#2529711yescharacteristic, complemented, Sylow
338C22×Q16—731no—
339C22×SD16—731no—
340C2×#124C2×D7311yescharacteristic
341C2×C4×Q8—731no—
342C2×C4×Q8C2×D7311yescharacteristic
343#464—1464no—
344C8×Q8—1464no—
345C2×#107—731no—
346C2×#106—731no—
347C2×C4⋊C8—731no—
348C2×C4⋊C8—731no—
349C2×C4⋊C8—731no—
350C2×#103—731no—
351C2×#103—731no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#102—731no—
356C2×#102—731no—
357C2×#102—731no—
358C2×C4×C8—731no—
359C2×C4×C8—731no—
360C2×C4×C8—731no—
361#400—1461no—
362#397—21no—
363#395—1461no—
364#393—21no—
365#391—731no—
366#391—731no—
367#391C2×D7311yescharacteristic
368#391C2×D7311yescharacteristic
369#390C2×D7311yescharacteristic
370#390C2×D7311yescharacteristic
371#389—731no—
372#389—731no—
373#388—21no—
374#388—21no—
375#387—1461no—
376#387—1461no—
377#382—731no—
378C4×C4⋊C4C2×D7311yescharacteristic
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383C22×Q8—731no—
384C22×Q8C22⋊D7311yescharacteristic
385C22×Q8C22×D7311yescharacteristic
386C22×D4C22×D7311yescharacteristic
387C2×Q16—2921no—
388C2×Q16—1462no—
389C2×Q16—1462no—
390C2×SD16—2921no—
391C2×SD16—1462no—
392C2×SD16—1462no—
393C22×C8—731no—
394C22×C8—731no—
395C22×C8—731no—
396C22×C8—731no—
397#124—22no—
398#124—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—1462no—
402C4×Q8—1462no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4C22⋊D7311yescharacteristic
410C2×C4⋊C4C22⋊D7311yescharacteristic
411C2×C4⋊C4C22⋊D7311yescharacteristic
412C2×C4⋊C4C22×D7311yescharacteristic
413C2×C4⋊C4C22×D7311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C42—21no—
421C2×C42—1461no—
422C2×C42C22×D7311yescharacteristic
423C2×C42C22×D7311yescharacteristic
424C2×C42C22×D7311yescharacteristic
425#107—2921no—
426#106—2921no—
427C4⋊C8—734no—
428C4⋊C8—1462no—
429C4⋊C8—1462no—
430#103—2921no—
431#103—1462no—
432#103—1462no—
433#103—1462no—
434#103—1462no—
435#102—2921no—
436#102—1462no—
437#102—1462no—
438C4×C8—1462no—
439C4×C8—1462no—
440C4×C8—1462no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1461no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454C24—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—732no—
465C2×Q8—1461no—
466C2×Q8—1461no—
467C2×Q8—1461no—
468C2×Q8—1461no—
469C2×Q8—1461no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4C2×C22⋊D7311yescharacteristic
484C22×C4C2×C22⋊D7311yescharacteristic
485C22×C4C2×C22⋊D7311yescharacteristic
486C22×C4C2×C22⋊D7311yescharacteristic
487C22×C4C2×C22⋊D7311yescharacteristic
488C22×C4C2×C22⋊D7311yescharacteristic
489C22×C4C23×D7311yescharacteristic
490Q16—2924no—
491SD16—2924no—
492C2×C8—732no—
493C2×C8—732no—
494C2×C8—732no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—2922no—
512C4⋊C4—2922no—
513C4⋊C4—2922no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—81no—
531C42—2922no—
532C42D4×D7312yes—
533C42#1192402712yes—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D7311yescharacteristic
542Q8—28no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—2921no—
548Q8—2921no—
549Q8—1462no—
550Q8—1462no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
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—1462no—
590C2×C4#1455656312yes—
591C2×C4#1455656312yes—
592C2×C4#1455661112yes—
593C2×C4#1455661112yes—
594C2×C4C2×D4×D7311yescharacteristic
595C2×C4#41×D7311yescharacteristic
596C2×C4#41×D7311yescharacteristic
597C2×C4C2×#1192402711yescharacteristic
598C2×C4#1455663711yescharacteristic
599C2×C4#1455663811yescharacteristic
600C8—2921no—
601C8—1462no—
602C8—1462no—
603C8—1462no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#1455656311yescharacteristic
612C22#1798924411yescharacteristic
613C22#1798925611yescharacteristic
614C22C2×#1455661111yescharacteristic
615C22C2×#1455661111yescharacteristic
616C22#1798939811yescharacteristic
617C22#1798940211yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—2922no—
628C4—5841no—
629C4—5841no—
630C4—5841no—
631C4#1798911611yescharacteristic
632C4#1798920611yescharacteristic
633C4#1798923011yescharacteristic
634C4#1798924511yescharacteristic
635C2—161no—
636C2#2261350011yescharacteristic
637C2#2261950911yescharacteristic
638C2#2261962411yescharacteristic
639C2#2262261211yescharacteristic
640C2#2262262311yescharacteristic
641C2#2262291511yescharacteristic
642C2#2262292311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2592C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2592D73
hypocenterC73#25297
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#10596831C2semidirect
3. #22615297C2semidirect
5. #22611312C2semidirect
6. #22611253C2semidirect
8. #22610791C2semidirect
11. #22610699C2semidirect
13. #22610610C2semidirect
14. C4×#14556514C2semidirect
337. C73#25297semidirect

Maximal subgroups

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

subgroupindex
C2×#179888062
C2×Q8×#105968312
#226152972
#226113192
#226113122
#226112532
#226112482
#226107912
#226107842
#226107432
#226106992
#226106172
#226106102
C4×#145565142
C73×#25922
#2529773