Finite Group Databasebeta

← #62861⋊C3 #11710752

Subgroups — #62861⋊C3 #11710752

Order 768.

1479 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1479 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3523⋊C3C214yesmaximal, complemented
2#3516⋊C3C214yesmaximal, complemented
3#3504⋊C3C211yesmaximal, characteristic, complemented
4#3499⋊C3C211yesmaximal, characteristic, complemented
5#3489⋊C3C212yesmaximal, complemented
6C4×2−1+4⋊C3C212yesmaximal, complemented
7#3428⋊C3C211yesmaximal, characteristic, complemented
8#62861C311yesmaximal, characteristic, complemented, Sylow
9C2×2−1+4⋊C3C2212yes—
10C22×#41⋊C3C2211yescharacteristic
11C3×#540—41nomaximal
12#557⋊C3C2214yescomplemented
13D4×SL(2,𝔽3)C2214yes—
14D4×SL(2,𝔽3)C2214yescomplemented
15#548⋊C3C2214yes—
16#520⋊C3C2212yes—
17#520⋊C3C2212yes—
18#519⋊C3C2212yes—
19#519⋊C3C2214yescomplemented
20C4×#41⋊C3C2211yescharacteristic
21C4×#41⋊C3C2211yescharacteristic
22C2×C4×SL(2,𝔽3)C2214yescomplemented
23#3523C2×C314yescomplemented
24#3521—34no—
25#3516C2×C314yescomplemented
26#3515—34no—
27#3512—31no—
28#3509—31no—
29#3504C2×C311yescharacteristic, complemented
30#3499C2×C311yescharacteristic, complemented
31#3495—32no—
32#3489C2×C312yescomplemented
33#3461—32no—
34#3455—31no—
35C2×#541—31no—
36C4×2−1+4C2×C312yescomplemented
37#3428C2×C311yescharacteristic, complemented
382−1+4⋊C3—24no—
39C2×#41⋊C3—22no—
40C2×#41⋊C3C2311yescharacteristic
41C2×#41⋊C3C2311yescharacteristic
42C22×SL(2,𝔽3)C2311yescharacteristic
43C22×SL(2,𝔽3)C2314yes—
44C3×#127—41no—
45C3×#125—41no—
46C3×#123—44no—
47C3×#121—42no—
48C3×#121—44no—
49C3×C4×D4—42no—
50C3×#117—41no—
51C4×SL(2,𝔽3)C2312yes—
52C4×SL(2,𝔽3)C2314yes—
53C4×SL(2,𝔽3)C2312yes—
54C2×2−1+4C22×C312yes—
55C22×#41C22×C311yescharacteristic
56#557—34no—
57#557C22×C314yes—
58#556—34no—
59#556—34no—
60#554—34no—
61#554—34no—
62#553—38no—
63#553—34no—
64#552—34no—
65#552—38no—
66#552—38no—
67#552—34no—
68#552—34no—
69#551—38no—
70D4×Q8C22×C314yes—
71D4×Q8C22×C314yes—
72#548C22×C314yes—
73#547—34no—
74#547—34no—
75#546—34no—
76#544—32no—
77#542—34no—
78#542—32no—
79#542—32no—
80#541—32no—
81#541—32no—
82#541—32no—
83#541—34no—
84#541—34no—
85#541—31no—
86#541—34no—
87#541—31no—
88#540—31no—
89#540A411yescharacteristic
90#539—38no—
91#539—34no—
92#538—34no—
93#537—32no—
94#536—32no—
95#536—32no—
96#533—34no—
97#533—31no—
98#533—34no—
99#532—31no—
100C2×C4⋊Q8—31no—
101#529—34no—
102#529—38no—
103#529—34no—
104#529—34no—
105#529—34no—
106C2×#126—34no—
107C2×#125—31no—
108C2×#125—34no—
109#525—32no—
110C2×C22⋊Q8—32no—
111#520—32no—
112#520—34no—
113#520—32no—
114#520C22×C312yes—
115#520C22×C312yes—
116#519—32no—
117#519C22×C312yes—
118#519C22×C314yes—
119#518—32no—
120C4×#41—32no—
121C4×#41—34no—
122C4×#41—31no—
123C4×#41—31no—
124C4×#41C22×C311yescharacteristic
125C4×#41C22×C311yescharacteristic
126C2×C4×Q8—32no—
127C2×C4×Q8C22×C314yes—
128C2×#117—31no—
129C2×#117—31no—
130C2×C3×D4—42no—
131C2×C3×D4—44no—
132C2×C3×D4—44no—
133C22×C3×C4—41no—
134C22×C3×C4—44no—
135#41⋊C3—21no—
136#41⋊C3#4112yes—
137C2×SL(2,𝔽3)—21no—
138C2×SL(2,𝔽3)—24no—
139C2×SL(2,𝔽3)C2411yescharacteristic
140C3×C4⋊C4—42no—
141C3×C4⋊C4—42no—
142C3×C4⋊C4—44no—
143C3×C22⋊C4—42no—
144C3×C22⋊C4—44no—
145C3×C22⋊C4—44no—
146C3×C42—41no—
147C3×C42—41no—
1482−1+4—24no—
1492−1+4—64no—
150C2×#41—22no—
151C2×#41—32no—
152C2×#41—34no—
153C2×#41—31no—
154C2×#41—31no—
155C2×#41—62no—
156C2×#41C23×C311yescharacteristic
157C2×#41C23×C311yescharacteristic
158C22×Q8—32no—
159C22×Q8C23×C314yes—
160C22×Q8C23×C311yescharacteristic
161C22×D4—31no—
162C23×C4—31no—
163C4⋊Q8—34no—
164C4⋊Q8—32no—
165C4⋊Q8—32no—
166C4⋊Q8—34no—
167C4⋊Q8—34no—
168C4⋊Q8—31no—
169#127C2×A411yescharacteristic
170#126—38no—
171#126—38no—
172#126—38no—
173#126—38no—
174#126—34no—
175#126—34no—
176#126—34no—
177#126—34no—
178#125—34no—
179#125—34no—
180#125—32no—
181#125—38no—
182#125—38no—
183#125—38no—
184#125—38no—
185#125—34no—
186#125—34no—
187#125—31no—
188#125C2×A411yescharacteristic
189#124—32no—
190#124—34no—
191#124—34no—
192#124—34no—
193#123—38no—
194#123—38no—
195#123—38no—
196#123—34no—
197#123—34no—
198#123—34no—
199#123—34no—
200#123C2×A414yes—
201C22⋊Q8—34no—
202C22⋊Q8—38no—
203C22⋊Q8—38no—
204C22⋊Q8—32no—
205C22⋊Q8—34no—
206C22⋊Q8—32no—
207C22⋊Q8—34no—
208C22⋊Q8—34no—
209C22⋊Q8—34no—
210C22⋊Q8—34no—
211C22⋊Q8—34no—
212C22⋊Q8—34no—
213C22⋊Q8—34no—
214#121—34no—
215#121—38no—
216#121—32no—
217#121—34no—
218#121—34no—
219#121C2×A412yes—
220#121C2×A414yes—
221#120—34no—
222C4×Q8—34no—
223C4×Q8—38no—
224C4×Q8—32no—
225C4×Q8—34no—
226C4×Q8—34no—
227C4×Q8—34no—
228C4×Q8—32no—
229C4×Q8—32no—
230C4×Q8—32no—
231C4×Q8—34no—
232C4×Q8—34no—
233C4×Q8C23×C312yes—
234C4×Q8C23×C314yes—
235C4×Q8C23×C312yes—
236C4×D4—34no—
237C4×D4—34no—
238C4×D4—38no—
239C4×D4—38no—
240C4×D4—32no—
241C4×D4—34no—
242C4×D4—34no—
243C4×D4—34no—
244C4×D4—32no—
245C4×D4—32no—
246C4×D4—34no—
247C4×D4—34no—
248C4×D4—32no—
249C4×D4—32no—
250C4×D4—34no—
251C4×D4C2×A412yes—
252#117—32no—
253#117—38no—
254#117—38no—
255#117—38no—
256#117—31no—
257#117—32no—
258#117—32no—
259#117—32no—
260#117—34no—
261#117—32no—
262#117—34no—
263#117—31no—
264#117—31no—
265#117—31no—
266#117—31no—
267#117C2×A411yescharacteristic
268C2×C4⋊C4—34no—
269C2×C4⋊C4—34no—
270C2×C4⋊C4—34no—
271C2×C4⋊C4—34no—
272C2×C4⋊C4—34no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—32no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—32no—
279C2×C42—32no—
280C2×C42—34no—
281C2×C42—34no—
282C2×C42—31no—
283C2×C42—31no—
284C23×C3—41no—
285C23×C3—44no—
286C3×D4—84no—
287C3×D4—164no—
288C3×D4—164no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—44no—
293C2×C3×C4—41no—
294C2×C3×C4—82no—
295C2×C3×C4—164no—
296SL(2,𝔽3)C2×#4111yescharacteristic
297C24—31no—
298#41—21no—
299#41—32no—
300#41—64no—
301#41—64no—
302#41—62no—
303#41—62no—
304#41—62no—
305#41—68no—
306#41—68no—
307#41—62no—
308#41—61no—
309#41C3×#4112yes—
310C2×Q8—21no—
311C2×Q8—24no—
312C2×Q8—32no—
313C2×Q8—34no—
314C2×Q8—34no—
315C2×Q8—31no—
316C2×Q8—64no—
317C2×Q8—64no—
318C2×Q8—61no—
319C2×Q8—64no—
320C2×Q8C24×C311yescharacteristic
321C2×D4—31no—
322C2×D4—31no—
323C2×D4—32no—
324C2×D4—34no—
325C2×D4—34no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4C22×A412yes—
329C2×D4C22×A414yes—
330C2×D4C22×A414yes—
331C22×C4—31no—
332C22×C4—34no—
333C22×C4—34no—
334C22×C4—31no—
335C22×C4—34no—
336C22×C4—34no—
337C22×C4—34no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4C22×A411yescharacteristic
346C22×C4C22×A414yes—
347C4⋊C4—32no—
348C4⋊C4—34no—
349C4⋊C4—32no—
350C4⋊C4—38no—
351C4⋊C4—38no—
352C4⋊C4—38no—
353C4⋊C4—38no—
354C4⋊C4—32no—
355C4⋊C4—34no—
356C4⋊C4—34no—
357C4⋊C4—34no—
358C4⋊C4—34no—
359C4⋊C4—32no—
360C4⋊C4—32no—
361C4⋊C4—32no—
362C4⋊C4—34no—
363C4⋊C4—34no—
364C4⋊C4—34no—
365C4⋊C4C22×A412yes—
366C4⋊C4C22×A412yes—
367C4⋊C4C22×A414yes—
368C22⋊C4—32no—
369C22⋊C4—32no—
370C22⋊C4—38no—
371C22⋊C4—32no—
372C22⋊C4—34no—
373C22⋊C4—34no—
374C22⋊C4—34no—
375C22⋊C4—34no—
376C22⋊C4—34no—
377C22⋊C4—32no—
378C22⋊C4—32no—
379C22⋊C4—34no—
380C22⋊C4—34no—
381C22⋊C4C22×A412yes—
382C22⋊C4C22×A414yes—
383C22⋊C4C22×A414yes—
384C42—34no—
385C42—38no—
386C42—32no—
387C42—31no—
388C42—31no—
389C42—34no—
390C42—34no—
391C42—32no—
392C42—34no—
393C42C22×A411yescharacteristic
394C42C22×A411yescharacteristic
395C22×C3—41no—
396C22×C3—81no—
397C22×C3—84no—
398C22×C3—84no—
399C22×C3—84no—
400C22×C3—82no—
401C3×C4—42no—
402C3×C4—81no—
403C3×C4—82no—
404C3×C4—84no—
405C3×C4—82no—
406C23—31no—
407C23—31no—
408C23—62no—
409C23—122no—
410C23C23×A411yescharacteristic
411C23C23×A414yes—
412Q8—31no—
413Q8—64no—
414Q8—61no—
415Q8—64no—
416Q8—61no—
417Q8—64no—
418Q8—124no—
419Q8C2×C3×#4111yescharacteristic
420D4—24no—
421D4—32no—
422D4—32no—
423D4—44no—
424D4—44no—
425D4—61no—
426D4—61no—
427D4—68no—
428D4—64no—
429D4—124no—
430C2×C4—22no—
431C2×C4—31no—
432C2×C4—32no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—34no—
437C2×C4—34no—
438C2×C4—31no—
439C2×C4—44no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—62no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—64no—
446C2×C4—64no—
447C2×C4—68no—
448C2×C4—62no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—124no—
452C2×C4—124no—
453C2×C4—124no—
454C2×C4C23×A412yes—
455C2×C4C23×A414yes—
456C2×C4C23×A412yes—
457C2×C4C23×A411yescharacteristic
458C2×C4C23×A411yescharacteristic
459C2×C3—41no—
460C2×C3—42no—
461C2×C3—161no—
462C2×C3—164no—
463C22—21no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—22no—
468C22—32no—
469C22—62no—
470C22—61no—
471C22—62no—
472C22—242no—
473C22C24×A411yescharacteristic
474C4—21no—
475C4—22no—
476C4—24no—
477C4—22no—
478C4—31no—
479C4—31no—
480C4—61no—
481C4—64no—
482C4—62no—
483C4—64no—
484C4—62no—
485C4A4×#4112yes—
486C3—41noSylow
487C2—41no—
488C2—44no—
489C2—62no—
490C2—121no—
491C2C2×A4×#4111yescharacteristic
492C2#8753712yes—
subgroupG/H
center, Frattini, socleC22C24×A4
commutator, cosocleC2×Q8C24×C3
Fitting#62861C3
hypercenter#540A4
hypocenterQ8C2×C3×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #3523⋊C3C2semidirect
2. #3516⋊C3C2semidirect
3. #3504⋊C3C2semidirect
4. #3499⋊C3C2semidirect
5. #3489⋊C3C2semidirect
6. C4×2−1+4⋊C3C2semidirect
7. #3428⋊C3C2semidirect
8. #62861C3semidirect
12. #557⋊C3C22semidirect
14. D4×SL(2,𝔽3)C22semidirect
19. #519⋊C3C22semidirect
22. C2×C4×SL(2,𝔽3)C22semidirect
23. #3523C2×C3semidirect
25. #3516C2×C3semidirect
29. #3504C2×C3semidirect
30. #3499C2×C3semidirect
32. #3489C2×C3semidirect
36. C4×2−1+4C2×C3semidirect
37. #3428C2×C3semidirect

Maximal subgroups

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

subgroupindex
#3523⋊C32
#3516⋊C32
#3504⋊C32
#3499⋊C32
#3489⋊C32
C4×2−1+4⋊C32
#3428⋊C32
#628613
C3×#5404