Finite Group Databasebeta

← #10697653

Subgroups — #10697653

Order 768.

504 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 103 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#82478C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic
3#79272C211yesmaximal, characteristic
4C3×#1479C211yesmaximal, characteristic
5#70792C211yesmaximal, characteristic
6#70780C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic
8#10253—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5229—21no—
15#5225—21no—
16#5022—22no—
17#5021—21no—
18#5020—21no—
19#4979—22no—
20#4975—21no—
21#4964—21no—
22C4×C4⋊Dic3C2211yescharacteristic
23#4928—22no—
24#4924C2211yescharacteristic
25#4850—22no—
26#4722—22no—
27#4682—21no—
28#4681—21no—
29C3×Q8⋊C8—22no—
30#4482—22no—
31#4467—22no—
32#3230—31no—
33#3078—31no—
34#2567—31no—
35#1549—31no—
36#1503—31no—
37#1479D311yescharacteristic
38#1303—31no—
39C22×C3×Q8D411yescharacteristic
40C22×Dic6—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49#926—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D412yes—
53#924—21no—
54#924—21no—
55C2×Dic3⋊C4—24no—
56C2×C4×Dic3—21no—
57C2×#208—21no—
58#870—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73#835—22no—
74#835—22no—
75C3⋊#95—24no—
76C3⋊#95D412yes—
77#818—24no—
78#816—24no—
79#808—22no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×Q8C2×D311yescharacteristic
86#500—61no—
87#487—61no—
88#484—62no—
89#479—62no—
90#477—61no—
91#477—62no—
92#475—61no—
93#431C2×D311yescharacteristic
94#431C2×D311yescharacteristic
95#428—61no—
96#423—31no—
97#419—61no—
98#395—62no—
99#391—61no—
100#389—62no—
101#384—61no—
102C4×C4⋊C4—31no—
103#332—62no—
104#328—62no—
105Q8⋊C8—22no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8Q1612yes—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—22no—
118C22×Dic3—22no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—41no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42—41no—
131C3×C42—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#217—41no—
137C4⋊Dic3—24no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—84no—
149C4×Dic3—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152#208—41no—
153#208—41no—
154C2×C3⋊C8—22no—
155C22×Q8—61no—
156C22×Q8D1211yescharacteristic
157C2×M16—61no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164#125—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—64no—
179C2×C4⋊C4D1211yescharacteristic
180C2×C42—21no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#107—64no—
185#106—62no—
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—61no—
189C4⋊C8#6712yes—
190#103—64no—
191#103—62no—
192#103—62no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C4×C8—21no—
196C4×C8—21no—
197#95—32no—
198#95—64no—
199C23×C3#12011yescharacteristic
200C3×Q8—22no—
201C3×Q8—22no—
202C3×Q8—41no—
203C3×Q8—41no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12711yescharacteristic
217C2×C3×C4C2×Q1611yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4#13711yescharacteristic
220C2×C3×C4#13711yescharacteristic
221C2×Dic3—24no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—44no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228Dic6—42no—
229Dic6—82no—
230Dic6—82no—
231Dic6—81no—
232C3×C8—42no—
233C3×C8—81no—
234C3⋊C8—42no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—121no—
243C2×Q8—121no—
244C2×Q8—242no—
245C2×Q8Dic1212yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—62no—
249C22×C4—62no—
250C22×C4C2×D1211yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253M16—121no—
254M16—121no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—62no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—64no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—62no—
268C4⋊C4—61no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—244no—
274C22⋊C4—121no—
275C42—41no—
276C42—41no—
277C42—121no—
278C42—121no—
279C42—121no—
280C42#23511yescharacteristic
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42C2×#6711yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3#39211yescharacteristic
287C22×C3#45111yescharacteristic
288C22×C3#45211yescharacteristic
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—42no—
293C3×C4—81no—
294C3×C4#45111yescharacteristic
295C3×C4#45211yescharacteristic
296C3×C4#45612yes—
297C3×C4#46711yescharacteristic
298C3×C4#47011yescharacteristic
299Dic3—44no—
300Dic3—81no—
301Dic3—82no—
302Dic3—81no—
303C23#88211yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—41no—
307Q8—41no—
308Q8—122no—
309Q8—241no—
310Q8—242no—
311Q8—242no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—64no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—124no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—122no—
326C2×C4#88211yescharacteristic
327C2×C4#88411yescharacteristic
328C2×C4#88411yescharacteristic
329C2×C4#90411yescharacteristic
330C2×C4C2×Dic1211yescharacteristic
331C2×C4#90911yescharacteristic
332C2×C4#90911yescharacteristic
333C2×C4#93011yescharacteristic
334C2×C4#93711yescharacteristic
335C2×C4#94011yescharacteristic
336C8—42no—
337C8—81no—
338C8—122no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#200111yescharacteristic
342C2×C3#200511yescharacteristic
343C2×C3#200811yescharacteristic
344C22—21no—
345C22—21no—
346C22#473311yescharacteristic
347C22#473911yescharacteristic
348C22#496011yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—42no—
353C4—81no—
354C4—124no—
355C4—242no—
356C4—241no—
357C4—241no—
358C4#473311yescharacteristic
359C4#473911yescharacteristic
360C4#511611yescharacteristic
361C4#513711yescharacteristic
362C4#520512yes—
363C3#1025311yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2#6996411yescharacteristic
367C2#7034611yescharacteristic
368C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1479D3
hypocenterC3#10253
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
363. C3#10253semidirect

Maximal subgroups

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

subgroupindex
#824782
#824312
#792722
C3×#14792
#707922
#707802
#677462
#102533