Finite Group Databasebeta

← #11714440

Subgroups — #11714440

Order 768.

1579 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1579 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#5919C211yesmaximal, characteristic, complemented
2C2×Q8×S4C211yesmaximal, characteristic, complemented
3C4×Q8×A4C211yesmaximal, characteristic, complemented
4#85312C213yesmaximal, complemented
5#85310C213yesmaximal, complemented
6#85290C213yesmaximal, complemented
7C4×C4⋊S4C213yesmaximal, complemented
8#36434—31nomaximal, Sylow
9C2×Q8×A4C2211yescharacteristic
10#5919—24no—
11Q8×S4—24no—
12C2×C4⋊S4C2213yes—
13C2×C4⋊S4C2213yes—
14C2×C4×S4C2213yes—
15C2×C4×S4C2213yes—
16C2×A4⋊Q8C2213yes—
17#5576—41nomaximal
18A4×C4⋊C4C2213yes—
19C42×A4C2213yes—
20(C2×S4)⋊C4C2211yescharacteristic
21(C2×S4)⋊C4C2213yes—
22(C2×S4)⋊C4C2216yes—
23C4⋊SL(2,ℤ/4)C2213yes—
24C2×D4×Q8—31no—
25C2×#548—31no—
26C2×#548—31no—
27C22×C4×Q8D311yescharacteristic, complemented
28#2850—33no—
29#2545—33no—
30#2536—33no—
31#2429—33no—
32#2426—33no—
33#2417—33no—
34#2416—33no—
35#2351—31no—
36#2341—31no—
37Q8×C22⋊C4—31no—
38C4×#121—33no—
39C22×S4C2313yes—
40C22×S4C2311yescharacteristic
41C2×#239—41no—
42C2×D3×Q8—41no—
43Q8×A4D414yescomplemented
44C2×C4×A4C2313yes—
45C2×C4×A4C2311yescharacteristic
46C2×C4×A4C2313yes—
47C2×SL(2,ℤ/4)C2313yes—
48C2×SL(2,ℤ/4)C2311yescharacteristic
49C4⋊S4—26no—
50C4⋊S4—26no—
51C4×S4—26no—
52C4×S4—26no—
53A4⋊Q8—26no—
54C3×C4×Q8—41no—
55#897—43no—
56#895—43no—
57#875—43no—
58C4×D12—43no—
59C22×#41—31no—
60C23×Q8—31no—
61C23×Q8C2×D311yescharacteristic
62D4×Q8—64no—
63D4×Q8—64no—
64#548—61no—
65#548—63no—
66#548—63no—
67#548—64no—
68#548—64no—
69#548—61no—
70C2×C4⋊Q8—33no—
71C2×#124—33no—
72C2×#124—33no—
73C2×C22⋊Q8—33no—
74C2×C22⋊Q8—33no—
75C2×C22⋊Q8—33no—
76C2×C22⋊Q8—33no—
77C2×#121—33no—
78C2×#121—33no—
79C2×#121—33no—
80C2×C4×Q8—31no—
81C2×C4×Q8—31no—
82C2×C4×Q8—33no—
83C2×C4×Q8—33no—
84C2×C4×Q8—31no—
85C2×C4×Q8—31no—
86C2×C4×D4—33no—
87C2×C4×D4—33no—
88C2×C4×D4—33no—
89C22×C4⋊C4C2×D313yes—
90C22×C42C2×D313yes—
91#398—36no—
92#396—36no—
93#396—36no—
94#394—36no—
95#391—33no—
96#391—33no—
97#390—33no—
98#390—33no—
99#390—33no—
100#389—33no—
101#389—33no—
102#389—33no—
103#388—36no—
104#388—36no—
105#388—33no—
106#388—33no—
107#386—33no—
108#386—36no—
109#386—31no—
110#382—33no—
111#380—33no—
112#380—33no—
113#380—33no—
114C4×C4⋊C4—33no—
115C4×C22⋊C4—33no—
116C4×C22⋊C4—33no—
117C22×A4C2411yescharacteristic
118C2×S4—21no—
119C2×S4—21no—
120C2×S4—23no—
121C2×S4—23no—
122C2×S4—23no—
123C2×S4#4112yescomplemented
124C2×C3×Q8—41no—
125#239—84no—
126D3×Q8—84no—
127C2×D12—43no—
128C2×D12—43no—
129C2×C4×D3—43no—
130C2×C4×D3—43no—
131C2×Dic6—43no—
132C4×A4—23no—
133C4×A4—21no—
134C4×A4C2×D416yes—
135SL(2,ℤ/4)—23no—
136SL(2,ℤ/4)—21no—
137C3×C4⋊C4—43no—
138C3×C42—43no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—43no—
141(C2×D3)⋊C4—46no—
142C4⋊Dic3—43no—
143C2×#41—64no—
144C2×#41—63no—
145C2×#41—64no—
146C2×#41—61no—
147C22×Q8—31no—
148C22×Q8—33no—
149C22×Q8—34no—
150C22×Q8—33no—
151C22×Q8—63no—
152C22×Q8—64no—
153C22×Q8—64no—
154C22×Q8—61no—
155C22×Q8D1214yescomplemented
156C22×D4—31no—
157C22×D4—33no—
158C22×D4—33no—
159C22×D4—33no—
160C23×C4—33no—
161C23×C4—33no—
162C23×C4C22×D313yes—
163C23×C4C22×D311yescharacteristic
164C23×C4C22×D313yes—
165C4⋊Q8—612no—
166#124—63no—
167#124—63no—
168#124—63no—
169#124—612no—
170#124—63no—
171C22⋊Q8—63no—
172C22⋊Q8—66no—
173C22⋊Q8—66no—
174C22⋊Q8—612no—
175C22⋊Q8—612no—
176C22⋊Q8—66no—
177C22⋊Q8—63no—
178#121—63no—
179#121—66no—
180#121—66no—
181#121—66no—
182#121—66no—
183#121—66no—
184#121—63no—
185C4×Q8—31no—
186C4×Q8—33no—
187C4×Q8—33no—
188C4×Q8—61no—
189C4×Q8—63no—
190C4×Q8—63no—
191C4×Q8—64no—
192C4×Q8—64no—
193C4×Q8S411yescharacteristic, complemented
194C4×D4—63no—
195C4×D4—63no—
196C4×D4—63no—
197C4×D4—66no—
198C4×D4—66no—
199C4×D4—66no—
200C4×D4—66no—
201C4×D4—63no—
202C2×C4⋊C4—33no—
203C2×C4⋊C4—33no—
204C2×C4⋊C4—36no—
205C2×C4⋊C4—33no—
206C2×C4⋊C4—33no—
207C2×C4⋊C4—33no—
208C2×C4⋊C4—33no—
209C2×C4⋊C4—33no—
210C2×C4⋊C4—33no—
211C2×C4⋊C4—36no—
212C2×C4⋊C4—33no—
213C2×C22⋊C4—31no—
214C2×C22⋊C4—33no—
215C2×C22⋊C4—33no—
216C2×C22⋊C4—33no—
217C2×C22⋊C4—36no—
218C2×C22⋊C4—36no—
219C2×C22⋊C4—33no—
220C2×C22⋊C4—33no—
221C2×C22⋊C4—31no—
222C2×C22⋊C4—33no—
223C2×C22⋊C4—31no—
224C2×C22⋊C4—33no—
225C2×C22⋊C4—36no—
226C2×C42—33no—
227C2×C42—33no—
228C2×C42—33no—
229C2×C42—33no—
230C2×C42—33no—
231C2×C42—33no—
232#95—31no—
233#95—33no—
234#95—33no—
235#95—33no—
236#95—31no—
237#95—33no—
238#95—36no—
239#95—36no—
240C22×D3—41no—
241C22×D3—43no—
242C2×A4C22×D411yescharacteristic
243C2×A4C2×#4111yescharacteristic
244C2×A42−1+411yescharacteristic
245S4—22no—
246S4—43no—
247C3×Q8—44no—
248C2×C3×C4—43no—
249C2×C3×C4—43no—
250C2×C3×C4—41no—
251C2×Dic3—41no—
252C2×Dic3—43no—
253D12—86no—
254D12—86no—
255C4×D3—86no—
256C4×D3—86no—
257Dic6—86no—
258C24—31no—
259C24—33no—
260C24C23×D311yescharacteristic
261#41—1212no—
262#41—124no—
263C2×Q8—34no—
264C2×Q8—34no—
265C2×Q8—312no—
266C2×Q8—33no—
267C2×Q8—61no—
268C2×Q8—66no—
269C2×Q8—63no—
270C2×Q8—66no—
271C2×Q8—66no—
272C2×Q8—63no—
273C2×Q8—1212no—
274C2×Q8—124no—
275C2×Q8C2×S411yescharacteristic
276C2×D4—32no—
277C2×D4—32no—
278C2×D4—63no—
279C2×D4—63no—
280C2×D4—66no—
281C2×D4—66no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—63no—
287C2×D4—66no—
288C2×D4—63no—
289C2×D4—63no—
290C2×D4—63no—
291C2×D4—63no—
292C2×D4—63no—
293C2×D4—63no—
294C2×D4—63no—
295C2×D4—123no—
296C22×C4—21no—
297C22×C4—23no—
298C22×C4—33no—
299C22×C4—31no—
300C22×C4—33no—
301C22×C4—36no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—33no—
305C22×C4—33no—
306C22×C4—33no—
307C22×C4—31no—
308C22×C4—33no—
309C22×C4—33no—
310C22×C4—31no—
311C22×C4—61no—
312C22×C4—63no—
313C22×C4—66no—
314C22×C4—63no—
315C22×C4—66no—
316C22×C4—66no—
317C22×C4—66no—
318C22×C4—63no—
319C22×C4—63no—
320C22×C4—63no—
321C22×C4C2×D1216yes—
322C4⋊C4—36no—
323C4⋊C4—33no—
324C4⋊C4—63no—
325C4⋊C4—66no—
326C4⋊C4—63no—
327C4⋊C4—66no—
328C4⋊C4—66no—
329C4⋊C4—66no—
330C4⋊C4—66no—
331C4⋊C4—612no—
332C4⋊C4—63no—
333C4⋊C4—123no—
334C4⋊C4C2×S413yes—
335C22⋊C4—62no—
336C22⋊C4—66no—
337C22⋊C4—61no—
338C22⋊C4—66no—
339C22⋊C4—66no—
340C22⋊C4—63no—
341C22⋊C4—66no—
342C22⋊C4—612no—
343C22⋊C4—66no—
344C22⋊C4—61no—
345C22⋊C4—63no—
346C22⋊C4—66no—
347C22⋊C4—61no—
348C22⋊C4—63no—
349C22⋊C4—61no—
350C22⋊C4—63no—
351C22⋊C4—66no—
352C22⋊C4—126no—
353C42—33no—
354C42—33no—
355C42—33no—
356C42—63no—
357C42—66no—
358C42—66no—
359C42C2×S413yes—
360C22×C3—41no—
361C2×D3—42no—
362C2×D3—81no—
363C2×D3—81no—
364C2×D3—83no—
365C2×D3—83no—
366C2×D3—83no—
367A4#54811yescharacteristic, complemented
368C3×C4—46no—
369C3×C4—83no—
370C3×C4—81no—
371Dic3—83no—
372Dic3—81no—
373C23—31no—
374C23—31no—
375C23—31no—
376C23—31no—
377C23—32no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—63no—
384C23—63no—
385C23—63no—
386C23—63no—
387C23—63no—
388C23—63no—
389C23—61no—
390C23—63no—
391C23C22×D1211yescharacteristic
392C23#100711yescharacteristic
393C23D3×#4111yescharacteristic
394Q8—312no—
395Q8—64no—
396Q8—126no—
397Q8—126no—
398Q8C4⋊S414yescomplemented
399D4—62no—
400D4—62no—
401D4—62no—
402D4—62no—
403D4—126no—
404D4—126no—
405D4—126no—
406D4—123no—
407D4—123no—
408D4—123no—
409D4—123no—
410D4—126no—
411C2×C4—33no—
412C2×C4—36no—
413C2×C4—36no—
414C2×C4—36no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—31no—
418C2×C4—33no—
419C2×C4—63no—
420C2×C4—63no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—63no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—63no—
427C2×C4—63no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—63no—
431C2×C4—63no—
432C2×C4—61no—
433C2×C4—63no—
434C2×C4—63no—
435C2×C4—61no—
436C2×C4—63no—
437C2×C4—63no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—63no—
441C2×C4—63no—
442C2×C4—63no—
443C2×C4—61no—
444C2×C4—126no—
445C2×C4—126no—
446C2×C4—126no—
447C2×C4—126no—
448C2×C4C22×S411yescharacteristic
449C2×C4C22×S413yes—
450C2×C4C22×S413yes—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—41no—
454D3—82no—
455D3—163no—
456C22—31no—
457C22—31no—
458C22—31no—
459C22—31no—
460C22—31no—
461C22—31no—
462C22—31no—
463C22—31no—
464C22—31no—
465C22—61no—
466C22—62no—
467C22—62no—
468C22—62no—
469C22—121no—
470C22—121no—
471C22—123no—
472C22—123no—
473C22—123no—
474C22—123no—
475C22—121no—
476C22—121no—
477C22—123no—
478C22—123no—
479C22—123no—
480C22#557611yescharacteristic, complemented
481C22C23×S411yescharacteristic
482C4—21no—
483C4—23no—
484C4—36no—
485C4—62no—
486C4—61no—
487C4—63no—
488C4—121no—
489C4—123no—
490C4—123no—
491C4—121no—
492C4—123no—
493C4C2×C4⋊S416yes—
494C3—41noSylow
495C2—31no—
496C2—31no—
497C2—31no—
498C2—31no—
499C2—122no—
500C2—243no—
501C2C22×C4⋊S411yescharacteristic
502C2#8748311yescharacteristic
503C2#41×S411yescharacteristic
subgroupG/H
center, FrattiniC22C23×S4
commutator, cosocleC22×A4C24
FittingC22×C4×Q8D3
socleC24C23×D3
hypercenterC4×Q8S4
hypocenterA4#548
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#5919C2semidirect
2. C2×Q8×S4C2semidirect
3. C4×Q8×A4C2semidirect
4. #85312C2semidirect
5. #85310C2semidirect
6. #85290C2semidirect
7. C4×C4⋊S4C2semidirect
27. C22×C4×Q8D3semidirect
43. Q8×A4D4semidirect
123. C2×S4#41semidirect
155. C22×Q8D12semidirect
193. C4×Q8S4semidirect
367. A4#548semidirect
398. Q8C4⋊S4semidirect
480. C22#5576semidirect

Maximal subgroups

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

subgroupindex
C2×#59192
C2×Q8×S42
C4×Q8×A42
#853122
#853102
#852902
C4×C4⋊S42
#364343
#55764