Finite Group Databasebeta

← #33068825

Subgroups — #33068825

Order 18688.

728 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620324C211yesmaximal, characteristic, complemented
2C2×#17988792C211yesmaximal, characteristic, complemented
3D4×#11924012C211yesmaximal, characteristic, complemented
4#22609490C212yesmaximal, complemented
5#22609476C212yesmaximal, complemented
6#22609366C212yesmaximal, complemented
7#22609352C212yesmaximal, complemented
8#22608684C211yesmaximal, characteristic, complemented
9#22608615C211yesmaximal, characteristic, complemented
10#22608608C211yesmaximal, characteristic, complemented
11C73×#3100C211yesmaximal, characteristic
12C22×#11924016—21no—
13C22×#11924012—21no—
14C2×#14556591C2211yescharacteristic
15#17988946—22no—
16#17988937C2212yescomplemented
17#17988933C2212yescomplemented
18#17988792—21no—
19#17988792—21no—
20#17988792C2212yes—
21#17988792C2212yes—
22#17988665C2212yes—
23#17988659—22no—
24D4×#10596831C2212yes—
25#17988611C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27#17988588—21no—
28#17988588—21no—
29C2×#14556514—21no—
30#17988551—21no—
31C4×#11924012C2211yescharacteristic
32#17988339C2212yescomplemented
33#17988337C2212yescomplemented
34#17988334C2212yescomplemented
35#17988334C2212yescomplemented
36#17988333C2212yescomplemented
37#17988331C2212yescomplemented
38#532×C73C2211yescharacteristic
39C2×C4×D4×C73C2211yescharacteristic
40#480×C73—21no—
41#474×C73C2211yescharacteristic
42#474×C73C2211yescharacteristic
43#459×C73C2211yescharacteristic
44#459×C73C2211yescharacteristic
45#447×C73—21no—
46C4×D8×C73—21no—
47C4×D8×C73—21no—
48#431×C73C2211yescharacteristic
49C22×Dic146—21no—
50C2×#11924016—22no—
51C2×#11924016—22no—
52C2×#11924016—44no—
53C2×#11924012—22no—
54C2×#11924012—21no—
55C2×#11924012—44no—
56C2×#11924012D411yescharacteristic
57C2×#11924012D411yescharacteristic
58C22×#10596831—22no—
59C22×#10596831—21no—
60#14556591—21no—
61#14556591—21no—
62#14556591—21no—
63#14556591C2312yes—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C4⋊#10596831—21no—
67C4⋊#10596831—21no—
68C4⋊#10596831C2312yes—
69C4⋊#10596831C2312yes—
70#14556515—22no—
71#14556515—22no—
72#14556514—22no—
73#14556514—22no—
74#14556512—22no—
75#14556512—22no—
76C22⋊#10596831—22no—
77C22⋊#10596831—22no—
78#14556502D412yes—
79C4×#10596831C2312yes—
80C2×#41×C73—21no—
81C22×D4×C73D411yescharacteristic
82C23×C4×C73—21no—
83C2×D8×C73—21no—
84C2×D8×C73—21no—
85C4⋊Q8×C73C2311yescharacteristic
86#127×C73C2311yescharacteristic
87#124×C73—21no—
88#121×C73—21no—
89#121×C73—21no—
90C4×D4×C73—21no—
91C4×D4×C73—21no—
92C4×D4×C73—21no—
93C4×D4×C73—21no—
94C4×D4×C73D411yescharacteristic
95C4×D4×C73D411yescharacteristic
96C4×D4×C73C2311yescharacteristic
97C4×D4×C73C2311yescharacteristic
98C2×C4⋊C4×C73D411yescharacteristic
99C2×C22⋊C4×C73—21no—
100C2×C42×C73C2311yescharacteristic
101#107×C73—21no—
102#107×C73—21no—
103C4⋊C8×C73C2311yescharacteristic
104C4⋊C8×C73C2311yescharacteristic
105#102×C73—21no—
106#102×C73—21no—
107#102×C73—21no—
108#102×C73—21no—
109C22⋊C8×C73—21no—
110C4×C8×C73—21no—
111C2×Dic146—22no—
112C2×Dic146—42no—
113C2×Dic146—42no—
114C2×Dic146—41no—
115C22×Dic73—41no—
116#11924016—44no—
117#11924016—44no—
118#11924012—22no—
119#11924012—22no—
120#11924012—44no—
121#11924012—42no—
122C2×#10596831—22no—
123C2×#10596831—22no—
124C2×#10596831—44no—
125C2×#10596831—42no—
126C2×#10596831C2×D412yes—
127C2×#10596831C2×D412yes—
128C4⋊Dic73—22no—
129C4⋊Dic73—22no—
130C4⋊Dic73—41no—
131C4⋊Dic73—41no—
132C24×C73—21no—
133#41×C73—42no—
134#41×C73—42no—
135C2×Q8×C73—21no—
136C2×D4×C73—22no—
137C2×D4×C73—21no—
138C2×D4×C73—21no—
139C2×D4×C73—21no—
140C2×D4×C73—21no—
141C2×D4×C73—41no—
142C2×D4×C73—41no—
143C2×D4×C73C2×D411yescharacteristic
144C2×D4×C73C2×D411yescharacteristic
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—42no—
151C22×C4×C73—42no—
152C22×C4×C73—41no—
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73#4111yescharacteristic
156D8×C73—42no—
157D8×C73—42no—
158C2×C8×C73—21no—
159C2×C8×C73—21no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73C2×D411yescharacteristic
164C4⋊C4×C73C2×D411yescharacteristic
165C22⋊C4×C73—21no—
166C22⋊C4×C73—21no—
167C22⋊C4×C73—21no—
168C22⋊C4×C73—21no—
169C22⋊C4×C73—41no—
170C22⋊C4×C73—41no—
171C42×C73C2×D411yescharacteristic
172C42×C73C2×D411yescharacteristic
173C42×C73#4111yescharacteristic
174C42×C73C2411yescharacteristic
175C23×C73—21no—
176C23×C73—21no—
177C23×C73—42no—
178C23×C73—42no—
179C23×C73—41no—
180C23×C73—81no—
181C23×C73#12111yescharacteristic
182Q8×C73—41no—
183Q8×C73—41no—
184D4×C73—21no—
185D4×C73—21no—
186D4×C73—21no—
187D4×C73—42no—
188D4×C73—41no—
189D4×C73—41no—
190D4×C73—41no—
191D4×C73—41no—
192D4×C73—81no—
193D4×C73—82no—
194D4×C73AGL(1,ℤ/8)12yes—
195C2×C4×C73—21no—
196C2×C4×C73—21no—
197C2×C4×C73—21no—
198C2×C4×C73—21no—
199C2×C4×C73—41no—
200C2×C4×C73—42no—
201C2×C4×C73—41no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73—41no—
208C2×C4×C73—82no—
209C2×C4×C73#12111yescharacteristic
210C2×C4×C73#12111yescharacteristic
211C2×C4×C73#12111yescharacteristic
212C2×C4×C73#13512yes—
213C2×C4×C73AGL(1,ℤ/8)12yes—
214C2×C4×C73C22×D411yescharacteristic
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C2×#4111yescharacteristic
217C2×Dic73—42no—
218C2×Dic73—82no—
219Dic146—42no—
220Dic146—42no—
221Dic146—82no—
222C8×C73—41no—
223C8×C73—41no—
224#10596831—22no—
225#10596831—22no—
226#10596831—42no—
227#10596831—42no—
228C22×C73—21no—
229C22×C73—21no—
230C22×C73—22no—
231C22×C73—41no—
232C22×C73—41no—
233C22×C73—41no—
234C22×C73—41no—
235C22×C73—41no—
236C22×C73—84no—
237C22×C73—81no—
238C22×C73—81no—
239C22×C73—81no—
240C22×C73#46312yes—
241C22×C73C2×#12111yescharacteristic
242C4×C73—21no—
243C4×C73—21no—
244C4×C73—21no—
245C4×C73—41no—
246C4×C73—41no—
247C4×C73—81no—
248C4×C73#46312yes—
249C4×C73C2×#13511yescharacteristic
250C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C73#57511yescharacteristic
253Dic73—82no—
254#19169—731nomaximal, Sylow
255C2×C73—22no—
256C2×C73—42no—
257C2×C73—81no—
258C2×C73—161no—
259C2×C73C2×#46311yescharacteristic
260C2×C73#305511yescharacteristic
261C2×C73#306911yescharacteristic
262#3141—731no—
263#3100D7311yescharacteristic
264C2×#461—731no—
265D4×M16—731no—
266#1701—732no—
267#1699—732no—
268#1681—732no—
269#1679—732no—
270#1565—731no—
271#1549—731no—
272#1547—731no—
273C73#1916911yescharacteristic, complemented, Sylow
274C22×SD16—1461no—
275C22×M16—1461no—
276#532C2×D7311yescharacteristic
277C2×C4⋊Q8—731no—
278C2×C4×D4C2×D7311yescharacteristic
279#489—1462no—
280#487—732no—
281#486—732no—
282#480—21no—
283#474C2×D7311yescharacteristic
284#474C2×D7311yescharacteristic
285#461—732no—
286#461—732no—
287#461—1461no—
288#461—1461no—
289#459C2×D7311yescharacteristic
290#459C2×D7311yescharacteristic
291#447—21no—
292C4×D8—21no—
293C4×D8—21no—
294#436—732no—
295#435—1462no—
296C8×D4—732no—
297#431C2×D7311yescharacteristic
298#423—731no—
299C2×C4⋊C8—731no—
300#417—1461no—
301#417—1461no—
302C2×#103—1461no—
303#407—1461no—
304C4×M16—731no—
305#335—732no—
306#334—732no—
307#332—732no—
308#332—732no—
309#332—732no—
310#331—732no—
311C2×#41—21no—
312C22×Q8—1461no—
313C22×D4C22⋊D7311yescharacteristic
314C23×C4—21no—
315C2×SD16—2924no—
316C2×SD16—1462no—
317C2×SD16—1462no—
318C2×D8—21no—
319C2×D8—21no—
320C2×M16—2924no—
321C2×M16—731no—
322C2×M16—731no—
323C2×M16—1462no—
324C2×M16—1461no—
325C22×C8—1462no—
326C22×C8—1461no—
327C4⋊Q8—732no—
328C4⋊Q8—1461no—
329C4⋊Q8—1461no—
330C4⋊Q8—1461no—
331C4⋊Q8C22×D7311yescharacteristic
332#127C22×D7311yescharacteristic
333#124—21no—
334#121—21no—
335#121—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4C22⋊D7311yescharacteristic
341C4×D4C22⋊D7311yescharacteristic
342C4×D4C22×D7311yescharacteristic
343C4×D4C22×D7311yescharacteristic
344C2×C4⋊C4—1461no—
345C2×C4⋊C4—1461no—
346C2×C4⋊C4C22⋊D7311yescharacteristic
347C2×C22⋊C4—21no—
348C2×C42C22×D7311yescharacteristic
349#107—21no—
350#107—21no—
351C4⋊C8—732no—
352C4⋊C8—732no—
353C4⋊C8—1461no—
354C4⋊C8—1461no—
355C4⋊C8C22×D7311yescharacteristic
356C4⋊C8C22×D7311yescharacteristic
357#103—1462no—
358#103—1462no—
359#103—1462no—
360#103—1462no—
361#102—21no—
362#102—21no—
363#102—21no—
364#102—21no—
365#102—1462no—
366#102—1462no—
367C22⋊C8—21no—
368C22⋊C8—1462no—
369C22⋊C8—1462no—
370#97—732no—
371C4×C8—21no—
372C4×C8—732no—
373C24—21no—
374#41—42no—
375#41—42no—
376C2×Q8—21no—
377C2×Q8—2922no—
378C2×Q8—2922no—
379C2×Q8—2921no—
380C2×Q8—1462no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4C2×C22⋊D7311yescharacteristic
389C2×D4C2×C22⋊D7311yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—2921no—
399C22×C4D4×D7311yescharacteristic
400C22×C4C2×C22⋊D7311yescharacteristic
401C22×C4#1192402711yescharacteristic
402SD16—2924no—
403SD16—2924no—
404D8—42no—
405D8—42no—
406M16—2922no—
407M16—2924no—
408M16—1462no—
409M16—1462no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—2922no—
413C2×C8—2924no—
414C2×C8—732no—
415C2×C8—732no—
416C2×C8—1462no—
417C2×C8—1462no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—2921no—
422C4⋊C4—2921no—
423C4⋊C4—1462no—
424C4⋊C4—1462no—
425C4⋊C4C2×C22⋊D7311yescharacteristic
426C4⋊C4C2×C22⋊D7311yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42D4×D7311yescharacteristic
434C42C2×C22⋊D7311yescharacteristic
435C42#1192402711yescharacteristic
436C42C23×D7311yescharacteristic
437C23—21no—
438C23—21no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—81no—
443C23#1455656311yescharacteristic
444Q8—41no—
445Q8—41no—
446Q8—2922no—
447Q8—2922no—
448Q8—5842no—
449D4—21no—
450D4—21no—
451D4—21no—
452D4—42no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—81no—
458D4—82no—
459D4#1455661612yes—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—2922no—
475C2×C4—5842no—
476C2×C4#1455656311yescharacteristic
477C2×C4#1455656311yescharacteristic
478C2×C4#1455656311yescharacteristic
479C2×C4#1455661012yes—
480C2×C4#1455661512yes—
481C2×C4C2×D4×D7311yescharacteristic
482C2×C4C22×C22⋊D7311yescharacteristic
483C2×C4C2×#1192402711yescharacteristic
484C8—41no—
485C8—41no—
486C8—2922no—
487C8—2922no—
488C8—1462no—
489C8—1462no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—84no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22#1798880012yes—
503C22C2×#1455656311yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—41no—
508C4—41no—
509C4—81no—
510C4—5842no—
511C4#1798880012yes—
512C4C2×#1455661011yescharacteristic
513C4C2×#1455661511yescharacteristic
514C4C2×#1455661611yescharacteristic
515C4#1798938511yescharacteristic
516C2—22no—
517C2—42no—
518C2—81no—
519C2—161no—
520C2C2×#1798880011yescharacteristic
521C2#2261955211yescharacteristic
522C2#2261967011yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3100C2
socleC22×C73C2×#121
hypercenter#3100D73
hypocenterC73#19169
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #22620324C2semidirect
2. C2×#17988792C2semidirect
3. D4×#11924012C2semidirect
4. #22609490C2semidirect
5. #22609476C2semidirect
6. #22609366C2semidirect
7. #22609352C2semidirect
8. #22608684C2semidirect
9. #22608615C2semidirect
10. #22608608C2semidirect
16. #17988937C22semidirect
17. #17988933C22semidirect
32. #17988339C22semidirect
33. #17988337C22semidirect
34. #17988334C22semidirect
35. #17988334C22semidirect
36. #17988333C22semidirect
37. #17988331C22semidirect
273. C73#19169semidirect

Maximal subgroups

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

subgroupindex
#226203242
C2×#179887922
D4×#119240122
#226094902
#226094762
#226093662
#226093522
#226086842
#226086152
#226086082
C73×#31002
#1916973