Finite Group Databasebeta

← #10690401

Subgroups — #10690401

Order 768.

488 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2992C211yesmaximal, characteristic, complemented
2#83141C211yesmaximal, characteristic
3#71746C211yesmaximal, characteristic, complemented
4#70799C211yesmaximal, characteristic, complemented
5#70471C211yesmaximal, characteristic, complemented
6#67751C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#9052—31nomaximal, Sylow
9C3×#529C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C3×#446—21no—
12C3×#435—21no—
13C3×#435—21no—
14C3×C8×D4—21no—
15C3×#433—21no—
16C3×#432C2211yescharacteristic
17C3×#424—21no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×(C2×D3)⋊C8—21no—
20C2×(C2×D3)⋊C8—21no—
21#5066—21no—
22#5057—21no—
23#5046C411yescharacteristic
24#5044C411yescharacteristic
25#5036C411yescharacteristic
26#5034C411yescharacteristic
27C4×#215—21no—
28C4×#213—21no—
29#4925C2211yescharacteristic
30C4×#208C2211yescharacteristic
31C2×#803C2211yescharacteristic
32#4543—21no—
33#4540—21no—
34#4483—21no—
35#4483—21no—
36#3105—31no—
37#2992D311yescharacteristic, complemented
38#1955—31no—
39#1524—31no—
40#1490—31no—
41#1302—31no—
42#1283—31no—
43C22×D12C2×C411yescharacteristic
44C22×C4×D3—21no—
45C2×C3×M16—21no—
46C22×C3×C8—21no—
47C22×C3×C8D411yescharacteristic
48C22×C3×C8D411yescharacteristic
49C3×#126—21no—
50C3×#125D411yescharacteristic
51C3×#124D411yescharacteristic
52C3×#123—21no—
53C3×C22⋊Q8C2×C411yescharacteristic
54C3×#121C2×C411yescharacteristic
55C3×C4×Q8D411yescharacteristic
56C3×C4×D4—21no—
57C3×C4×D4D411yescharacteristic
58C3×#117—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#215—21no—
61C2×#213—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×#208—21no—
65C22×C3⋊C8—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D411yescharacteristic
75C3×C4⋊C8D411yescharacteristic
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×#97—21no—
80C3×#97—21no—
81C3×C4×C8—21no—
82#835—21no—
83#832—21no—
84#830C2×C411yescharacteristic
85#830C2×C411yescharacteristic
86(C2×D3)⋊C8—42no—
87(C2×D3)⋊C8—42no—
88#809—21no—
89#809—21no—
90#808—21no—
91#807—21no—
92C4⋊(C3⋊C8)—21no—
93#803—21no—
94#803—21no—
95#803—21no—
96C4×C3⋊C8—21no—
97C4×C3⋊C8—21no—
98#529C2×D311yescharacteristic
99C2×C4×D4—31no—
100#484—31no—
101#481—31no—
102#478—61no—
103#463—61no—
104#450—31no—
105#447—31no—
106#446—21no—
107C4×SD16—61no—
108C4×D8—61no—
109#435—21no—
110#435—21no—
111C8×D4—21no—
112#433—21no—
113#432C2×D311yescharacteristic
114#431—31no—
115#424—21no—
116C2×C4⋊C8C2×D311yescharacteristic
117C2×C22⋊C8—61no—
118C2×C22⋊C8—61no—
119C4×M16—31no—
120C2×#97—31no—
121#330—61no—
122#329—61no—
123D4⋊C8—61no—
124D4⋊C8—61no—
125C23×D3—21no—
126C2×C3×Q8C22⋊C411yescharacteristic
127C2×C3×D4—21no—
128C2×C3×D4C22⋊C411yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C4×D3—41no—
143C3×M16—81no—
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8—41no—
149C2×C3×C8—41no—
150C2×C3×C8—81no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4C22⋊C411yescharacteristic
156C3×C4⋊C4C22⋊C411yescharacteristic
157C3×C22⋊C4—21no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—21no—
160C3×C22⋊C4—21no—
161C3×C22⋊C4—21no—
162C3×C42—21no—
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4111yescharacteristic
167#215—42no—
168#213—42no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173#208—22no—
174#208—41no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C22×D4—31no—
182C23×C4—61no—
183C2×SD16—61no—
184C2×D8—61no—
185C2×M16—21no—
186C2×M16—61no—
187C22×C8—21no—
188C22×C8—61no—
189C22×C8#6711yescharacteristic
190C22×C8#6711yescharacteristic
191#126—21no—
192#125D1211yescharacteristic
193#124D1211yescharacteristic
194#123—21no—
195C22⋊Q8C4×D311yescharacteristic
196#121C4×D311yescharacteristic
197C4×Q8#6711yescharacteristic
198C4×D4—21no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C4×D4#6711yescharacteristic
204#117—21no—
205C2×C4⋊C4—31no—
206C2×C22⋊C4—61no—
207C2×C42C22×D311yescharacteristic
208#107—61no—
209#106—61no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—61no—
215C4⋊C8#6711yescharacteristic
216C4⋊C8#6711yescharacteristic
217#103—61no—
218#102—61no—
219#102—61no—
220#102—61no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8—21no—
224C22⋊C8—31no—
225C22⋊C8—31no—
226C22⋊C8—122no—
227C22⋊C8—122no—
228#97—21no—
229#97—21no—
230#97—61no—
231#97—61no—
232#97—61no—
233C4×C8—21no—
234C4×C8—61no—
235C4×C8—61no—
236C23×C3—21no—
237C23×C3C4×D411yescharacteristic
238C22×D3—22no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C3×Q8—41no—
246C3×D4—41no—
247C3×D4—41no—
248C3×D4—41no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4C2×C22⋊C411yescharacteristic
260C2×C3×C4C4×D411yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12311yescharacteristic
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268D12—22no—
269D12—22no—
270D12—41no—
271D12—41no—
272C4×D3—81no—
273C4×D3—81no—
274C4×D3—81no—
275C4×D3—81no—
276C3×C8—41no—
277C3×C8—41no—
278C3×C8—41no—
279C3×C8—41no—
280C3⋊C8—42no—
281C3⋊C8—41no—
282C3⋊C8—41no—
283C3⋊C8—42no—
284C24—61no—
285C2×Q8(C2×D3)⋊C411yescharacteristic
286C2×D4—21no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—122no—
290C2×D4(C2×D3)⋊C411yescharacteristic
291C22×C4—21no—
292C22×C4—61no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4C2×C4×D311yescharacteristic
300C22×C4#23511yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—122no—
303D8—122no—
304M16—62no—
305M16—81no—
306M16—121no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—62no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—81no—
317C2×C8—121no—
318C2×C8—121no—
319C2×C8—121no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4(C2×D3)⋊C411yescharacteristic
327C4⋊C4(C2×D3)⋊C411yescharacteristic
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C42—21no—
336C42C2×D1211yescharacteristic
337C42#23511yescharacteristic
338C42C2×#6711yescharacteristic
339C42C2×#6711yescharacteristic
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3#38611yescharacteristic
346C22×C3#44311yescharacteristic
347C22×C3#44411yescharacteristic
348C2×D3—42no—
349C2×D3—42no—
350C2×D3—42no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4#45311yescharacteristic
364C3×C4#45411yescharacteristic
365Dic3—81no—
366C23—21no—
367C23—62no—
368C23—121no—
369C23—121no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—121no—
374C23C4×#6711yescharacteristic
375Q8—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—62no—
380D4—62no—
381D4—121no—
382D4—121no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—241no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4C2×(C2×D3)⋊C411yescharacteristic
401C2×C4C4×#6711yescharacteristic
402C2×C4#92911yescharacteristic
403C2×C4#93011yescharacteristic
404C2×C4#93011yescharacteristic
405C2×C4#95311yescharacteristic
406C8—41no—
407C8—41no—
408C8—41no—
409C8—41no—
410C8—122no—
411C8—121no—
412C8—121no—
413C8—122no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—81no—
417C2×C3#187611yescharacteristic
418C2×C3#188311yescharacteristic
419C2×C3#189111yescharacteristic
420D3—82no—
421C22—21no—
422C22—21no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—122no—
427C22—122no—
428C22—122no—
429C22—241no—
430C22—241no—
431C22—241no—
432C22—241no—
433C22#514411yescharacteristic
434C22#514511yescharacteristic
435C22#521311yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—241no—
445C4#524111yescharacteristic
446C4#524511yescharacteristic
447C3#905211yescharacteristic, complemented, Sylow
448C2—21no—
449C2—21no—
450C2—81no—
451C2—242no—
452C2#7166711yescharacteristic
453C2#7201011yescharacteristic
454C2#7202911yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2992C2
socleC22×C3#386
prime radical#5036C4
cosocleC2×C3×C42C23
hypercenter#2992D3
hypocenterC3#9052
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2992

Complements

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

NH ≅ G/Ntype
1. C3×#2992C2semidirect
3. #71746C2semidirect
4. #70799C2semidirect
5. #70471C2semidirect
6. #67751C2semidirect
7. #67706C2semidirect
37. #2992D3semidirect
447. C3#9052semidirect

Maximal subgroups

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

subgroupindex
C3×#29922
#831412
#717462
#707992
#704712
#677512
#677062
#90523