Finite Group Databasebeta

← #10693208

Subgroups — #10693208

Order 768.

644 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#3004C211yesmaximal, characteristic
2#82919C211yesmaximal, characteristic
3#72133C211yesmaximal, characteristic, complemented
4#72132C211yesmaximal, characteristic, complemented
5#71709C211yesmaximal, characteristic
6#67687C211yesmaximal, characteristic, complemented
7#67685C211yesmaximal, characteristic, complemented
8#9519—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×C3×#137—21no—
11C3×#525C2211yescharacteristic
12C3×#451—21no—
13C3×#451C412yescomplemented
14C3×#448—21no—
15C3×#448C412yescomplemented
16C3×#419—21no—
17C3×#407C2211yescharacteristic
18#5234—21no—
19C2×#836—21no—
20C2×#836—21no—
21C2×#835—21no—
22C2×#835—21no—
23#5212C2211yescharacteristic
24#5207C2211yescharacteristic
25#5207C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#5012—21no—
28#5011—21no—
29#5011—21no—
30D4×C3⋊C8—21no—
31#5000—21no—
32#4538—21no—
33#4537—21no—
34#3004D311yescharacteristic
35#2976—31no—
36#1787—31no—
37#1784—31no—
38#1513—31no—
39#1305—31no—
40#1293—31no—
41C2×C3×#41D411yescharacteristic
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C22×C3×Q8—21no—
47C22×C3×Q8—42no—
48C22×C3×Q8—42no—
49C22×C3×Q8C2×C412yes—
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C3×#137—42no—
53C3×#137—42no—
54C2×C3×Q16—22no—
55C2×C3×SD16—22no—
56C2×C3×M16—21no—
57C3×#123—21no—
58C3×C22⋊Q8C2×C411yescharacteristic
59C3×C22⋊Q8D411yescharacteristic
60C3×#121C2×C411yescharacteristic
61C3×#121D411yescharacteristic
62C3×#120—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C3×#117D411yescharacteristic
66C2×#217—21no—
67C2×C4⋊Dic3—21no—
68C2×#208—21no—
69C2×#208—21no—
70C2×#208D411yescharacteristic
71C2×#208D411yescharacteristic
72C22×C3⋊C8—21no—
73C22×C3⋊C8D411yescharacteristic
74C22×C3⋊C8D411yescharacteristic
75C3×#103—22no—
76C3×#103—22no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8C2×C412yes—
79#836—42no—
80#836—42no—
81#835—42no—
82#835—42no—
83C3⋊#95—21no—
84#830—21no—
85#830—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830—21no—
91#830—21no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)—21no—
94#803—21no—
95C4×C3⋊C8—21no—
96C23×Q8C2×D311yescharacteristic
97C2×#137—21no—
98#525C2×D311yescharacteristic
99#451—21no—
100#451Dic312yescomplemented
101#448—21no—
102#448Dic312yescomplemented
103#436—61no—
104#435—61no—
105#435—61no—
106C8×D4—61no—
107#433—61no—
108#419—21no—
109C2×#103—61no—
110C2×#103—61no—
111C2×#101—61no—
112C2×#101—61no—
113#408—61no—
114#407—31no—
115#407—31no—
116#407C2×D311yescharacteristic
117C2×C22⋊C8—31no—
118#380—31no—
119#328—61no—
120#327—61no—
121C24×C3#4111yescharacteristic
122C3×#41—41no—
123C3×#41—41no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—44no—
133C2×C3×Q8—41no—
134C2×C3×Q8—41no—
135C2×C3×Q8—41no—
136C2×C3×Q8—41no—
137C2×C3×Q8—41no—
138C2×C3×Q8—41no—
139C2×C3×Q8—44no—
140C2×C3×Q8—44no—
141C2×C3×Q8—41no—
142C2×C3×Q8—41no—
143C2×C3×Q8C22⋊C411yescharacteristic
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4C22⋊C411yescharacteristic
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4—44no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4C22×C411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4#4111yescharacteristic
161C22×Dic3—41no—
162C3×Q16—44no—
163C3×SD16—44no—
164C3×M16—41no—
165C3×M16—41no—
166C2×C3×C8—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4C22⋊C411yescharacteristic
170C3×C4⋊C4C22⋊C411yescharacteristic
171C3×C22⋊C4—21no—
172C3×C22⋊C4—21no—
173C3×C22⋊C4—21no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—21no—
176C3×C42—21no—
177#217—81no—
178C4⋊Dic3—41no—
179C4⋊Dic3—41no—
180#208—41no—
181#208—41no—
182#208—41no—
183#208—41no—
184#208—81no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—41no—
190C2×C3⋊C8—41no—
191C2×C3⋊C8—41no—
192C2×C3⋊C8—81no—
193C2×#41#6711yescharacteristic
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8C2×Dic312yes—
202C23×C4—21no—
203C23×C4C22×D311yescharacteristic
204#137—42no—
205#137—42no—
206C2×Q16—22no—
207C2×SD16—22no—
208C2×M16—21no—
209C2×M16—31no—
210C2×M16—31no—
211C2×M16—61no—
212C2×M16—61no—
213C22×C8—31no—
214C22×C8—31no—
215C22×C8—61no—
216#123—21no—
217C22⋊Q8C2×Dic311yescharacteristic
218C22⋊Q8#6711yescharacteristic
219#121C2×Dic311yescharacteristic
220#121#6711yescharacteristic
221#120—21no—
222C4×D4—21no—
223C4×D4—21no—
224#117#6711yescharacteristic
225C2×C4⋊C4—61no—
226C2×C22⋊C4—61no—
227C4⋊C8—61no—
228C4⋊C8—61no—
229#103—22no—
230#103—22no—
231#103—122no—
232#103—122no—
233#101—122no—
234#101—122no—
235C22⋊C8—21no—
236C22⋊C8—61no—
237C22⋊C8—61no—
238C22⋊C8—61no—
239C22⋊C8—61no—
240C22⋊C8—61no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244C22⋊C8C2×Dic312yes—
245#97—61no—
246C4×C8—61no—
247#95—61no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3C4×D411yescharacteristic
254C23×C3#12111yescharacteristic
255C23×C3#12311yescharacteristic
256C3×Q8—24no—
257C3×Q8—24no—
258C3×Q8—41no—
259C3×Q8—42no—
260C3×Q8—42no—
261C3×Q8—42no—
262C3×Q8—42no—
263C3×Q8—42no—
264C3×Q8—42no—
265C3×D4—41no—
266C3×D4—41no—
267C3×D4—41no—
268C3×D4—81no—
269C3×D4—81no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4C2×C22⋊C411yescharacteristic
292C2×C3×C4C4×D411yescharacteristic
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4#12111yescharacteristic
295C2×Dic3—81no—
296C2×Dic3—81no—
297C2×Dic3—81no—
298C3×C8—42no—
299C3⋊C8—41no—
300C3⋊C8—41no—
301C3⋊C8—41no—
302C3⋊C8—41no—
303C3⋊C8—81no—
304C24#23711yescharacteristic
305#41—41no—
306#41—41no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—44no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8#21711yescharacteristic
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4#21711yescharacteristic
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—44no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—121no—
339C22×C4D3×D411yescharacteristic
340C22×C4D3×D411yescharacteristic
341C22×C4#23711yescharacteristic
342C22×C4C22×Dic311yescharacteristic
343C22×C4C2×#6711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345Q16—44no—
346SD16—44no—
347M16—41no—
348M16—41no—
349M16—121no—
350M16—121no—
351M16—121no—
352M16—121no—
353M16—241no—
354C2×C8—22no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—121no—
360C2×C8—121no—
361C2×C8—121no—
362C2×C8—241no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4#21711yescharacteristic
368C4⋊C4#21711yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—241no—
375C42—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3#38611yescharacteristic
390C22×C3#47111yescharacteristic
391C22×C3#47311yescharacteristic
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—44no—
396C3×C4—41no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4B(4,𝔽2)11yescharacteristic
400C3×C4B(4,𝔽2)11yescharacteristic
401Dic3—161no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—41no—
406C23—41no—
407C23D4×Dic311yescharacteristic
408C23#93511yescharacteristic
409C23#93911yescharacteristic
410Q8—24no—
411Q8—24no—
412Q8—41no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—42no—
418Q8—42no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—81no—
423D4—81no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—241no—
446C2×C4—241no—
447C2×C4—241no—
448C2×C4D4×Dic311yescharacteristic
449C2×C4#93711yescharacteristic
450C2×C4#93911yescharacteristic
451C2×C4C2×#21711yescharacteristic
452C8—42no—
453C8—121no—
454C8—121no—
455C8—121no—
456C8—121no—
457C8—241no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—81no—
463C2×C3#188911yescharacteristic
464C2×C3#190211yescharacteristic
465C2×C3#190811yescharacteristic
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#520311yescharacteristic
480C22#520411yescharacteristic
481C22#522011yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—44no—
488C4—41no—
489C4—481no—
490C4#503211yescharacteristic
491C4#503211yescharacteristic
492C3#951911yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—41no—
496C2—41no—
497C2—81no—
498C2#7166911yescharacteristic
499C2#7210211yescharacteristic
500C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3004C2
socleC22×C3#386
prime radicalC3×#525C22
cosocleC23×C3×C4C23
hypercenter#3004D3
hypocenterC3#9519
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3004

Complements

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

NH ≅ G/Ntype
3. #72133C2semidirect
4. #72132C2semidirect
6. #67687C2semidirect
7. #67685C2semidirect
13. C3×#451C4semidirect
15. C3×#448C4semidirect
100. #451Dic3semidirect
102. #448Dic3semidirect
492. C3#9519semidirect

Maximal subgroups

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

subgroupindex
C3×#30042
#829192
#721332
#721322
#717092
#676872
#676852
#95193