Finite Group Databasebeta

← #10695849

Subgroups — #10695849

Order 768.

502 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 79 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#82808C211yesmaximal, characteristic
2#82727C211yesmaximal, characteristic
3#78625C211yesmaximal, characteristic
4C3×#1511C211yesmaximal, characteristic
5#69534C211yesmaximal, characteristic
6#69533C211yesmaximal, characteristic
7#67686C211yesmaximal, characteristic
8#9985—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#381C2211yescharacteristic
14#5230—21no—
15#5225—21no—
16#5211C2211yescharacteristic
17#5207C2211yescharacteristic
18#5121—21no—
19#5120—21no—
20#5045—22no—
21#5044—21no—
22#5043—21no—
23#4986—21no—
24#4985—21no—
25#4979—21no—
26#4964—21no—
27#4952—22no—
28#4723—22no—
29#4722—21no—
30#4721—21no—
31C3×#328—22no—
32#4527—22no—
33#4452—22no—
34#3095—31no—
35#3095—31no—
36#2817—31no—
37#1820—31no—
38#1511—31no—
39#1511D311yescharacteristic
40#1306—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217D412yes—
56#926—21no—
57#926—21no—
58#926—21no—
59#926—21no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—22no—
69C2×C4×Dic3—21no—
70C2×#208—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#95—21no—
74C3×#95—21no—
75#835—22no—
76#835—22no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#830D412yes—
84#818—22no—
85#817—22no—
86#816—22no—
87#816—22no—
88#808—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#117—31no—
93#485—62no—
94#485—62no—
95#484—61no—
96#484—61no—
97#483—61no—
98#483—61no—
99#428—61no—
100#428—61no—
101#419—61no—
102#419—61no—
103#407—31no—
104#407C2×D311yescharacteristic
105#400—61no—
106#400—61no—
107#397—62no—
108#395—61no—
109#395—61no—
110#381—31no—
111#381C2×D311yescharacteristic
112#342—62no—
113#328—22no—
114#328—62no—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×Dic3—22no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—42no—
137C3×M16—42no—
138C2×C3×C8—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—42no—
146#217—41no—
147#217—41no—
148#217—41no—
149#217—41no—
150#217—42no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—82no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162#208—42no—
163C2×C3⋊C8—22no—
164C22×Q8—31no—
165C22×Q8D1211yescharacteristic
166C23×C4C22×D311yescharacteristic
167C2×M16—21no—
168C2×M16—61no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177#117—61no—
178#117—61no—
179#117—61no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4D1211yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C42—61no—
194#107—62no—
195#107—62no—
196#106—62no—
197#106—62no—
198#103—62no—
199#103—62no—
200#103—62no—
201#103—62no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C22⋊C8—61no—
205C22⋊C8#6712yes—
206#95—21no—
207#95—21no—
208#95—62no—
209#95—62no—
210#95—62no—
211#95—61no—
212#95—61no—
213C23×C3—22no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3#12011yescharacteristic
217C23×C3#12111yescharacteristic
218C23×C3#12711yescharacteristic
219C3×Q8—42no—
220C3×Q8—81no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—81no—
248C2×Dic3—81no—
249Dic6—42no—
250Dic6—81no—
251C3×C8—42no—
252C3⋊C8—42no—
253C24D3×D411yescharacteristic
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—42no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—122no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—62no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4C2×D1211yescharacteristic
270C22×C4#23511yescharacteristic
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4C2×#6711yescharacteristic
275M16—42no—
276M16—122no—
277C2×C8—22no—
278C2×C8—62no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—242no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—122no—
300C42—121no—
301C42—121no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—82no—
311C22×C3#39211yescharacteristic
312C22×C3#45812yes—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—81no—
318C3×C4—81no—
319C3×C4—161no—
320Dic3—82no—
321Dic3—81no—
322Dic3—81no—
323Dic3—81no—
324Dic3—81no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23#88211yescharacteristic
329C23#88411yescharacteristic
330C23#94011yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—122no—
334Q8—241no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—241no—
358C2×C4—241no—
359C2×C4#88211yescharacteristic
360C2×C4#88411yescharacteristic
361C2×C4#93011yescharacteristic
362C2×C4#93711yescharacteristic
363C8—42no—
364C8—122no—
365C2×C3—22no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3#200911yescharacteristic
369C2×C3#200911yescharacteristic
370C2×C3#202211yescharacteristic
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—82no—
380C22#474212yes—
381C22#496011yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—81no—
387C4—81no—
388C4—161no—
389C4—242no—
390C4—241no—
391C4—241no—
392C4—241no—
393C4—241no—
394C3#998511yescharacteristic, complemented, Sylow
395C2—22no—
396C2—41no—
397C2—41no—
398C2#7003111yescharacteristic
399C2#7035111yescharacteristic
400C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1511C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1511D3
hypocenterC3#9985
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
394. C3#9985semidirect

Maximal subgroups

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

subgroupindex
#828082
#827272
#786252
C3×#15112
#695342
#695332
#676862
#99853