Finite Group Databasebeta

← #10707751

Subgroups — #10707751

Order 768.

412 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 57 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#78088C211yesmaximal, characteristic, complemented
2C3×#1613C211yesmaximal, characteristic
3#71850C211yesmaximal, characteristic
4#71844C211yesmaximal, characteristic
5#67651C211yesmaximal, characteristic, complemented
6#67451C211yesmaximal, characteristic, complemented
7#67446C211yesmaximal, characteristic, complemented
8#12179—31nomaximal, Sylow
9C3×#534C2211yescharacteristic
10C3×#481—21no—
11C3×#399C2211yescharacteristic
12#5220—21no—
13C2×#834—21no—
14#5034—21no—
15#4985—21no—
16#4984—21no—
17#4981—21no—
18#4950—21no—
19#4949C2211yescharacteristic
20#4946C2211yescharacteristic
21#4945—21no—
22#4647—21no—
23C3×#327—21no—
24C3×#323C2211yescharacteristic
25#4537—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2807—31no—
29#2044—31no—
30#1613—31no—
31#1613D311yescharacteristic
32#1352—31no—
33#1352—31no—
34#1326—31no—
35C22×C3×D4—21no—
36C23×Dic3D411yescharacteristic
37C3×#123—41no—
38C3×#121—21no—
39C3×#121—21no—
40C3×#120—41no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4C2311yescharacteristic
45C2×#217—21no—
46C2×#217—21no—
47C2×#217—41no—
48C2×#217D411yescharacteristic
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3—21no—
56C3×#106—41no—
57C3×#102—41no—
58C3×C22⋊C8—21no—
59C3×#95D411yescharacteristic
60#834—42no—
61#832—41no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—41no—
68C3⋊#95D411yescharacteristic
69#830—21no—
70#808—41no—
71#534C2×D311yescharacteristic
72#481—21no—
73#481—61no—
74C2×#99—61no—
75#400—61no—
76#399—31no—
77#399C2×D311yescharacteristic
78#398—61no—
79#389—61no—
80#389—61no—
81#388—61no—
82#386—61no—
83#385—31no—
84#385—61no—
85#342—31no—
86#327—21no—
87#327—61no—
88#323—31no—
89#323C2×D311yescharacteristic
90C24×C3—21no—
91C24×C3C2×D411yescharacteristic
92C2×C3×D4—21no—
93C2×C3×D4—21no—
94C2×C3×D4—41no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—41no—
110C22×Dic3—41no—
111C22×Dic3—81no—
112C2×C3×C8—41no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—42no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121#217—21no—
122#217—41no—
123#217—41no—
124#217—81no—
125#217—82no—
126#217SD1612yes—
127C4⋊Dic3—41no—
128C4⋊Dic3—81no—
129C4⋊Dic3—81no—
130Dic3⋊C4—42no—
131Dic3⋊C4—81no—
132Dic3⋊C4—81no—
133C4×Dic3—81no—
134C2×C3⋊C8—41no—
135C22×D4—21no—
136C23×C4—31no—
137#123—41no—
138#121—21no—
139#121—21no—
140#120—41no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4#6711yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—121no—
154C2×C22⋊C4C22×D311yescharacteristic
155C2×C42—61no—
156#106—41no—
157#106—121no—
158#102—41no—
159#102—121no—
160#99—122no—
161C22⋊C8—21no—
162C22⋊C8—61no—
163#95—31no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—121no—
170#95#6711yescharacteristic
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—81no—
179C23×C3—81no—
180C23×C3#12011yescharacteristic
181C23×C3C2×SD1611yescharacteristic
182C23×C3#13711yescharacteristic
183C3×D4—81no—
184C3×D4—81no—
185C3×D4—81no—
186C3×D4—81no—
187C3×D4—81no—
188C3×D4—81no—
189C2×C3×C4—22no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×Dic3—22no—
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—42no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C3×C8—81no—
220C3⋊C8—81no—
221C24—21no—
222C24D3×D411yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—121no—
239C22×C4—121no—
240C22×C4—241no—
241C22×C4D3×D411yescharacteristic
242C22×C4C2×#6711yescharacteristic
243C2×C8—41no—
244C2×C8—121no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—121no—
250C4⋊C4—122no—
251C4⋊C4—241no—
252C4⋊C4—241no—
253C4⋊C4—241no—
254C4⋊C4—241no—
255C22⋊C4—32no—
256C22⋊C4—41no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—61no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—241no—
264C22⋊C4—242no—
265C42—241no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—81no—
275C22×C3—82no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3—161no—
280C22×C3#44811yescharacteristic
281C22×C3#45411yescharacteristic
282C22×C3B(4,𝔽2)11yescharacteristic
283C3×C4—42no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—81no—
287C3×C4—81no—
288Dic3—42no—
289Dic3—81no—
290Dic3—81no—
291Dic3—81no—
292Dic3—161no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—81no—
301C23—81no—
302C23D3×SD1611yescharacteristic
303C23#91511yescharacteristic
304C23#93711yescharacteristic
305D4—81no—
306D4—81no—
307D4—81no—
308D4—81no—
309D4—81no—
310D4—81no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—62no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—241no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4—241no—
341C8—81no—
342C8—241no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—161no—
348C2×C3#159911yescharacteristic
349C2×C3#219811yescharacteristic
350C2×C3#220011yescharacteristic
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—81no—
360C22—82no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—161no—
365C22#503211yescharacteristic
366C22#517011yescharacteristic
367C22#520111yescharacteristic
368C4—42no—
369C4—41no—
370C4—41no—
371C4—81no—
372C4—81no—
373C4—122no—
374C4—241no—
375C4—241no—
376C4—241no—
377C4—481no—
378C3#1217911yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2#7097911yescharacteristic
385C2#7098611yescharacteristic
386C2#7100111yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1613C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#534C22
hypercenter#1613D3
hypocenterC3#12179
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1613

Complements

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

NH ≅ G/Ntype
1. #78088C2semidirect
5. #67651C2semidirect
6. #67451C2semidirect
7. #67446C2semidirect
378. C3#12179semidirect

Maximal subgroups

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

subgroupindex
#780882
C3×#16132
#718502
#718442
#676512
#674512
#674462
#121793