Finite Group Databasebeta

← #10696219

Subgroups — #10696219

Order 768.

476 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3110C211yesmaximal, characteristic
2#83372C211yesmaximal, characteristic, complemented
3#82314C211yesmaximal, characteristic
4#71827C211yesmaximal, characteristic, complemented
5#71603C211yesmaximal, characteristic
6#70815C211yesmaximal, characteristic, complemented
7#67756C211yesmaximal, characteristic, complemented
8#10043—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#484—21no—
13C3×#474—21no—
14C3×#461—21no—
15C3×#450—21no—
16C3×#440—22no—
17C3×#432C2211yescharacteristic
18#5218—21no—
19#5094—21no—
20#5091—21no—
21#5069—21no—
22#5064—21no—
23#5019—21no—
24C4×#214—22no—
25#5008—21no—
26#4996—22no—
27#4971—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30#4925C2211yescharacteristic
31#4535—22no—
32#4525—22no—
33#4465—22no—
34#3237—31no—
35#3110D311yescharacteristic
36#3101—31no—
37#1537—31no—
38#1497—31no—
39#1466—31no—
40#1287—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×#117—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#214—22no—
60#926—21no—
61#924—21no—
62#924—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#107—22no—
69C3×C4⋊C8—21no—
70C3×#103—22no—
71C3×#102—22no—
72C3×C22⋊C8—21no—
73C3×#97D412yes—
74#832—22no—
75#830—21no—
76#830—21no—
77#830D412yes—
78#810—22no—
79#810—22no—
80#810—22no—
81#808—22no—
82#808—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#533C2×D311yescharacteristic
92C2×C4×D4C2×D311yescharacteristic
93#501—61no—
94#491—61no—
95#485—62no—
96#484—21no—
97#483—61no—
98#477—61no—
99#476—61no—
100#475—61no—
101#474—21no—
102#461—21no—
103#461—61no—
104#450—21no—
105#450—61no—
106#440—22no—
107C4×SD16—62no—
108#432—31no—
109#432—31no—
110#432C2×D311yescharacteristic
111#431—31no—
112#330—62no—
113Q8⋊C8—62no—
114#323—62no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—82no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×Dic6—21no—
129C2×Dic6—21no—
130C3×SD16—44no—
131C2×C3×C8—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148#217—41no—
149#214—44no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C22×D4#6711yescharacteristic
161C23×C4—21no—
162C2×SD16—22no—
163C2×SD16—62no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8#6711yescharacteristic
167C4⋊Q8#6711yescharacteristic
168#125—21no—
169#125—61no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182#117—21no—
183#117—61no—
184C2×C4⋊C4#6711yescharacteristic
185C2×C22⋊C4—21no—
186C2×C42C22×D311yescharacteristic
187#107—22no—
188#107—62no—
189#106—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195#103—22no—
196#103—62no—
197#103—62no—
198#103—62no—
199#102—22no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205#97—61no—
206#97—61no—
207#97#6712yes—
208C4×C8—32no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3#12111yescharacteristic
215C3×Q8—42no—
216C3×Q8—42no—
217C3×D4—22no—
218C3×D4—22no—
219C3×D4—41no—
220C3×D4—41no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—82no—
228C2×C3×C4—82no—
229C2×C3×C4—81no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×Dic3—41no—
237C2×Dic3—41no—
238Dic6—42no—
239Dic6—42no—
240C3×C8—42no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243C3⋊C8—81no—
244C24—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—42no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4D3×D411yescharacteristic
257C22×C4#23711yescharacteristic
258C22×C4C2×#6711yescharacteristic
259SD16—44no—
260SD16—124no—
261C2×C8—22no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—121no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—121no—
283C42—41no—
284C42—121no—
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C42C2×#6711yescharacteristic
288C42C2×#6711yescharacteristic
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—42no—
293C22×C3—82no—
294C22×C3—82no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#39211yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4—81no—
306C3×C4#47112yes—
307Dic3—81no—
308Dic3—81no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—81no—
313C23—81no—
314C23#93911yescharacteristic
315Q8—42no—
316Q8—42no—
317Q8—122no—
318Q8—122no—
319D4—22no—
320D4—22no—
321D4—41no—
322D4—41no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4#93711yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#93811yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4#94011yescharacteristic
339C2×C4#95311yescharacteristic
340C8—42no—
341C8—122no—
342C8—122no—
343C8—241no—
344C2×C3—41no—
345C2×C3—82no—
346C2×C3#200911yescharacteristic
347C2×C3#201011yescharacteristic
348C2×C3#201911yescharacteristic
349C22—21no—
350C22—21no—
351C22—21no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—81no—
356C22—81no—
357C22#522311yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—241no—
367C4—241no—
368C4#520012yes—
369C3#1004311yescharacteristic, complemented, Sylow
370C2—41no—
371C2—82no—
372C2#7171511yescharacteristic
373C2#7209711yescharacteristic
374C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3110C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3110D3
hypocenterC3#10043
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
2. #83372C2semidirect
4. #71827C2semidirect
6. #70815C2semidirect
7. #67756C2semidirect
369. C3#10043semidirect

Maximal subgroups

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

subgroupindex
C3×#31102
#833722
#823142
#718272
#716032
#708152
#677562
#100433