Finite Group Databasebeta

← #10691653

Subgroups — #10691653

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#217C211yesmaximal, characteristic, complemented
2C2×#5227C211yesmaximal, characteristic
3C3×#1787C211yesmaximal, characteristic, complemented
4#69944C211yesmaximal, characteristic
5#69528C211yesmaximal, characteristic
6#67678C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9262—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×Q8×Dic3—21no—
11C22×#215C412yescomplemented
12C2×#930C2211yescharacteristic
13C2×C3×#103—22no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5229—21no—
17#5229—22no—
18#5227—21no—
19#5227—21no—
20#5227—22no—
21C2×#835C412yescomplemented
22#5212—21no—
23#5212C2211yescharacteristic
24C4×#217—21no—
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27C2×#806—21no—
28C2×#806—21no—
29#4470—22no—
30C2×#448—31no—
31Q8×C22⋊C4—31no—
32#1964—31no—
33#1787D311yescharacteristic, complemented
34#1519—31no—
35#1289—31no—
36#1282—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×Q8—21no—
40C22×C3×Q8—44no—
41C22×C3×Q8C2×C412yes—
42C22×C3×Q8D411yescharacteristic
43C22×C3×Q8D411yescharacteristic
44C22×C3×Q8D412yes—
45C23×C3×C4—21no—
46C23×C3×C4C2311yescharacteristic
47C22×#67—21no—
48C22×D12C2×C411yescharacteristic
49C22×C3×C8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×#217—21no—
53C2×#217D411yescharacteristic
54C2×#217D411yescharacteristic
55Q8×Dic3—44no—
56C2×#215—22no—
57C2×#215—22no—
58C2×#215—44no—
59#930—21no—
60#930—21no—
61#930—21no—
62#930—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3C2×C411yescharacteristic
67C2×C4×Dic3—22no—
68C2×C4×Dic3—21no—
69C22×C3⋊C8C2×C412yes—
70C3×#103—44no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95—21no—
74#835—22no—
75#835—22no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78C3⋊#95—21no—
79C3⋊#95D412yes—
80#830—21no—
81#830—21no—
82#806—42no—
83#806—42no—
84C23×Q8C2×D311yescharacteristic
85C22×SD16—32no—
86C2×#121—31no—
87C2×C4×Q8—61no—
88#448—61no—
89#448—61no—
90#448—62no—
91C2×#103—22no—
92C2×#103—32no—
93C2×#99—61no—
94C2×#99—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97#391—62no—
98#391—61no—
99#380—31no—
100#380—61no—
101#380C2×D311yescharacteristic
102C4×C22⋊C4—31no—
103C4×C22⋊C4—61no—
104#327—62no—
105C24×C3C2×D411yescharacteristic
106C23×D3—21no—
107C2×C3×Q8—24no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—24no—
113C2×C3×Q8—24no—
114C2×C3×Q8—42no—
115C2×C3×Q8—42no—
116C2×C3×Q8—42no—
117C2×C3×Q8—44no—
118C2×C3×Q8—44no—
119C2×C3×Q8—44no—
120C2×C3×Q8C22⋊C412yes—
121C2×C3×Q8C22⋊C412yes—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—42no—
127C22×C3×C4—44no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—41no—
138C2×#67—41no—
139C2×#67—41no—
140C2×#67—41no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—42no—
148C2×C3×C8—42no—
149C2×C3×C8—41no—
150C3×C4⋊C4—42no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153#217—21no—
154#217—21no—
155#217—21no—
156#217—41no—
157#217—41no—
158#217SD1612yescomplemented
159#215—44no—
160#215—44no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—44no—
166C4⋊Dic3—42no—
167C4×Dic3—44no—
168C4×Dic3—42no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—42no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—44no—
175C22×Q8C4×D312yes—
176C22×Q8D1211yescharacteristic
177C22×Q8D1211yescharacteristic
178C22×Q8#6712yes—
179C22×D4—31no—
180C22×D4—61no—
181C23×C4—21no—
182C23×C4C22×D311yescharacteristic
183C2×SD16—62no—
184C2×SD16—62no—
185C2×SD16—124no—
186C22×C8—21no—
187C22×C8—32no—
188#121—61no—
189#121—61no—
190#121—61no—
191#121—61no—
192C4×Q8—124no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—61no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—31no—
199C2×C22⋊C4—31no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C42—62no—
203C2×C42—61no—
204#103—44no—
205#103—62no—
206#103—62no—
207#99—122no—
208#99—122no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#95—21no—
214#95—32no—
215#95—62no—
216#95—62no—
217#95—61no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3C4×D411yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3C2×SD1611yescharacteristic
224C23×C3#13511yescharacteristic
225C23×C3AGL(1,ℤ/8)11yescharacteristic
226C23×C3#13711yescharacteristic
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—81no—
231C22×D3—81no—
232C3×Q8—24no—
233C3×Q8—22no—
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—44no—
237C3×Q8—44no—
238C3×Q8—42no—
239C3×Q8#9914yescomplemented
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—44no—
250C2×C3×C4—44no—
251C2×C3×C4—42no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4C2×C22⋊C411yescharacteristic
256C2×C3×C4C4×D411yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12311yescharacteristic
259#67—81no—
260#67—81no—
261#67—81no—
262#67—81no—
263C2×Dic3—22no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—42no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275D12—42no—
276D12—81no—
277C3×C8—81no—
278C3⋊C8—42no—
279C24—61no—
280C24D3×D411yescharacteristic
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8(C2×D3)⋊C412yes—
295C2×Q8(C2×D3)⋊C412yes—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—122no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—121no—
302C2×D4—121no—
303C2×D4—121no—
304C2×D4—121no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—62no—
315C22×C4—61no—
316C22×C4C2×C4×D311yescharacteristic
317C22×C4C2×D1211yescharacteristic
318C22×C4D3×D411yescharacteristic
319C22×C4#23711yescharacteristic
320C22×C4#23711yescharacteristic
321C22×C4C2×#6711yescharacteristic
322SD16—124no—
323SD16—124no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—62no—
327C2×C8—122no—
328C4⋊C4—42no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—124no—
332C4⋊C4—122no—
333C22⋊C4—32no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C22⋊C4—61no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C42—124no—
344C42—122no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3#38611yescharacteristic
356C22×C3C4×SD1611yescharacteristic
357C22×C3#44011yescharacteristic
358C22×C3#44811yescharacteristic
359C22×C3#44911yescharacteristic
360C22×C3#46011yescharacteristic
361C22×C3#46411yescharacteristic
362C2×D3—81no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—161no—
366C2×D3—161no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—24no—
370C3×C4—21no—
371C3×C4—42no—
372C3×C4—161no—
373C3×C4C2×#9912yes—
374Dic3—42no—
375Dic3—81no—
376Dic3—82no—
377Dic3—81no—
378C23—41no—
379C23—41no—
380C23—121no—
381C23—121no—
382C23—121no—
383C23—241no—
384C23—241no—
385C23#67⋊C411yescharacteristic
386C23#88211yescharacteristic
387C23D3×SD1611yescharacteristic
388C23#91411yescharacteristic
389C23#91811yescharacteristic
390C23#91911yescharacteristic
391C23#93911yescharacteristic
392Q8—24no—
393Q8—22no—
394Q8—22no—
395Q8—22no—
396Q8—44no—
397Q8—44no—
398Q8—42no—
399Q8#80614yescomplemented
400D4—122no—
401D4—241no—
402D4—241no—
403D4—241no—
404D4—241no—
405D4—241no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—62no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—122no—
425C2×C4—121no—
426C2×C4—122no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—121no—
433C2×C4#67⋊C411yescharacteristic
434C2×C4#88611yescharacteristic
435C2×C4C2×(C2×D3)⋊C411yescharacteristic
436C2×C4#93911yescharacteristic
437C8—81no—
438C8—122no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3#188011yescharacteristic
446C2×C3#189811yescharacteristic
447C2×C3#190511yescharacteristic
448D3—161no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22—481no—
463C22—481no—
464C22#478811yescharacteristic
465C22#480611yescharacteristic
466C22#481011yescharacteristic
467C22#481711yescharacteristic
468C22#495811yescharacteristic
469C22#516311yescharacteristic
470C22#518511yescharacteristic
471C4—21no—
472C4—24no—
473C4—21no—
474C4—21no—
475C4—42no—
476C4—122no—
477C4—161no—
478C4—241no—
479C4—242no—
480C4—241no—
481C4C2×#80612yes—
482C3#926211yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—481no—
490C2#6949611yescharacteristic
491C2#7024011yescharacteristic
492C2#7033411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1787C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1787D3
hypocenterC3#9262
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. Q8×#217C2semidirect
3. C3×#1787C2semidirect
6. #67678C2semidirect
7. #67625C2semidirect
11. C22×#215C4semidirect
21. C2×#835C4semidirect
33. #1787D3semidirect
158. #217SD16semidirect
239. C3×Q8#99semidirect
399. Q8#806semidirect
482. C3#9262semidirect

Maximal subgroups

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

subgroupindex
Q8×#2172
C2×#52272
C3×#17872
#699442
#695282
#676782
#676252
#92623