Finite Group Databasebeta

← #10691765

Subgroups — #10691765

Order 768.

564 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82749C211yesmaximal, characteristic, complemented
2#82736C211yesmaximal, characteristic
3C3×#1520C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic
5#69528C211yesmaximal, characteristic
6#67677C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9278—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5229—21no—
14#5227—21no—
15C4×#217C2211yescharacteristic
16C2×C3⋊#95C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5048—22no—
19#5046C412yescomplemented
20#5043C412yescomplemented
21#4980—21no—
22#4980—21no—
23Dic3×C4⋊C4—21no—
24#4968—21no—
25C2×#806—21no—
26C2×#806—21no—
27#4950—21no—
28Dic3×C22⋊C4—21no—
29C3×#328—22no—
30#4470—21no—
31#4470—21no—
32#3187—31no—
33#2368—31no—
34#1964—31no—
35#1520D311yescharacteristic, complemented
36C2×#327—31no—
37#1294—31no—
38#1282—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C23×Dic3—21no—
43C22×D12C2×C411yescharacteristic
44C22×C3×C8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8C2×C412yes—
48C3×C22⋊Q8D412yes—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×#217D411yescharacteristic
55C2×#217D411yescharacteristic
56C2×#215—22no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3C2×C411yescharacteristic
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C22×C3⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72#835—22no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95D411yescharacteristic
80C3⋊#95D411yescharacteristic
81#830—21no—
82#830C2×C412yes—
83#809—22no—
84#808—22no—
85#806—42no—
86#806—42no—
87C2×C22⋊Q8C2×D311yescharacteristic
88C2×#121—31no—
89#483—32no—
90#481—61no—
91#465—62no—
92#450—32no—
93#448—61no—
94C2×#99—61no—
95C2×#99—61no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8C2×D311yescharacteristic
98#391—61no—
99#387—61no—
100#387—61no—
101#385—61no—
102#380C2×D311yescharacteristic
103C4×C4⋊C4—61no—
104C4×C22⋊C4—31no—
105C4×C22⋊C4—61no—
106C2×#95—31no—
107#328—22no—
108#327—61no—
109#327—61no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—81no—
114C2×C3×Q8C22⋊C412yes—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—81no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—42no—
137C22×Dic3—41no—
138C22×Dic3—41no—
139C22×Dic3—81no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—81no—
143C2×C3×C8—41no—
144C2×C3×C8—81no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—81no—
149C3×C4⋊C4C22⋊C412yes—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154#217—21no—
155#217—21no—
156#217—21no—
157#217SD1612yescomplemented
158#215—82no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—42no—
164C4⋊Dic3—81no—
165C4×Dic3—42no—
166C4×Dic3—81no—
167C4×Dic3—81no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—81no—
170C22×Q8D1211yescharacteristic
171C22×D4—31no—
172C22×D4—61no—
173C23×C4—61no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C22×C8—21no—
177C22×C8—61no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8C4×D312yes—
181C22⋊Q8#6712yes—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4D1211yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—31no—
196C2×C22⋊C4—61no—
197C2×C42—61no—
198C2×C42—61no—
199C2×C42—61no—
200#106—62no—
201#103—62no—
202#102—62no—
203#99—122no—
204#99—122no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—32no—
208C22⋊C8—61no—
209#95—21no—
210#95—31no—
211#95—31no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216#95—61no—
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—42no—
233C3×Q8—81no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4C2×C22⋊C411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12311yescharacteristic
253#67—81no—
254#67—81no—
255#67—81no—
256#67—81no—
257C2×Dic3—22no—
258C2×Dic3—22no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—82no—
273C2×Dic3—82no—
274D12—42no—
275D12—81no—
276C3×C8—81no—
277C3⋊C8—42no—
278C24—61no—
279C24D3×D411yescharacteristic
280C2×Q8—21no—
281C2×Q8—81no—
282C2×Q8(C2×D3)⋊C412yes—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—241no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—81no—
301C22×C4—122no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—241no—
305C22×C4C2×C4×D311yescharacteristic
306C22×C4C2×D1211yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4#23711yescharacteristic
309C22×C4#23711yescharacteristic
310C22×C4C2×#6711yescharacteristic
311SD16—242no—
312C2×C8—41no—
313C2×C8—62no—
314C2×C8—81no—
315C2×C8—241no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—81no—
322C4⋊C4—122no—
323C4⋊C4—241no—
324C4⋊C4(C2×D3)⋊C412yes—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—32no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C42—122no—
336C42—241no—
337C42—241no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3#38611yescharacteristic
349C22×C3C4×SD1611yescharacteristic
350C22×C3#44011yescharacteristic
351C22×C3#44811yescharacteristic
352C22×C3#44911yescharacteristic
353C22×C3#46011yescharacteristic
354C22×C3#46411yescharacteristic
355C2×D3—81no—
356C2×D3—81no—
357C2×D3—81no—
358C2×D3—161no—
359C2×D3—161no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—161no—
367Dic3—42no—
368Dic3—42no—
369Dic3—81no—
370Dic3—81no—
371Dic3—81no—
372C23—41no—
373C23—41no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23—241no—
378C23—241no—
379C23#67⋊C411yescharacteristic
380C23#88211yescharacteristic
381C23D3×SD1611yescharacteristic
382C23#91411yescharacteristic
383C23#91811yescharacteristic
384C23#91911yescharacteristic
385C23#93911yescharacteristic
386Q8—42no—
387Q8—81no—
388D4—122no—
389D4—241no—
390D4—241no—
391D4—241no—
392D4—241no—
393D4—241no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—242no—
425C2×C4—242no—
426C2×C4#67⋊C411yescharacteristic
427C2×C4#88611yescharacteristic
428C2×C4C2×(C2×D3)⋊C411yescharacteristic
429C2×C4#93911yescharacteristic
430C8—81no—
431C8—122no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3#188011yescharacteristic
439C2×C3#190611yescharacteristic
440C2×C3#191311yescharacteristic
441D3—161no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22—481no—
456C22—481no—
457C22#478811yescharacteristic
458C22#480611yescharacteristic
459C22#481011yescharacteristic
460C22#481711yescharacteristic
461C22#495811yescharacteristic
462C22#516311yescharacteristic
463C22#518511yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—42no—
470C4—122no—
471C4—122no—
472C4—161no—
473C4—241no—
474C4—241no—
475C4—241no—
476C3#927811yescharacteristic, complemented, Sylow
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—481no—
484C2#7002411yescharacteristic
485C2#7024011yescharacteristic
486C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1520D3
hypocenterC3#9278
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
1. #82749C2semidirect
3. C3×#1520C2semidirect
6. #67677C2semidirect
7. #67624C2semidirect
19. #5046C4semidirect
20. #5043C4semidirect
35. #1520D3semidirect
157. #217SD16semidirect
476. C3#9278semidirect

Maximal subgroups

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

subgroupindex
#827492
#827362
C3×#15202
C2×#44702
#695282
#676772
#676242
#92783