Finite Group Databasebeta

← #10696659

Subgroups — #10696659

Order 768.

552 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3152C211yesmaximal, characteristic
2#83321C211yesmaximal, characteristic
3#83222C211yesmaximal, characteristic, complemented
4#71801C211yesmaximal, characteristic
5#71790C211yesmaximal, characteristic, complemented
6#69862C211yesmaximal, characteristic, complemented
7#67669C211yesmaximal, characteristic, complemented
8#10107—31nomaximal, Sylow
9C2×C3×#137—22no—
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#497—22no—
14C3×#494—21no—
15C3×#492—21no—
16C3×C4×M16C2211yescharacteristic
17#5228—22no—
18#5092—21no—
19#5091—21no—
20#5090—22no—
21#5079—21no—
22#5075—21no—
23#5042—22no—
24#5019—21no—
25#5007—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C4×#208C2211yescharacteristic
29D4⋊(C3⋊C8)—22no—
30#4530—22no—
31#4465—22no—
32#3254—31no—
33#3209—31no—
34#3152D311yescharacteristic
35#1997—31no—
36#1526—31no—
37#1484—31no—
38#1299—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D412yes—
41C3×#137—48no—
42C2×C3×Q16—24no—
43C2×C3×SD16—24no—
44C2×C3×M16—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8D411yescharacteristic
50C3×#127D411yescharacteristic
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#216—24no—
60C2×#214—22no—
61#926—21no—
62#924—21no—
63#924—21no—
64C2×#208—21no—
65C2×#208D412yes—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#97D412yes—
72C3×C4×C8—21no—
73#835—24no—
74#832—22no—
75#830—21no—
76#810—22no—
77#807—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84C2×#137—22no—
85#533—31no—
86#532C2×D311yescharacteristic
87C2×C4⋊Q8C2×D311yescharacteristic
88#497—22no—
89#494—21no—
90#494—61no—
91#492—21no—
92#487—61no—
93#477—61no—
94#474—61no—
95#468—62no—
96#462—62no—
97#461—61no—
98#460—61no—
99#451—62no—
100#431—31no—
101#423—31no—
102C4×M16—31no—
103C4×M16C2×D311yescharacteristic
104#343—62no—
105#330—62no—
106D4⋊C8—62no—
107C3×#41—44no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—44no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C22×C3×C4—22no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×Dic6—21no—
122C2×Dic6—21no—
123C3×Q16—48no—
124C3×SD16—48no—
125C3×M16—24no—
126C2×C3×C8—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139#217—41no—
140#216—84no—
141#214—82no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—41no—
149#208—22no—
150#208—41no—
151#208—42no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×#41—21no—
156C22×Q8#6712yes—
157#137—48no—
158C2×Q16—24no—
159C2×Q16—64no—
160C2×SD16—24no—
161C2×SD16—62no—
162C2×M16—21no—
163C2×M16—32no—
164C2×M16—61no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—31no—
170C4⋊Q8—31no—
171C4⋊Q8#6711yescharacteristic
172#127#6711yescharacteristic
173#125—61no—
174#124—21no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177#121—21no—
178#121—21no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×D4—21no—
182C4×D4—21no—
183#117—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C42C22×D311yescharacteristic
187#107—62no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#103—62no—
192#103—64no—
193#102—62no—
194C22⋊C8—61no—
195#97—61no—
196#97#6712yes—
197C4×C8—21no—
198C4×C8—61no—
199C4×C8—61no—
200C23×C3—41no—
201C23×C3#12111yescharacteristic
202C3×Q8—44no—
203C3×Q8—42no—
204C3×Q8—44no—
205C3×D4—42no—
206C3×D4—41no—
207C3×D4—41no—
208C3×D4—42no—
209C3×D4—82no—
210C2×C3×C4—22no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—82no—
217C2×C3×C4—82no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×Dic3—41no—
225C2×Dic3—41no—
226Dic6—42no—
227Dic6—41no—
228Dic6—41no—
229C3×C8—24no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C3⋊C8—42no—
233#41—44no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—41no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4D3×D411yescharacteristic
247C22×C4#23711yescharacteristic
248C22×C4C2×#6711yescharacteristic
249Q16—48no—
250Q16—244no—
251SD16—48no—
252SD16—242no—
253M16—24no—
254M16—62no—
255M16—121no—
256M16—122no—
257C2×C8—22no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—121no—
276C42—121no—
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C42C2×#6711yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3#39211yescharacteristic
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—81no—
292C3×C4—82no—
293C3×C4#45312yes—
294C3×C4#45511yescharacteristic
295C3×C4#45611yescharacteristic
296C3×C4#47312yes—
297Dic3—81no—
298Dic3—81no—
299C23—41no—
300C23#93911yescharacteristic
301Q8—44no—
302Q8—42no—
303Q8—44no—
304Q8—121no—
305Q8—121no—
306Q8—122no—
307D4—42no—
308D4—41no—
309D4—42no—
310D4—41no—
311D4—82no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93811yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#94011yescharacteristic
327C2×C4#95311yescharacteristic
328C8—24no—
329C8—122no—
330C8—122no—
331C8—122no—
332C2×C3—41no—
333C2×C3—161no—
334C2×C3#200211yescharacteristic
335C2×C3#200711yescharacteristic
336C2×C3#202011yescharacteristic
337C22—21no—
338C22—21no—
339C22—21no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#522311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—82no—
349C4—241no—
350C4—241no—
351C4#507712yes—
352C4#520211yescharacteristic
353C4#520412yes—
354C4#520511yescharacteristic
355C3#1010711yescharacteristic, complemented, Sylow
356C2—41no—
357C2—161no—
358C2#7176911yescharacteristic
359C2#7210911yescharacteristic
360C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3152C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3152D3
hypocenterC3#10107
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
3. #83222C2semidirect
5. #71790C2semidirect
6. #69862C2semidirect
7. #67669C2semidirect
355. C3#10107semidirect

Maximal subgroups

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

subgroupindex
C3×#31522
#833212
#832222
#718012
#717902
#698622
#676692
#101073