Finite Group Databasebeta

← #10697685

Subgroups — #10697685

Order 768.

474 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C3×#3078C211yesmaximal, characteristic
2#83373C211yesmaximal, characteristic
3#82431C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic
5#71634C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic
7#67711C211yesmaximal, characteristic
8#10259—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#479—22no—
13C3×#477—21no—
14C3×#475—21no—
15C3×#428—21no—
16C3×#423C2211yescharacteristic
17C3×#419—21no—
18#5225—21no—
19#5091—21no—
20#5090—21no—
21#5051—22no—
22#5031—21no—
23#5022—22no—
24#5021—21no—
25#5021—21no—
26#5020—21no—
27#4964—21no—
28#4924C2211yescharacteristic
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30C2×C4×C3⋊C8C2211yescharacteristic
31#4535—22no—
32#4524—22no—
33#4484—22no—
34#3249—31no—
35#3078—31no—
36#3078D311yescharacteristic
37#1984—31no—
38#1493—31no—
39#1493—31no—
40#1292—31no—
41C22×C3×Q8D411yescharacteristic
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C42—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#216—22no—
60#926—21no—
61#924—21no—
62#924—21no—
63C2×#208—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D412yes—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#107—22no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×#103—22no—
76C3×#103—22no—
77#835—22no—
78#835—22no—
79#835—22no—
80#810—22no—
81#808—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90#533—31no—
91#533C2×D311yescharacteristic
92C2×C4×Q8C2×D311yescharacteristic
93#479—22no—
94#479—62no—
95#477—21no—
96#477—61no—
97#477—61no—
98#477—61no—
99#475—21no—
100#475—61no—
101#467—62no—
102#462—61no—
103#462—61no—
104#428—21no—
105#428—61no—
106#423—31no—
107#423C2×D311yescharacteristic
108C2×C4⋊C8—31no—
109#419—21no—
110#419—61no—
111C2×C4×C8—31no—
112#336—62no—
113Q8⋊C8—62no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8Q1612yescomplemented
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×Dic6—21no—
129C2×Dic6—21no—
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151#217—41no—
152#216—82no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—41no—
160#208—41no—
161#208—41no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—42no—
168C22×Q8#6711yescharacteristic
169C2×Q16—62no—
170C2×M16—21no—
171C2×M16—61no—
172C22×C8—32no—
173C22×C8—61no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8#6711yescharacteristic
177C4⋊Q8#6711yescharacteristic
178#125—21no—
179#125—61no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—61no—
191C4×Q8—61no—
192#117—21no—
193#117—61no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4#6711yescharacteristic
196C2×C42—21no—
197C2×C42C22×D311yescharacteristic
198#107—22no—
199#107—62no—
200#107—62no—
201#106—22no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208C4⋊C8#6712yes—
209#103—22no—
210#103—22no—
211#103—62no—
212#103—62no—
213#103—62no—
214#103—62no—
215C4×C8—61no—
216C4×C8—61no—
217C23×C3#12111yescharacteristic
218C3×Q8—22no—
219C3×Q8—22no—
220C3×Q8—42no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—41no—
224C3×Q8—41no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×C3×C4C2×Q1611yescharacteristic
242C2×C3×C4#13511yescharacteristic
243C2×C3×C4#13711yescharacteristic
244C2×C3×C4#13711yescharacteristic
245C2×Dic3—41no—
246C2×Dic3—41no—
247Dic6—42no—
248Dic6—41no—
249Dic6—41no—
250C3×C8—42no—
251C3⋊C8—42no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8#21612yescomplemented
264C22×C4—21no—
265C22×C4—21no—
266C22×C4D3×D411yescharacteristic
267C22×C4#23711yescharacteristic
268C22×C4C2×#6711yescharacteristic
269Q16—242no—
270M16—41no—
271M16—41no—
272M16—121no—
273M16—121no—
274C2×C8—22no—
275C2×C8—62no—
276C2×C8—62no—
277C2×C8—62no—
278C2×C8—121no—
279C2×C8—121no—
280C2×C8—122no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—41no—
298C22⋊C4—121no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—121no—
303C42D3×D411yescharacteristic
304C42D3×D411yescharacteristic
305C42C2×#6711yescharacteristic
306C42C2×#6711yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3#39211yescharacteristic
310C22×C3#46711yescharacteristic
311C22×C3#47011yescharacteristic
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—42no—
316C3×C4—81no—
317C3×C4—81no—
318C3×C4—81no—
319C3×C4#45111yescharacteristic
320C3×C4#45111yescharacteristic
321C3×C4#45211yescharacteristic
322C3×C4#45211yescharacteristic
323C3×C4#45511yescharacteristic
324C3×C4#45611yescharacteristic
325Dic3—81no—
326Dic3—81no—
327C23#93911yescharacteristic
328Q8—22no—
329Q8—22no—
330Q8—42no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—41no—
335Q8—122no—
336Q8—121no—
337Q8—121no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4#93711yescharacteristic
351C2×C4#93711yescharacteristic
352C2×C4#93811yescharacteristic
353C2×C4#93911yescharacteristic
354C2×C4#94011yescharacteristic
355C2×C4#94211yescharacteristic
356C2×C4C2×#21611yescharacteristic
357C2×C4#95011yescharacteristic
358C2×C4#95111yescharacteristic
359C2×C4#95311yescharacteristic
360C8—42no—
361C8—122no—
362C8—122no—
363C8—122no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3#200111yescharacteristic
367C2×C3#200711yescharacteristic
368C2×C3#201711yescharacteristic
369C22—21no—
370C22—21no—
371C22#505111yescharacteristic
372C22#505211yescharacteristic
373C22#522311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—241no—
382C4—241no—
383C4#504711yescharacteristic
384C4#505011yescharacteristic
385C4#520211yescharacteristic
386C4#520511yescharacteristic
387C4#522811yescharacteristic
388C4#523911yescharacteristic
389C3#1025911yescharacteristic, complemented, Sylow
390C2—21no—
391C2—21no—
392C2#7170011yescharacteristic
393C2#7211011yescharacteristic
394C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3078C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3078D3
hypocenterC3#10259
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
122. C2×C3×Q8Q16semidirect
263. C2×Q8#216semidirect
389. C3#10259semidirect

Maximal subgroups

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

subgroupindex
C3×#30782
#833732
#824312
#718262
#716342
#705012
#677112
#102593