Finite Group Databasebeta

← #33068842

Subgroups — #33068842

Order 18688.

766 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#22620384C211yesmaximal, characteristic
2C2×#17988787C211yesmaximal, characteristic, complemented
3Q8×#11924012C211yesmaximal, characteristic, complemented
4#22609413C212yesmaximal, complemented
5#22609405C212yesmaximal, complemented
6#22609289C212yesmaximal, complemented
7#22609281C212yesmaximal, complemented
8#22608684C211yesmaximal, characteristic, complemented
9#22608608C211yesmaximal, characteristic, complemented
10#22608598C211yesmaximal, characteristic, complemented
11C73×#3103C211yesmaximal, characteristic, complemented
12C2×#14556616—21no—
13C2×#14556616—21no—
14C22×#11924015—21no—
15C2×#14556588C2211yescharacteristic
16#17988977—22no—
17#17988963C2212yes—
18#17988961C2212yes—
19#17988787—21no—
20#17988787—21no—
21#17988787C2212yes—
22#17988787C2212yes—
23#17988718—22no—
24#17988718C2212yescomplemented
25Q8×#10596831C2212yescomplemented
26#17988611—21no—
27#17988611C2211yescharacteristic
28C2×C4⋊#10596831C2211yescharacteristic
29C2×#14556511—21no—
30C4×#11924012—21no—
31C4×#11924012C2211yescharacteristic
32#17988339C2212yescomplemented
33#17988337C2212yescomplemented
34#17988333C2212yescomplemented
35#17988331C2212yescomplemented
36#17988330C2212yes—
37#17988330C2212yes—
38#532×C73C2211yescharacteristic
39C2×C4×Q8×C73C2211yescharacteristic
40#481×C73—21no—
41#475×C73C2211yescharacteristic, complemented
42#475×C73C2211yescharacteristic, complemented
43#460×C73C2211yescharacteristic, complemented
44#460×C73C2211yescharacteristic, complemented
45#448×C73—21no—
46C4×SD16×C73—21no—
47C4×SD16×C73—21no—
48#431×C73C2211yescharacteristic
49C22×D292—21no—
50C22×D292—21no—
51C22×D292—21no—
52#14556616—44no—
53#14556616—44no—
54C2×#11924015—22no—
55C2×#11924015—22no—
56C2×#11924015—22no—
57C2×#11924015—22no—
58C2×#11924015—44no—
59C2×#11924013—22no—
60C2×#11924013—22no—
61C2×#11924012—21no—
62C2×#11924012D411yescharacteristic
63C2×#11924012D411yescharacteristic
64C22×#10596831—21no—
65#14556588—21no—
66#14556588—21no—
67#14556588—21no—
68#14556588C2312yes—
69C4⋊#10596831—22no—
70C4⋊#10596831—22no—
71C4⋊#10596831—21no—
72C4⋊#10596831—21no—
73C4⋊#10596831C2312yes—
74C4⋊#10596831C2312yes—
75#14556511—22no—
76#14556511—22no—
77#14556502—22no—
78#14556502D412yes—
79C4×#10596831—22no—
80C4×#10596831C2312yes—
81C2×#41×C73—21no—
82C22×Q8×C73D411yescharacteristic
83C2×SD16×C73—21no—
84C2×SD16×C73—21no—
85C4⋊Q8×C73C2311yescharacteristic
86#127×C73C2311yescharacteristic
87#124×C73—21no—
88#121×C73—21no—
89#121×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73—21no—
92C4×Q8×C73D411yescharacteristic
93C4×Q8×C73D411yescharacteristic
94C4×Q8×C73C2311yescharacteristic
95C4×Q8×C73C2311yescharacteristic
96C4×D4×C73—21no—
97C4×D4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73D411yescharacteristic
100C2×C42×C73—21no—
101C2×C42×C73C2311yescharacteristic
102#106×C73—21no—
103#106×C73—21no—
104C4⋊C8×C73C2311yescharacteristic
105C4⋊C8×C73C2311yescharacteristic
106#103×C73—21no—
107#103×C73—21no—
108#102×C73—21no—
109#102×C73—21no—
110C22⋊C8×C73—21no—
111C4×C8×C73—21no—
112C23×D73—41no—
113C2×D292—22no—
114C2×D292—22no—
115C2×D292—22no—
116C2×D292—42no—
117C2×D292—42no—
118C2×D292—42no—
119C2×D292—42no—
120C2×D292—41no—
121C2×D292—41no—
122C2×D292—41no—
123C2×D292—82no—
124#11924015—44no—
125#11924015—44no—
126#11924015—44no—
127#11924015—44no—
128#11924013—44no—
129#11924013—44no—
130#11924012—22no—
131#11924012—42no—
132#11924012SD1614yescomplemented
133C2×#10596831—22no—
134C2×#10596831—22no—
135C2×#10596831—42no—
136C2×#10596831C2×D412yes—
137C2×#10596831C2×D412yes—
138#41×C73—42no—
139#41×C73—42no—
140C2×Q8×C73—21no—
141C2×Q8×C73—22no—
142C2×Q8×C73—21no—
143C2×Q8×C73—41no—
144C2×Q8×C73C2×D411yescharacteristic
145C2×Q8×C73C2×D411yescharacteristic
146C2×D4×C73—21no—
147C2×D4×C73—21no—
148C2×D4×C73—21no—
149C2×D4×C73—41no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—41no—
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73#4111yescharacteristic
156SD16×C73—42no—
157SD16×C73—42no—
158C2×C8×C73—21no—
159C2×C8×C73—21no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—21no—
165C4⋊C4×C73—21no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73C2×D411yescharacteristic
168C4⋊C4×C73C2×D411yescharacteristic
169C22⋊C4×C73—41no—
170C22⋊C4×C73—41no—
171C42×C73—21no—
172C42×C73—21no—
173C42×C73—21no—
174C42×C73—21no—
175C42×C73C2×D411yescharacteristic
176C42×C73C2×D411yescharacteristic
177C42×C73#4111yescharacteristic
178C42×C73C2411yescharacteristic
179C23×C73—41no—
180C23×C73#12111yescharacteristic
181C22×D73—42no—
182C22×D73—82no—
183C22×D73—82no—
184C22×D73—82no—
185Q8×C73—21no—
186Q8×C73—21no—
187Q8×C73—21no—
188Q8×C73—42no—
189Q8×C73—41no—
190Q8×C73—41no—
191Q8×C73AGL(1,ℤ/8)12yescomplemented
192D4×C73—41no—
193D4×C73—41no—
194D4×C73—41no—
195D4×C73—41no—
196D4×C73—81no—
197D4×C73—82no—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—41no—
205C2×C4×C73—42no—
206C2×C4×C73—41no—
207C2×C4×C73—81no—
208C2×C4×C73—82no—
209C2×C4×C73#12111yescharacteristic
210C2×C4×C73#12111yescharacteristic
211C2×C4×C73#12111yescharacteristic
212C2×C4×C73C2×SD1612yes—
213C2×C4×C73AGL(1,ℤ/8)12yes—
214C2×C4×C73C22×D411yescharacteristic
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C2×#4111yescharacteristic
217D292—42no—
218D292—42no—
219D292—42no—
220D292—42no—
221D292—82no—
222D292—82no—
223D292—82no—
224D292—82no—
225C8×C73—41no—
226C8×C73—41no—
227#10596831—22no—
228#10596831—42no—
229#10596831—42no—
230#10596831C2×SD1614yescomplemented
231C22×C73—21no—
232C22×C73—21no—
233C22×C73—81no—
234C22×C73—81no—
235C22×C73—81no—
236C22×C73#46012yes—
237C22×C73C2×#12111yescharacteristic
238C2×D73—82no—
239C2×D73—82no—
240C2×D73—82no—
241C2×D73—164no—
242C4×C73—21no—
243C4×C73—22no—
244C4×C73—21no—
245C4×C73—21no—
246C4×C73—41no—
247C4×C73—41no—
248C4×C73—41no—
249C4×C73—81no—
250C4×C73#46012yes—
251C4×C73C22×SD1611yescharacteristic
252C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C73#57511yescharacteristic
255#19170—731nomaximal, Sylow
256C2×C73—22no—
257C2×C73—161no—
258C2×C73C2×#46011yescharacteristic
259C2×C73#305511yescharacteristic
260C2×C73#306811yescharacteristic
261D73—162no—
262#3151—731no—
263#3103D7311yescharacteristic, complemented
264C2×#460—731no—
265Q8×M16—731no—
266#1688—732no—
267#1686—732no—
268#1668—732no—
269#1666—732no—
270#1565—731no—
271#1547—731no—
272#1545—731no—
273C73#1917011yescharacteristic, complemented, Sylow
274C2×AGL(1,ℤ/8)—1461no—
275C2×AGL(1,ℤ/8)—1461no—
276C22×SD16—1461no—
277#532C2×D7311yescharacteristic
278C2×#127—731no—
279C2×C4×Q8C2×D7311yescharacteristic
280#496—1462no—
281#493—732no—
282#492—732no—
283#481—21no—
284#475C2×D7311yescharacteristic, complemented
285#475C2×D7311yescharacteristic, complemented
286#460—732no—
287#460—732no—
288#460—1461no—
289#460—1461no—
290#460C2×D7311yescharacteristic, complemented
291#460C2×D7311yescharacteristic, complemented
292#448—21no—
293#446—732no—
294#446—1462no—
295C8×Q8—732no—
296C4×SD16—21no—
297C4×SD16—21no—
298#431C2×D7311yescharacteristic
299#423—731no—
300#423—1461no—
301C2×C4⋊C8—731no—
302C2×#102—1461no—
303C4×M16—731no—
304C4×M16—1461no—
305#335—732no—
306#334—732no—
307#332—732no—
308#331—732no—
309#331—732no—
310#331—732no—
311C2×#41—21no—
312C22×Q8C22⋊D7311yescharacteristic
313C22×D4—1461no—
314C22×D4—1461no—
315C22×D4—1461no—
316AGL(1,ℤ/8)—2924no—
317AGL(1,ℤ/8)—2924no—
318C2×SD16—21no—
319C2×SD16—21no—
320C2×SD16—2924no—
321C2×SD16—1462no—
322C2×SD16—1462no—
323C2×SD16—1462no—
324C2×SD16—1462no—
325C2×D8—1462no—
326C2×D8—1462no—
327C2×M16—731no—
328C2×M16—731no—
329C2×M16—1461no—
330C22×C8—1461no—
331C4⋊Q8C22×D7311yescharacteristic
332#127—732no—
333#127—1461no—
334#127—1461no—
335#127—1461no—
336#127C22×D7311yescharacteristic
337#124—21no—
338#121—21no—
339#121—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8C22⋊D7311yescharacteristic
343C4×Q8C22⋊D7311yescharacteristic
344C4×Q8C22×D7311yescharacteristic
345C4×Q8C22×D7311yescharacteristic
346C4×D4—21no—
347C4×D4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D7311yescharacteristic
350C2×C42—21no—
351C2×C42C22×D7311yescharacteristic
352#106—21no—
353#106—21no—
354C4⋊C8—732no—
355C4⋊C8—732no—
356C4⋊C8—1462no—
357C4⋊C8—1462no—
358C4⋊C8—1461no—
359C4⋊C8—1461no—
360C4⋊C8C22×D7311yescharacteristic
361C4⋊C8C22×D7311yescharacteristic
362#103—21no—
363#103—21no—
364#102—21no—
365#102—21no—
366#102—1462no—
367#102—1462no—
368C22⋊C8—21no—
369#97—732no—
370#97—1462no—
371C4×C8—21no—
372C4×C8—732no—
373C4×C8—1462no—
374C24—2921no—
375#41—42no—
376#41—42no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—41no—
381C2×Q8C2×C22⋊D7311yescharacteristic
382C2×Q8C2×C22⋊D7311yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—2922no—
388C2×D4—2922no—
389C2×D4—2921no—
390C2×D4—2921no—
391C2×D4—2921no—
392C2×D4—2922no—
393C2×D4—2922no—
394C2×D4—5842no—
395C2×D4—1462no—
396C2×D4—1462no—
397C2×D4—1462no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4D4×D7311yescharacteristic
402C22×C4C2×C22⋊D7311yescharacteristic
403C22×C4#1192402711yescharacteristic
404SD16—42no—
405SD16—42no—
406SD16—2924no—
407SD16—2924no—
408SD16—2924no—
409SD16—2924no—
410D8—2924no—
411D8—2924no—
412M16—2922no—
413M16—734no—
414M16—1462no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—2922no—
418C2×C8—732no—
419C2×C8—732no—
420C2×C8—1462no—
421C2×C8—1462no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4C2×C22⋊D7311yescharacteristic
430C4⋊C4C2×C22⋊D7311yescharacteristic
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42D4×D7311yescharacteristic
438C42C2×C22⋊D7311yescharacteristic
439C42#1192402711yescharacteristic
440C42C23×D7311yescharacteristic
441C23—41no—
442C23—2922no—
443C23—5842no—
444C23—5842no—
445C23—5842no—
446C23#1455656311yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—41no—
452Q8—41no—
453Q8#1455661612yescomplemented
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—81no—
459D4—82no—
460D4—2922no—
461D4—2922no—
462D4—2922no—
463D4—2922no—
464D4—5842no—
465D4—5842no—
466D4—5842no—
467D4—5842no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4#1455656311yescharacteristic
480C2×C4#1455656311yescharacteristic
481C2×C4#1455656311yescharacteristic
482C2×C4SD16×D7312yes—
483C2×C4#1455661812yes—
484C2×C4C2×D4×D7311yescharacteristic
485C2×C4C22×C22⋊D7311yescharacteristic
486C2×C4C2×#1192402711yescharacteristic
487C8—41no—
488C8—41no—
489C8—2922no—
490C8—2922no—
491C8—734no—
492C8—1462no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—5842no—
499C22—5842no—
500C22—5842no—
501C22—11684no—
502C22#1798878512yes—
503C22C2×#1455656311yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—81no—
512C4#1798878512yes—
513C4C2×SD16×D7311yescharacteristic
514C4C2×#1455661611yescharacteristic
515C4C2×#1455661811yescharacteristic
516C4#1798938511yescharacteristic
517C2—22no—
518C2—161no—
519C2—11682no—
520C2C2×#1798878511yescharacteristic
521C2#2261955211yescharacteristic
522C2#2261966111yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3103C2
socleC22×C73C2×#121
prime radical#22620384C2
hypercenter#3103D73
hypocenterC73#19170
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
2. C2×#17988787C2semidirect
3. Q8×#11924012C2semidirect
4. #22609413C2semidirect
5. #22609405C2semidirect
6. #22609289C2semidirect
7. #22609281C2semidirect
8. #22608684C2semidirect
9. #22608608C2semidirect
10. #22608598C2semidirect
11. C73×#3103C2semidirect
24. #17988718C22semidirect
25. Q8×#10596831C22semidirect
32. #17988339C22semidirect
33. #17988337C22semidirect
34. #17988333C22semidirect
35. #17988331C22semidirect
41. #475×C73C22semidirect
42. #475×C73C22semidirect
43. #460×C73C22semidirect
44. #460×C73C22semidirect
132. #11924012SD16semidirect
191. Q8×C73AGL(1,ℤ/8)semidirect
230. #10596831C2×SD16semidirect
263. #3103D73semidirect
273. C73#19170semidirect
284. #475C2×D73semidirect
285. #475C2×D73semidirect
290. #460C2×D73semidirect
291. #460C2×D73semidirect
453. Q8#14556616semidirect

Maximal subgroups

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

subgroupindex
#226203842
C2×#179887872
Q8×#119240122
#226094132
#226094052
#226092892
#226092812
#226086842
#226086082
#226085982
C73×#31032
#1917073