Finite Group Databasebeta

← #10697417

Subgroups — #10697417

Order 768.

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

normal subgroups only

This group has 430 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#83374C211yesmaximal, characteristic
2#83372C211yesmaximal, characteristic
3#79321C211yesmaximal, characteristic
4C3×#1506C211yesmaximal, characteristic
5#69441C211yesmaximal, characteristic
6#69433C211yesmaximal, characteristic
7#67758C211yesmaximal, characteristic
8#10217—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5094—21no—
15#5094—21no—
16#5091—21no—
17#5090—21no—
18#5069—21no—
19#5064—21no—
20#5052—22no—
21#5031—21no—
22#5020—21no—
23#4996—22no—
24#4926C2211yescharacteristic
25#4924C2211yescharacteristic
26#4851—21no—
27#4850—21no—
28#4723—22no—
29#4703—21no—
30#4702—21no—
31C3×#328—22no—
32#4523—22no—
33#4452—22no—
34#3255—31no—
35#3237—31no—
36#3237—31no—
37#1995—31no—
38#1506—31no—
39#1506D311yescharacteristic
40#1307—31no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—22no—
52#926—21no—
53#926—21no—
54#924—21no—
55#924—21no—
56#924—21no—
57#924—21no—
58C2×#208D412yes—
59#870—21no—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8D412yes—
71C3×#97—21no—
72C3×#95—21no—
73C3×#95—21no—
74#835—22no—
75#830D412yes—
76#818—22no—
77#817—22no—
78#816—22no—
79#816—22no—
80#810—22no—
81#810—22no—
82#810—22no—
83#808—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89#803—21no—
90#533—31no—
91#533—31no—
92#533C2×D311yescharacteristic
93#501—61no—
94#501—61no—
95#491—61no—
96#491—61no—
97#485—62no—
98#485—62no—
99#477—61no—
100#477—61no—
101#475—61no—
102#475—61no—
103#475—61no—
104#475—61no—
105#470—62no—
106#462—61no—
107#462—61no—
108#432—31no—
109#432C2×D311yescharacteristic
110#423—31no—
111#376C2×D311yescharacteristic
112#339—62no—
113#328—22no—
114#328—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42—22no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141#217—41no—
142#216—82no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157#208—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×Q16—62no—
161C2×M16—32no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8D1211yescharacteristic
167C4⋊Q8D1211yescharacteristic
168#125—21no—
169#125—61no—
170#125—61no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183#117—21no—
184#117—61no—
185#117—61no—
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#107—62no—
190#106—62no—
191#106—62no—
192#106—62no—
193C4⋊C8—21no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—62no—
198#103—62no—
199#103—62no—
200#103—62no—
201#103—62no—
202#103—62no—
203C22⋊C8—32no—
204C22⋊C8#6712yes—
205#97—21no—
206#97—61no—
207#95—21no—
208#95—21no—
209C23×C3#12711yescharacteristic
210C3×Q8—41no—
211C3×Q8—41no—
212C3×Q8—81no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232Dic6—42no—
233Dic6—42no—
234Dic6—41no—
235Dic6—41no—
236Dic6—81no—
237C3×C8—42no—
238C3⋊C8—42no—
239C3⋊C8—42no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C22×C4—41no—
247C22×C4D3×D411yescharacteristic
248C22×C4D3×D411yescharacteristic
249C22×C4C2×#6711yescharacteristic
250Q16—242no—
251M16—122no—
252C2×C8—22no—
253C2×C8—62no—
254C2×C8—62no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—41no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—22no—
277C42—41no—
278C42—41no—
279C42—121no—
280C42—121no—
281C42C2×D1211yescharacteristic
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3#39211yescharacteristic
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—82no—
295C3×C4—81no—
296C3×C4—81no—
297C3×C4#45511yescharacteristic
298C3×C4#45611yescharacteristic
299Dic3—81no—
300Dic3—81no—
301Dic3—81no—
302Dic3—81no—
303C23#94011yescharacteristic
304Q8—41no—
305Q8—41no—
306Q8—81no—
307Q8—122no—
308Q8—122no—
309Q8—121no—
310Q8—121no—
311Q8—241no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4#88211yescharacteristic
326C2×C4#88211yescharacteristic
327C2×C4#88411yescharacteristic
328C2×C4#88411yescharacteristic
329C2×C4#93011yescharacteristic
330C2×C4#93711yescharacteristic
331C8—42no—
332C8—122no—
333C8—122no—
334C2×C3—41no—
335C2×C3#200711yescharacteristic
336C2×C3#200911yescharacteristic
337C2×C3#200911yescharacteristic
338C22—21no—
339C22—21no—
340C22—21no—
341C22#496011yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—81no—
349C4—81no—
350C4—241no—
351C4—241no—
352C4—241no—
353C4—241no—
354C4#482511yescharacteristic
355C4#482611yescharacteristic
356C3#1021711yescharacteristic, complemented, Sylow
357C2—41no—
358C2#7005311yescharacteristic
359C2#7035111yescharacteristic
360C2#7039311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1506C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1506D3
hypocenterC3#10217
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
356. C3#10217semidirect

Maximal subgroups

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

subgroupindex
#833742
#833722
#793212
C3×#15062
#694412
#694332
#677582
#102173