Finite Group Databasebeta

← #10696599

Subgroups — #10696599

Order 768.

512 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2640C211yesmaximal, characteristic
2#83198C211yesmaximal, characteristic, complemented
3#82949C211yesmaximal, characteristic
4#71777C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic
6#69433C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10098—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#397—22no—
13C3×#393—22no—
14C3×#391—21no—
15C3×#390—21no—
16C3×#376C2211yescharacteristic
17#5228—21no—
18#5225—22no—
19#5218—21no—
20#5082—22no—
21#5079—21no—
22#5079—21no—
23#5060—21no—
24#5055—21no—
25#5054—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4546—22no—
30#4540—22no—
31#4452—22no—
32#3215—31no—
33#3139—31no—
34#2640D311yescharacteristic
35#1558—31no—
36#1527—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8D412yes—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—24no—
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#216—22no—
58C2×#214—22no—
59#926—21no—
60#924—21no—
61#924—21no—
62C2×#208—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#95—24no—
69C3×#95—21no—
70C3×#95D412yes—
71#835—24no—
72#835—24no—
73#835—22no—
74#832—22no—
75#830—21no—
76#830—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124C2×D311yescharacteristic
86#497—61no—
87#491—62no—
88#489—61no—
89#487—61no—
90#487—61no—
91#464—62no—
92#451—61no—
93#450—61no—
94#432—31no—
95#432—31no—
96#419—62no—
97C4×M16—31no—
98#397—22no—
99#393—22no—
100#391—21no—
101#390—21no—
102#376C2×D311yescharacteristic
103#333—62no—
104#329—62no—
105#328—62no—
106C24×C3—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—42no—
112C2×C3×Q8—44no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—42no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×Dic6—21no—
123C2×Dic6—21no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—84no—
133C3×C42—42no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—41no—
139#216—82no—
140#214—82no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147C4×Dic3—41no—
148#208—42no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C22×Q8#6711yescharacteristic
153C22×Q8#6712yes—
154C22×D4#6711yescharacteristic
155C2×Q16—62no—
156C2×SD16—62no—
157C2×M16—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164#125—61no—
165#124—21no—
166#124—21no—
167#124—21no—
168#124—21no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—24no—
179C2×C42—21no—
180C2×C42C22×D311yescharacteristic
181C4⋊C8—32no—
182C4⋊C8—61no—
183#103—64no—
184#103—64no—
185#103—62no—
186#102—62no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189#97—32no—
190#97—61no—
191#97—61no—
192C4×C8—61no—
193#95—24no—
194#95—21no—
195#95#6712yes—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3#12011yescharacteristic
201C3×Q8—44no—
202C3×Q8—42no—
203C3×Q8—81no—
204C3×Q8—82no—
205C3×D4—42no—
206C3×D4—81no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—44no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12711yescharacteristic
223C2×Dic3—41no—
224C2×Dic3—41no—
225Dic6—42no—
226Dic6—81no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229C3⋊C8—81no—
230C24—21no—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4C2×#6711yescharacteristic
246C22×C4C2×#6711yescharacteristic
247C22×C4C2×#6711yescharacteristic
248Q16—242no—
249SD16—242no—
250M16—122no—
251C2×C8—62no—
252C2×C8—62no—
253C2×C8—121no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—84no—
269C22⋊C4—121no—
270C42—42no—
271C42—121no—
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
275C42#23711yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—84no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3#39211yescharacteristic
283C22×C3#45512yes—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—82no—
290C3×C4—81no—
291C3×C4—82no—
292C3×C4#45512yes—
293Dic3—81no—
294Dic3—81no—
295C23—42no—
296C23—42no—
297C23—41no—
298C23—42no—
299C23#95311yescharacteristic
300Q8—44no—
301Q8—42no—
302Q8—81no—
303Q8—82no—
304Q8—122no—
305Q8—241no—
306D4—42no—
307D4—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4#93711yescharacteristic
321C2×C4#93711yescharacteristic
322C2×C4#93811yescharacteristic
323C2×C4#93911yescharacteristic
324C2×C4#93911yescharacteristic
325C2×C4#94011yescharacteristic
326C8—122no—
327C8—122no—
328C8—241no—
329C2×C3—22no—
330C2×C3—161no—
331C2×C3#200411yescharacteristic
332C2×C3#200411yescharacteristic
333C2×C3#200911yescharacteristic
334C22—21no—
335C22—21no—
336C22—84no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#522311yescharacteristic
341C22#524212yes—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—81no—
349C4—82no—
350C4—241no—
351C4—241no—
352C4#506112yes—
353C3#1009811yescharacteristic, complemented, Sylow
354C2—22no—
355C2—161no—
356C2#7173111yescharacteristic
357C2#7185711yescharacteristic
358C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2640C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#2640D3
hypocenterC3#10098
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2640

Complements

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

NH ≅ G/Ntype
2. #83198C2semidirect
4. #71777C2semidirect
6. #69433C2semidirect
7. #67615C2semidirect
353. C3#10098semidirect

Maximal subgroups

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

subgroupindex
C3×#26402
#831982
#829492
#717772
#717182
#694332
#676152
#100983