Finite Group Databasebeta

← #10684206

Subgroups — #10684206

Order 768.

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

normal subgroups only

This group has 454 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#76422C211yesmaximal, characteristic
2#69638C211yesmaximal, characteristic
3#68688C211yesmaximal, characteristic, complemented
4#68633C211yesmaximal, characteristic, complemented
5C3×#1270C211yesmaximal, characteristic
6#67480C211yesmaximal, characteristic, complemented
7#67477C211yesmaximal, characteristic, complemented
8#7785—31nomaximal, Sylow
9C22×Dic12—21no—
10C2×#869C2211yescharacteristic
11C3×#390C2211yescharacteristic
12#5222—21no—
13C2×#832—21no—
14C2×#817—21no—
15C2×#816—21no—
16#4978—21no—
17#4953—21no—
18#4951—21no—
19#4951—21no—
20#4947—21no—
21#4929C2211yescharacteristic
22#4928C2211yescharacteristic
23C3×#336C2211yescharacteristic
24C3×#336C2211yescharacteristic
25#4551—21no—
26#4529—21no—
27#4524C2211yescharacteristic
28#2808—31no—
29#2058—31no—
30#2023—31no—
31#2016—31no—
32#1272—31no—
33#1271—31no—
34#1270D311yescharacteristic
35C22×C3×D4D411yescharacteristic
36C22×Dic6—21no—
37C22×Dic6D411yescharacteristic
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42C2311yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47C2×#217—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56C2×Dic12—42no—
57C2×Dic12—42no—
58C2×Dic12—42no—
59#869—42no—
60#832—42no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67#817—42no—
68#816—42no—
69C22×Q16—61no—
70C2×C4⋊Q8—31no—
71C2×#106—61no—
72C2×#103—61no—
73C2×#102—61no—
74#397—61no—
75#394—61no—
76#393—61no—
77#391—31no—
78#390C2×D311yescharacteristic
79#389—31no—
80#388—61no—
81#388—61no—
82#387—61no—
83#340—61no—
84#340—61no—
85#336—31no—
86#336C2×D311yescharacteristic
87#336C2×D311yescharacteristic
88C24×C3—21no—
89C2×C3×D4—21no—
90C2×C3×D4—21no—
91C2×C3×D4—41no—
92C2×C3×D4—41no—
93C22×C3×C4—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—41no—
102C2×Dic6—22no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—41no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×Dic6—41no—
110C2×Dic6—41no—
111C2×Dic6—41no—
112C2×C3×C8—22no—
113C2×C3×C8—22no—
114C2×C3×C8—41no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
118C3×C4⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C42—41no—
124#217—81no—
125#217—81no—
126#217—81no—
127#217—81no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—81no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142Dic12—48no—
143C22×Q8—31no—
144C22×Q8—61no—
145C22×D4D1211yescharacteristic
146C2×Q16—122no—
147C2×Q16—122no—
148C2×Q16—122no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—61no—
152C4⋊Q8—122no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4D1211yescharacteristic
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C42—61no—
167C2×C42C22×D311yescharacteristic
168#106—122no—
169#103—122no—
170#102—122no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95—61no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3#12011yescharacteristic
185C3×D4—42no—
186C3×D4—81no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4#13511yescharacteristic
198C2×C3×C4#13511yescharacteristic
199C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
200C2×C3×C4#13711yescharacteristic
201C2×C3×C4#13711yescharacteristic
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—42no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217Dic6—42no—
218Dic6—42no—
219Dic6—42no—
220Dic6—81no—
221Dic6—81no—
222Dic6—81no—
223C3×C8—42no—
224C3×C8—42no—
225C3⋊C8—81no—
226C24—21no—
227C2×Q8—62no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—121no—
231C2×Q8—121no—
232C2×Q8—121no—
233C2×Q8—121no—
234C2×Q8—121no—
235C2×Q8—121no—
236C2×Q8—121no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—121no—
247C22×C4C2×D1211yescharacteristic
248C22×C4D3×D411yescharacteristic
249C22×C4D3×D411yescharacteristic
250Q16—128no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—121no—
258C2×C8—121no—
259C2×C8—121no—
260C4⋊C4—41no—
261C4⋊C4—62no—
262C4⋊C4—62no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—241no—
276C22⋊C4—241no—
277C22⋊C4—241no—
278C22⋊C4—241no—
279C42—41no—
280C42—241no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3#44911yescharacteristic
289C22×C3#45211yescharacteristic
290C22×C3#45211yescharacteristic
291C22×C3#45311yescharacteristic
292C22×C3#45511yescharacteristic
293C22×C3#45611yescharacteristic
294C22×C3B(4,𝔽2)11yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—81no—
302Dic3—82no—
303Dic3—81no—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23#88211yescharacteristic
315Q8—122no—
316Q8—122no—
317Q8—122no—
318Q8—241no—
319Q8—241no—
320Q8—241no—
321D4—42no—
322D4—81no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—122no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—241no—
346C2×C4—241no—
347C2×C4#90411yescharacteristic
348C2×C4#90911yescharacteristic
349C2×C4#91111yescharacteristic
350C2×C4#91211yescharacteristic
351C2×C4#91511yescharacteristic
352C2×C4#91611yescharacteristic
353C8—42no—
354C8—42no—
355C8—241no—
356C2×C3—161no—
357C2×C3#160011yescharacteristic
358C2×C3#160311yescharacteristic
359C2×C3#160611yescharacteristic
360C2×C3#163411yescharacteristic
361C2×C3#164011yescharacteristic
362C2×C3#165211yescharacteristic
363C2×C3#166111yescharacteristic
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#473311yescharacteristic
372C22#474111yescharacteristic
373C22#474611yescharacteristic
374C22#478111yescharacteristic
375C22#478311yescharacteristic
376C22#482211yescharacteristic
377C22#482611yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—81no—
385C4—241no—
386C4—241no—
387C4—241no—
388C4—241no—
389C4—242no—
390C3#778511yescharacteristic, complemented, Sylow
391C2—161no—
392C2#6810911yescharacteristic
393C2#6819311yescharacteristic
394C2#6820311yescharacteristic
395C2#6828511yescharacteristic
396C2#6829411yescharacteristic
397C2#6835711yescharacteristic
398C2#6840611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1270C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenter#1270D3
hypocenterC3#7785
LCSG ▷ C2×C3×C42 ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
3. #68688C2semidirect
4. #68633C2semidirect
6. #67480C2semidirect
7. #67477C2semidirect
390. C3#7785semidirect

Maximal subgroups

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

subgroupindex
#764222
#696382
#686882
#686332
C3×#12702
#674802
#674772
#77853