Finite Group Databasebeta

← #10691517

Subgroups — #10691517

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
1C2×C3×#482C211yesmaximal, characteristic
2#82210C211yesmaximal, characteristic, complemented
3#71676C211yesmaximal, characteristic
4#71673C211yesmaximal, characteristic
5#71590C211yesmaximal, characteristic, complemented
6#67679C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9242—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C3×#482—21no—
12C3×#482—21no—
13C3×#482—21no—
14C3×#482—21no—
15C2×C3×#107C411yescharacteristic, complemented
16C2×C3×#106C411yescharacteristic, complemented
17C2×C3×#102C411yescharacteristic, complemented
18C2×C3×#102C411yescharacteristic, complemented
19C2×C3×C22⋊C8C2211yescharacteristic
20C4×#217C2211yescharacteristic
21#5207C2211yescharacteristic
22#5207C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5000—22no—
25#4999—21no—
26#4997—21no—
27#4926—21no—
28#4925—21no—
29#4537—22no—
30#4532—21no—
31#4532—21no—
32#4525—21no—
33#4525—21no—
34C2×#482D311yescharacteristic
35#2924—31no—
36#1826—31no—
37#1507—31no—
38#1471—31no—
39#1293—31no—
40#1282—31no—
41C22×C3×D4—21no—
42C22×C3×D4C2×C411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×C3×C8C2×C411yescharacteristic
45C22×C3×C8C2×C411yescharacteristic
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#117—21no—
51C3×#117—21no—
52C3×#117D411yescharacteristic
53C3×#117D411yescharacteristic
54C3×#117D411yescharacteristic
55C3×#117D411yescharacteristic
56C2×C3×C4⋊C4C2×C411yescharacteristic
57C2×C3×C4⋊C4C2×C411yescharacteristic
58C2×C3×C4⋊C4C2×C411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42—21no—
62C2×#217—21no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C2×#208—21no—
66C22×C3⋊C8—21no—
67C3×#107—21no—
68C3×#107—21no—
69C3×#106—21no—
70C3×#106—21no—
71C3×#102—21no—
72C3×#102—21no—
73C3×#102—21no—
74C3×#102—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3⋊#95—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830D411yescharacteristic
83#830D411yescharacteristic
84#830D412yes—
85C4⋊(C3⋊C8)—22no—
86C4⋊(C3⋊C8)—22no—
87#803—22no—
88C4×C3⋊C8—22no—
89C2×#121C2×D311yescharacteristic
90C2×#117C2×D311yescharacteristic
91#482—21no—
92#482—21no—
93#482—21no—
94#482—21no—
95#433—62no—
96#432—61no—
97#431—61no—
98C2×#107Dic311yescharacteristic, complemented
99C2×#106Dic311yescharacteristic, complemented
100#424—61no—
101C2×#102Dic311yescharacteristic, complemented
102C2×#102Dic311yescharacteristic, complemented
103#407—31no—
104#407—31no—
105C2×C22⋊C8—31no—
106C2×C22⋊C8C2×D311yescharacteristic
107#405—61no—
108C4×C22⋊C4—31no—
109#339—61no—
110#339—61no—
111#327—62no—
112#323—61no—
113#323—61no—
114C24×C3—21no—
115C24×C3#4111yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—82no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—41no—
135C2×C3×C8—21no—
136C2×C3×C8—21no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—21no—
154C3×C42—21no—
155C3×C42—21no—
156C3×C42—21no—
157#217—22no—
158#217—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161#208—41no—
162#208—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×D4—21no—
172C22×D4C2×Dic311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C2×M16—61no—
175C2×M16—61no—
176C22×C8—61no—
177C22×C8C2×Dic311yescharacteristic
178C22×C8C2×Dic311yescharacteristic
179#121—21no—
180#121—21no—
181#121—21no—
182#121—21no—
183#117—21no—
184#117—21no—
185#117#6711yescharacteristic
186#117#6711yescharacteristic
187#117#6711yescharacteristic
188#117#6711yescharacteristic
189C2×C4⋊C4C2×Dic311yescharacteristic
190C2×C4⋊C4C2×Dic311yescharacteristic
191C2×C4⋊C4C2×Dic311yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—61no—
195C2×C42—21no—
196C2×C42—61no—
197#107—21no—
198#107—21no—
199#106—21no—
200#106—21no—
201C4⋊C8—62no—
202C4⋊C8—62no—
203#102—21no—
204#102—21no—
205#102—21no—
206#102—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—31no—
210C22⋊C8—31no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216#97—62no—
217C4×C8—62no—
218#95—61no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—81no—
227C23×C3—81no—
228C23×C3C4×D411yescharacteristic
229C23×C3#12111yescharacteristic
230C23×C3#12311yescharacteristic
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—81no—
234C3×D4—82no—
235C3×D4—82no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4C2×C22⋊C411yescharacteristic
255C2×C3×C4C4×D411yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×Dic3—42no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C3×C8—41no—
262C3×C8—41no—
263C3⋊C8—41no—
264C3⋊C8—41no—
265C3⋊C8—42no—
266C3⋊C8—81no—
267C24—21no—
268C24#23711yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—82no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—121no—
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4#23711yescharacteristic
285C22×C4C22×Dic311yescharacteristic
286C22×C4C2×#6711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288M16—121no—
289M16—121no—
290M16—121no—
291M16—121no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—61no—
297C2×C8—62no—
298C2×C8—61no—
299C2×C8—121no—
300C2×C8—121no—
301C2×C8—121no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4#21711yescharacteristic
307C4⋊C4#21711yescharacteristic
308C4⋊C4#21711yescharacteristic
309C4⋊C4#21711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—62no—
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—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—161no—
338C22×C3—161no—
339C22×C3#38611yescharacteristic
340C22×C3#44311yescharacteristic
341C22×C3#44411yescharacteristic
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—81no—
350C3×C4#41011yescharacteristic
351C3×C4#41011yescharacteristic
352Dic3—82no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—81no—
361C23—81no—
362C23D4×Dic311yescharacteristic
363C23#93511yescharacteristic
364C23#93911yescharacteristic
365D4—41no—
366D4—41no—
367D4—81no—
368D4—82no—
369D4—82no—
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—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—122no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4D4×Dic311yescharacteristic
392C2×C4#93711yescharacteristic
393C2×C4#93911yescharacteristic
394C2×C4C2×#21711yescharacteristic
395C8—41no—
396C8—41no—
397C8—121no—
398C8—121no—
399C8—122no—
400C8—241no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3—161no—
406C2×C3#187611yescharacteristic
407C2×C3#189911yescharacteristic
408C2×C3#190311yescharacteristic
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—161no—
423C22—161no—
424C22#519611yescharacteristic
425C22#519711yescharacteristic
426C22#522011yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—81no—
435C4—242no—
436C4#524011yescharacteristic
437C4#524011yescharacteristic
438C3#924211yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—161no—
444C2#7170511yescharacteristic
445C2#7208711yescharacteristic
446C2#7209011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#482C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenterC2×#482D3
hypocenterC3#9242
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#482

Complements

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

NH ≅ G/Ntype
2. #82210C2semidirect
5. #71590C2semidirect
6. #67679C2semidirect
7. #67677C2semidirect
15. C2×C3×#107C4semidirect
16. C2×C3×#106C4semidirect
17. C2×C3×#102C4semidirect
18. C2×C3×#102C4semidirect
98. C2×#107Dic3semidirect
99. C2×#106Dic3semidirect
101. C2×#102Dic3semidirect
102. C2×#102Dic3semidirect
438. C3#9242semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4822
#822102
#716762
#716732
#715902
#676792
#676772
#92423