Finite Group Databasebeta

← #10691515

Subgroups — #10691515

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2924C211yesmaximal, characteristic, complemented
2C2×#5003C211yesmaximal, characteristic
3#71590C211yesmaximal, characteristic, complemented
4#69977C211yesmaximal, characteristic
5#69921C211yesmaximal, characteristic
6#67687C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9242—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#433—22no—
12C3×#432—21no—
13C3×#431—21no—
14C3×#424—21no—
15C3×#407C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17C3×#405—21no—
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#5003—21no—
22#5003—21no—
23#5003—21no—
24#5003—21no—
25C2×#809C411yescharacteristic, complemented
26C2×#809C411yescharacteristic, complemented
27C2×#808C411yescharacteristic, complemented
28C2×#807C411yescharacteristic, complemented
29#4532—21no—
30#4532—21no—
31#4470—22no—
32#4468—21no—
33#4468—21no—
34C2×#482—31no—
35#2924D311yescharacteristic, complemented
36#1826—31no—
37#1507—31no—
38#1471—31no—
39#1293—31no—
40#1282—31no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12C2×C411yescharacteristic
44C2×C3×M16—21no—
45C22×C3×C8—21no—
46C3×#117—21no—
47C3×#117—21no—
48C3×#117D411yescharacteristic
49C3×#117D411yescharacteristic
50C3×#117D411yescharacteristic
51C3×#117D411yescharacteristic
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C4⋊C4C2×C411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42—21no—
56C2×#217—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C22×C3⋊C8C2×C411yescharacteristic
66C22×C3⋊C8C2×C411yescharacteristic
67C3×C4⋊C8—22no—
68C3×C4⋊C8—22no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8D411yescharacteristic
72C3×C22⋊C8D411yescharacteristic
73C3×C22⋊C8D412yes—
74C3×#97—22no—
75C3×C4×C8—22no—
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#809—21no—
82#809—21no—
83#809—21no—
84#809—21no—
85#808—21no—
86#808—21no—
87#807—21no—
88#807—21no—
89C2×#121—31no—
90C2×#117C2×D311yescharacteristic
91#482—61no—
92#482—61no—
93#482—61no—
94#482—61no—
95#433—22no—
96#432—21no—
97#431—21no—
98C2×#107—31no—
99C2×#106—31no—
100#424—21no—
101C2×#102—31no—
102C2×#102—31no—
103#407—31no—
104#407C2×D311yescharacteristic
105C2×C22⋊C8—31no—
106C2×C22⋊C8C2×D311yescharacteristic
107#405—21no—
108C4×C22⋊C4—31no—
109#339—61no—
110#339—61no—
111#327—62no—
112#323—61no—
113#323—61no—
114C24×C3#4111yescharacteristic
115C23×D3—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—82no—
129C22×Dic3—21no—
130C22×Dic3—41no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—41no—
134C2×D12—41no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—21no—
138C2×C3×C8—22no—
139C2×C3×C8—21no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42—21no—
155C3×C42—21no—
156#217—22no—
157#217—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×D4—31no—
172C22×D4—61no—
173C23×C4C22×D311yescharacteristic
174C2×M16—21no—
175C2×M16—61no—
176C22×C8—21no—
177C22×C8—31no—
178C22×C8—31no—
179#121—61no—
180#121—61no—
181#121—61no—
182#121—61no—
183#117—21no—
184#117—21no—
185#117D1211yescharacteristic
186#117D1211yescharacteristic
187#117#6711yescharacteristic
188#117#6711yescharacteristic
189C2×C4⋊C4—31no—
190C2×C4⋊C4C4×D311yescharacteristic
191C2×C4⋊C4C4×D311yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C42—21no—
196C2×C42—61no—
197#107—61no—
198#107—61no—
199#106—61no—
200#106—61no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203#102—61no—
204#102—61no—
205#102—61no—
206#102—61no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8#6711yescharacteristic
214C22⋊C8#6711yescharacteristic
215C22⋊C8#6712yes—
216#97—22no—
217C4×C8—22no—
218#95—61no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12311yescharacteristic
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—81no—
230C22×D3—81no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
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×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251#67—82no—
252#67—82no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258D12—41no—
259D12—41no—
260D12—81no—
261C3×C8—41no—
262C3×C8—41no—
263C3×C8—42no—
264C3⋊C8—41no—
265C3⋊C8—41no—
266C3⋊C8—81no—
267C24—61no—
268C24#23511yescharacteristic
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—242no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—121no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4C2×D1211yescharacteristic
284C22×C4#23511yescharacteristic
285C22×C4C2×#6711yescharacteristic
286C22×C4C2×#6711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288M16—41no—
289M16—41no—
290M16—121no—
291M16—121no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—121no—
300C2×C8—121no—
301C2×C8—121no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4(C2×D3)⋊C411yescharacteristic
307C4⋊C4(C2×D3)⋊C411yescharacteristic
308C4⋊C4(C2×D3)⋊C411yescharacteristic
309C4⋊C4(C2×D3)⋊C411yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—62no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—121no—
323C42—121no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#38611yescharacteristic
335C22×C3#44311yescharacteristic
336C22×C3#44411yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—161no—
341C2×D3—161no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4#41011yescharacteristic
350C3×C4#41011yescharacteristic
351Dic3—82no—
352Dic3—81no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—121no—
358C23—121no—
359C23—121no—
360C23—241no—
361C23—241no—
362C23C4×#6711yescharacteristic
363C23#92911yescharacteristic
364C23#93011yescharacteristic
365D4—121no—
366D4—121no—
367D4—241no—
368D4—242no—
369D4—242no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—122no—
387C2×C4—121no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4C2×(C2×D3)⋊C411yescharacteristic
392C2×C4C4×#6711yescharacteristic
393C2×C4#93011yescharacteristic
394C2×C4#95311yescharacteristic
395C8—41no—
396C8—41no—
397C8—42no—
398C8—121no—
399C8—121no—
400C8—241no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3#187611yescharacteristic
406C2×C3#189911yescharacteristic
407C2×C3#190311yescharacteristic
408D3—161no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—241no—
420C22—241no—
421C22—241no—
422C22—481no—
423C22—481no—
424C22#514411yescharacteristic
425C22#514511yescharacteristic
426C22#521311yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—242no—
435C4—241no—
436C4#500611yescharacteristic
437C4#500611yescharacteristic
438C3#924211yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—481no—
444C2#7159511yescharacteristic
445C2#7201011yescharacteristic
446C2#7202311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2924C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#2924D3
hypocenterC3#9242
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2924

Complements

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

NH ≅ G/Ntype
1. C3×#2924C2semidirect
3. #71590C2semidirect
6. #67687C2semidirect
7. #67678C2semidirect
25. C2×#809C4semidirect
26. C2×#809C4semidirect
27. C2×#808C4semidirect
28. C2×#807C4semidirect
35. #2924D3semidirect
438. C3#9242semidirect

Maximal subgroups

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

subgroupindex
C3×#29242
C2×#50032
#715902
#699772
#699212
#676872
#676782
#92423