Finite Group Databasebeta

← #10696083

Subgroups — #10696083

Order 768.

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

normal subgroups only

This group has 492 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#83316C211yesmaximal, characteristic
2#83284C211yesmaximal, characteristic
3#74278C211yesmaximal, characteristic, complemented
4C3×#1514C211yesmaximal, characteristic
5#70814C211yesmaximal, characteristic, complemented
6#70802C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#10021—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#383C2211yescharacteristic
14#5232—21no—
15#5232—21no—
16#5091—21no—
17#5082—21no—
18#5075—21no—
19#5071—21no—
20#5056—22no—
21#5040—21no—
22#5033—21no—
23#5018—22no—
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4661—21no—
27#4659—21no—
28#4659—21no—
29#4656—21no—
30#4653—22no—
31C3×#327—22no—
32#4484—22no—
33#4465—22no—
34#3142—31no—
35#3112—31no—
36#2634—31no—
37#1534—31no—
38#1514D311yescharacteristic
39#1491—31no—
40#1288—31no—
41C2×C3×#41—21no—
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#214—22no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D411yescharacteristic
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×C4×Dic3—21no—
60C2×#208—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95D412yes—
75#835—22no—
76#835—22no—
77#832—22no—
78#832—22no—
79#832—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83#830—21no—
84#810—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803D412yes—
89C4×C3⋊C8D412yes—
90#532C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×Q8—31no—
93#491—61no—
94#490—61no—
95#488—62no—
96#480—61no—
97#477—61no—
98#460—61no—
99#448—61no—
100#440—62no—
101#432—31no—
102#432C2×D311yescharacteristic
103#417—61no—
104#417—61no—
105C4×M16—31no—
106#400—61no—
107#400—61no—
108#398—61no—
109#395—61no—
110#387—62no—
111#383C2×D311yescharacteristic
112#330—62no—
113#327—22no—
114Q8⋊C8—62no—
115C3×#41—44no—
116C2×C3×Q8—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—22no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—42no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#214—44no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—44no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158#208—42no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×#41—21no—
162C22×Q8—31no—
163C2×SD16—62no—
164C2×M16—61no—
165C4⋊Q8D1211yescharacteristic
166#127D1211yescharacteristic
167#125—61no—
168#125—61no—
169#125—61no—
170#125—61no—
171#124—21no—
172#121—21no—
173#121—21no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190C4⋊C8—21no—
191C4⋊C8—61no—
192#103—62no—
193#103—62no—
194#103—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198C22⋊C8—21no—
199C22⋊C8—61no—
200#97—21no—
201#97—32no—
202#97—61no—
203C4×C8—32no—
204#95—21no—
205#95—21no—
206#95—62no—
207#95—62no—
208#95—62no—
209#95#6712yes—
210C23×C3—41no—
211C23×C3#12011yescharacteristic
212C3×Q8—42no—
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—81no—
216C3×D4—82no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—82no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12711yescharacteristic
231C2×Dic3—22no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240Dic6—22no—
241Dic6—22no—
242Dic6—41no—
243Dic6—41no—
244C3×C8—81no—
245C3⋊C8—42no—
246C3⋊C8—42no—
247#41—44no—
248C2×Q8—21no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—122no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4C2×#6711yescharacteristic
265SD16—124no—
266M16—122no—
267C2×C8—41no—
268C2×C8—62no—
269C2×C8—62no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—124no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—121no—
288C42—121no—
289C42C2×D1211yescharacteristic
290C42#23511yescharacteristic
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3#39211yescharacteristic
299C22×C3#45412yes—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—82no—
307C3×C4—81no—
308Dic3—42no—
309Dic3—81no—
310Dic3—81no—
311Dic3—81no—
312C23—41no—
313C23#93711yescharacteristic
314Q8—42no—
315Q8—62no—
316Q8—62no—
317Q8—121no—
318Q8—121no—
319D4—42no—
320D4—42no—
321D4—81no—
322D4—82no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—62no—
331C2×C4—82no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4#88211yescharacteristic
341C2×C4#88211yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4#93011yescharacteristic
345C2×C4#94011yescharacteristic
346C8—81no—
347C8—122no—
348C8—122no—
349C2×C3—22no—
350C2×C3—161no—
351C2×C3#200311yescharacteristic
352C2×C3#200911yescharacteristic
353C2×C3#201011yescharacteristic
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22#496011yescharacteristic
360C22#520112yes—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—82no—
368C4—81no—
369C4—122no—
370C4—241no—
371C4—241no—
372C4—241no—
373C3#1002111yescharacteristic, complemented, Sylow
374C2—22no—
375C2—161no—
376C2#7005311yescharacteristic
377C2#7032611yescharacteristic
378C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1514C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1514D3
hypocenterC3#10021
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1514

Complements

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

NH ≅ G/Ntype
3. #74278C2semidirect
5. #70814C2semidirect
6. #70802C2semidirect
7. #67633C2semidirect
373. C3#10021semidirect

Maximal subgroups

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

subgroupindex
#833162
#832842
#742782
C3×#15142
#708142
#708022
#676332
#100213