Finite Group Databasebeta

← #10691106

Subgroups — #10691106

Order 768.

494 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82338C211yesmaximal, characteristic
2#82296C211yesmaximal, characteristic
3C3×#1478C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic, complemented
5#70792C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9177—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#431C2211yescharacteristic
13#5221—21no—
14#5215—21no—
15#5211—21no—
16C4×#217—21no—
17#5009—22no—
18#5008C412yescomplemented
19#5007C412yescomplemented
20Dic3×C4⋊C4—21no—
21#4964—21no—
22#4951—22no—
23#4936C2211yescharacteristic
24C4×C4⋊Dic3C2211yescharacteristic
25#4924C2211yescharacteristic
26C3×D4⋊C8—22no—
27#4467—21no—
28#4467—21no—
29#4467—22no—
30#3077—31no—
31#2357—31no—
32#1518—31no—
33#1503—31no—
34#1478D311yescharacteristic
35#1303—31no—
36#1286—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C3×C4×D4—21no—
40C3×C4×D4—21no—
41C3×C4×D4C2×C412yes—
42C3×C4×D4D412yes—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#217—22no—
47C2×#217—22no—
48#926—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51#924—21no—
52#924—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3—22no—
55C2×#208—21no—
56#870—21no—
57#869C2×C411yescharacteristic
58#869C2×C411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×C4×C8—21no—
66C3×C4×C8—21no—
67#832—22no—
68#832—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73C3⋊#95D412yes—
74#808—22no—
75#807—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78#533—31no—
79C2×C4×D4C2×D311yescharacteristic
80#478—62no—
81#476—32no—
82#474—32no—
83#431C2×D311yescharacteristic
84#431C2×D311yescharacteristic
85#428—61no—
86#423—31no—
87#418—61no—
88#390—61no—
89#388—62no—
90#382—31no—
91#381—61no—
92C4×C4⋊C4—31no—
93C4×C4⋊C4—61no—
94C4×C22⋊C4—61no—
95#328—61no—
96#328—61no—
97#328—62no—
98D4⋊C8—22no—
99C24×C3—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4C22⋊C412yes—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—82no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4C22⋊C412yes—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128#217—44no—
129#217—41no—
130#217—82no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—42no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140C4×Dic3—41no—
141C4×Dic3—41no—
142C4×Dic3—41no—
143C4×Dic3—82no—
144#208—41no—
145#208—41no—
146C2×C3⋊C8—22no—
147C22×D4D1211yescharacteristic
148C23×C4—21no—
149C2×M16—61no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152#125—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155C4×Q8—61no—
156C4×Q8—61no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4C4×D312yes—
160C4×D4#6712yes—
161#117—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4D1211yescharacteristic
165C2×C22⋊C4—21no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C42—61no—
169C2×C42—62no—
170C2×C42C22×D311yescharacteristic
171#107—62no—
172#106—62no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—32no—
176C4⋊C8—61no—
177#102—62no—
178#102—62no—
179C22⋊C8—21no—
180C22⋊C8—21no—
181C4×C8—21no—
182C4×C8—21no—
183#95—31no—
184#95—31no—
185#95—32no—
186#95—62no—
187#95—61no—
188C23×C3—22no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3#12011yescharacteristic
194C3×D4—22no—
195C3×D4—22no—
196C3×D4—41no—
197C3×D4—41no—
198C2×C3×C4—22no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—82no—
203C2×C3×C4—82no—
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×C3×C4#13511yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4#13711yescharacteristic
213C2×C3×C4#13711yescharacteristic
214C2×Dic3—22no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225Dic6—81no—
226Dic6—81no—
227C3×C8—81no—
228C3×C8—81no—
229C3⋊C8—42no—
230C24—21no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4(C2×D3)⋊C412yes—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—61no—
243C22×C4—62no—
244C22×C4—61no—
245C22×C4—82no—
246C22×C4C2×D1211yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4D3×D411yescharacteristic
249M16—121no—
250M16—121no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—62no—
254C4⋊C4—21no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—122no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4(C2×D3)⋊C412yes—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—124no—
268C22⋊C4—121no—
269C22⋊C4—242no—
270C42—121no—
271C42—121no—
272C42—121no—
273C42—242no—
274C42C2×C4×D311yescharacteristic
275C42#23711yescharacteristic
276C42#23711yescharacteristic
277C42C2×#6711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—42no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3#38611yescharacteristic
286C22×C3#45211yescharacteristic
287C22×C3#45211yescharacteristic
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—42no—
293C3×C4#41611yescharacteristic
294C3×C4#41911yescharacteristic
295C3×C4#48511yescharacteristic
296C3×C4#48511yescharacteristic
297Dic3—42no—
298Dic3—81no—
299Dic3—82no—
300Dic3—81no—
301Dic3—81no—
302C23—22no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—41no—
307C23#88211yescharacteristic
308Q8—241no—
309Q8—241no—
310D4—22no—
311D4—22no—
312D4—41no—
313D4—41no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—62no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4#67⋊C411yescharacteristic
332C2×C4#67⋊C411yescharacteristic
333C2×C4#88611yescharacteristic
334C2×C4#90411yescharacteristic
335C2×C4#90411yescharacteristic
336C2×C4#90911yescharacteristic
337C2×C4#90911yescharacteristic
338C2×C4C2×(C2×D3)⋊C411yescharacteristic
339C2×C4#93911yescharacteristic
340C2×C4#93911yescharacteristic
341C8—81no—
342C8—81no—
343C8—122no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—82no—
347C2×C3#187311yescharacteristic
348C2×C3#190411yescharacteristic
349C2×C3#190611yescharacteristic
350C22—21no—
351C22—21no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—82no—
356C22—82no—
357C22#473311yescharacteristic
358C22#473311yescharacteristic
359C22#495811yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—122no—
366C4—241no—
367C4—242no—
368C4—241no—
369C4—241no—
370C4#472311yescharacteristic
371C4#472311yescharacteristic
372C4#511311yescharacteristic
373C4#512011yescharacteristic
374C3#917711yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—82no—
378C2#6993111yescharacteristic
379C2#7031411yescharacteristic
380C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1478D3
hypocenterC3#9177
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
4. #70793C2semidirect
5. #70792C2semidirect
6. #67746C2semidirect
7. #67738C2semidirect
18. #5008C4semidirect
19. #5007C4semidirect
374. C3#9177semidirect

Maximal subgroups

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

subgroupindex
#823382
#822962
C3×#14782
#707932
#707922
#677462
#677382
#91773