Finite Group Databasebeta

← #10707737

Subgroups — #10707737

Order 768.

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

normal subgroups only

This group has 432 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#76844C211yesmaximal, characteristic, complemented
2C3×#1601C211yesmaximal, characteristic
3#71850C211yesmaximal, characteristic
4#71846C211yesmaximal, characteristic
5#67646C211yesmaximal, characteristic, complemented
6#67449C211yesmaximal, characteristic, complemented
7#67446C211yesmaximal, characteristic, complemented
8#12177—31nomaximal, Sylow
9C3×#534C2211yescharacteristic
10C3×#448—21no—
11C3×#393C2211yescharacteristic
12#5230—21no—
13#5220—21no—
14C2×#834—21no—
15#5040—21no—
16#4947C2211yescharacteristic
17#4946—21no—
18#4946C2211yescharacteristic
19#4945—21no—
20#4658—21no—
21#4650—21no—
22#4646—21no—
23C3×#327—21no—
24C3×#323C2211yescharacteristic
25#4537—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2763—31no—
29#2044—31no—
30#1601—31no—
31#1601D311yescharacteristic
32#1351—31no—
33#1351—31no—
34#1326—31no—
35C22×C3×Q8D411yescharacteristic
36C22×C3×D4—21no—
37C23×Dic3D411yescharacteristic
38C22×Dic6D411yescharacteristic
39C2×C3×SD16—41no—
40C3×#123—41no—
41C3×#121—21no—
42C3×#121—21no—
43C3×#120—41no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4C2311yescharacteristic
46C2×#217—21no—
47C2×#217—21no—
48C2×#217—41no—
49C2×#217D411yescharacteristic
50C2×#214—41no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3—21no—
56C3×#103—41no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95D411yescharacteristic
60#834—42no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—41no—
68C3⋊#95D411yescharacteristic
69#830—21no—
70#810—41no—
71#534C2×D311yescharacteristic
72#448—21no—
73#448—61no—
74C2×#99—61no—
75#400—61no—
76#395—61no—
77#393—31no—
78#393C2×D311yescharacteristic
79#391—61no—
80#388—61no—
81#386—61no—
82#385—31no—
83#385—61no—
84#382—61no—
85#342—31no—
86#327—21no—
87#327—61no—
88#323—31no—
89#323C2×D311yescharacteristic
90C24×C3—21no—
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—41no—
93C2×C3×Q8—81no—
94C2×C3×D4—21no—
95C2×C3×D4—21no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113C22×Dic3—81no—
114C2×Dic6—41no—
115C2×Dic6—42no—
116C3×SD16—82no—
117C2×C3×C8—41no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—81no—
124#217—21no—
125#217—42no—
126#217—41no—
127#217—41no—
128#217—82no—
129#217SD1612yescomplemented
130#214—82no—
131Dic3⋊C4—81no—
132Dic3⋊C4—81no—
133Dic3⋊C4—81no—
134Dic3⋊C4—81no—
135C4×Dic3—42no—
136C2×C3⋊C8—41no—
137C22×Q8—31no—
138C22×Q8#6711yescharacteristic
139C22×D4—21no—
140C23×C4—31no—
141C2×SD16—41no—
142C2×SD16—121no—
143#123—41no—
144#121—21no—
145#121—21no—
146#120—41no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—121no—
156C2×C22⋊C4C22×D311yescharacteristic
157C2×C42—61no—
158#103—41no—
159#103—121no—
160#99—122no—
161C22⋊C8—21no—
162C22⋊C8—61no—
163#95—21no—
164#95—31no—
165#95—61no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—121no—
172#95#6711yescharacteristic
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—81no—
181C23×C3—81no—
182C23×C3#12011yescharacteristic
183C23×C3C2×SD1611yescharacteristic
184C23×C3#13711yescharacteristic
185C3×Q8—42no—
186C3×Q8—81no—
187C3×D4—81no—
188C3×D4—81no—
189C3×D4—81no—
190C3×D4—81no—
191C3×D4—81no—
192C3×D4—81no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
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×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223Dic6—42no—
224Dic6—81no—
225C3×C8—81no—
226C3⋊C8—81no—
227C24—21no—
228C24D3×D411yescharacteristic
229C2×Q8—41no—
230C2×Q8—81no—
231C2×Q8—121no—
232C2×Q8—122no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4—241no—
251C22×C4D3×D411yescharacteristic
252C22×C4C2×#6711yescharacteristic
253SD16—82no—
254SD16—242no—
255C2×C8—41no—
256C2×C8—121no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—241no—
260C4⋊C4—241no—
261C4⋊C4—241no—
262C4⋊C4—241no—
263C22⋊C4—32no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—61no—
268C22⋊C4—81no—
269C22⋊C4—122no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—242no—
273C42—122no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—81no—
283C22×C3—82no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—161no—
288C22×C3#44811yescharacteristic
289C22×C3#45411yescharacteristic
290C22×C3B(4,𝔽2)11yescharacteristic
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4—81no—
296Dic3—42no—
297Dic3—42no—
298Dic3—81no—
299Dic3—81no—
300Dic3—161no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—81no—
309C23—81no—
310C23D3×SD1611yescharacteristic
311C23#91511yescharacteristic
312C23#93711yescharacteristic
313Q8—42no—
314Q8—81no—
315Q8—122no—
316Q8—241no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—241no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4—241no—
353C8—81no—
354C8—241no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—161no—
360C2×C3#159911yescharacteristic
361C2×C3#219711yescharacteristic
362C2×C3#219911yescharacteristic
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—82no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—161no—
377C22#503211yescharacteristic
378C22#517011yescharacteristic
379C22#520111yescharacteristic
380C4—41no—
381C4—41no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—122no—
386C4—122no—
387C4—241no—
388C4—241no—
389C4—481no—
390C3#1217711yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—161no—
396C2#7097911yescharacteristic
397C2#7099011yescharacteristic
398C2#7099711yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1601C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#534C22
hypercenter#1601D3
hypocenterC3#12177
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1601

Complements

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

NH ≅ G/Ntype
1. #76844C2semidirect
5. #67646C2semidirect
6. #67449C2semidirect
7. #67446C2semidirect
129. #217SD16semidirect
390. C3#12177semidirect

Maximal subgroups

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

subgroupindex
#768442
C3×#16012
#718502
#718462
#676462
#674492
#674462
#121773