Finite Group Databasebeta

← #10692361

Subgroups — #10692361

Order 768.

500 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 93 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#78625C211yesmaximal, characteristic
2#73881C211yesmaximal, characteristic
3#69980C211yesmaximal, characteristic
4#69975C211yesmaximal, characteristic
5#69529C211yesmaximal, characteristic
6C3×#1305C211yesmaximal, characteristic
7#67683C211yesmaximal, characteristic
8#9363—31nomaximal, Sylow
9C2×#926C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C4×#217C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5121C412yescomplemented
17#5120C412yescomplemented
18Dic3×C4⋊C4—21no—
19#4972—21no—
20#4945—22no—
21#4723—22no—
22#4722—21no—
23#4721—21no—
24#4650—21no—
25#4648—21no—
26#4555—22no—
27#4527—22no—
28#4467—21no—
29#4467—21no—
30#3095—31no—
31#2346—31no—
32#1996—31no—
33#1826—31no—
34#1508—31no—
35#1305D311yescharacteristic
36#1297—31no—
37C23×C3×C4C2311yescharacteristic
38C22×Dic6C2×C411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×M16C2×C412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C22⋊C4D412yes—
43C2×#217—21no—
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47#926—21no—
48#926—21no—
49#926C2×C412yes—
50#926D412yes—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54#924—21no—
55#924—21no—
56#924—21no—
57#924—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—22no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—22no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C3×C22⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—21no—
72C3⋊#95D411yescharacteristic
73C3⋊#95D411yescharacteristic
74#818—22no—
75#817—22no—
76#816—22no—
77#816—22no—
78C2×C22⋊Q8—31no—
79C2×#117—31no—
80#485—62no—
81#484—61no—
82#483—61no—
83#428—32no—
84#419—32no—
85#407C2×D311yescharacteristic
86#407C2×D311yescharacteristic
87#389—61no—
88#388—62no—
89#382—61no—
90#380C2×D311yescharacteristic
91C4×C4⋊C4—61no—
92C4×C22⋊C4—31no—
93C4×C22⋊C4—31no—
94#376—61no—
95#342—62no—
96#339—62no—
97#328—61no—
98#328—61no—
99C24×C3C2×D411yescharacteristic
100C22×C3×C4—22no—
101C22×C3×C4—42no—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—41no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—42no—
118C3×M16—42no—
119C3×M16—42no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C4⋊C4—42no—
123C3×C22⋊C4—42no—
124#217—22no—
125#217—22no—
126#217—22no—
127#217—42no—
128#217—41no—
129#217—41no—
130#217—41no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3C22⋊C412yes—
136C4⋊Dic3C22⋊C412yes—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139Dic3⋊C4—82no—
140C4×Dic3—22no—
141C4×Dic3—22no—
142C4×Dic3—44no—
143C4×Dic3—42no—
144C4×Dic3—42no—
145C22×Q8—31no—
146C23×C4C22×D311yescharacteristic
147C2×M16—21no—
148C2×M16C4×D312yes—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152C22⋊Q8—61no—
153#117—32no—
154#117—32no—
155#117—61no—
156#117—61no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—62no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4D1212yes—
168C2×C42—61no—
169C2×C42—62no—
170C2×C42—61no—
171C2×C42—61no—
172#107—62no—
173#106—62no—
174#103—62no—
175#103—62no—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C22⋊C8—21no—
179C22⋊C8—21no—
180#95—21no—
181#95—31no—
182#95—31no—
183#95—62no—
184#95—62no—
185#95—61no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3C4×D411yescharacteristic
191C23×C3#12011yescharacteristic
192C23×C3#12111yescharacteristic
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4C2×C22⋊C411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12311yescharacteristic
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224Dic6—42no—
225Dic6—81no—
226C3×C8—42no—
227C3×C8—81no—
228C24C2×D1211yescharacteristic
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—122no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—121no—
241C22×C4C2×C4×D311yescharacteristic
242C22×C4#23511yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4#23911yescharacteristic
247M16—42no—
248M16—42no—
249C2×C8—22no—
250C2×C8—41no—
251C4⋊C4—32no—
252C4⋊C4—32no—
253C4⋊C4—42no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—242no—
261C22⋊C4—42no—
262C22⋊C4—62no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—122no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—62no—
270C42—62no—
271C42—122no—
272C42—122no—
273C42—124no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3#38611yescharacteristic
285C22×C3#45511yescharacteristic
286C22×C3#45611yescharacteristic
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—82no—
291C3×C4#41012yes—
292Dic3—42no—
293Dic3—42no—
294Dic3—42no—
295Dic3—81no—
296Dic3—82no—
297Dic3—81no—
298Dic3—81no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23C4×D1211yescharacteristic
304C23#88211yescharacteristic
305C23#89511yescharacteristic
306Q8—122no—
307Q8—241no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4—62no—
320C2×C4—121no—
321C2×C4—122no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—122no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4—241no—
335C2×C4D3×C22⋊C411yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#89311yescharacteristic
338C2×C4#89411yescharacteristic
339C8—42no—
340C8—81no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3#188211yescharacteristic
346C2×C3#189911yescharacteristic
347C2×C3#190611yescharacteristic
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22#466511yescharacteristic
359C22#474611yescharacteristic
360C22#475211yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—82no—
365C4—122no—
366C4—122no—
367C4—122no—
368C4—241no—
369C4—242no—
370C4—241no—
371C4—241no—
372C4#474012yes—
373C3#936311yescharacteristic, complemented, Sylow
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#6895411yescharacteristic
379C2#6900411yescharacteristic
380C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1305C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1305D3
hypocenterC3#9363
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
16. #5121C4semidirect
17. #5120C4semidirect
373. C3#9363semidirect

Maximal subgroups

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

subgroupindex
#786252
#738812
#699802
#699752
#695292
C3×#13052
#676832
#93633