Finite Group Databasebeta

← #10692497

Subgroups — #10692497

Order 768.

462 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3234C211yesmaximal, characteristic
2#83024C211yesmaximal, characteristic, complemented
3#71818C211yesmaximal, characteristic
4#71813C211yesmaximal, characteristic
5#71735C211yesmaximal, characteristic, complemented
6#67752C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9382—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C3×#501C412yescomplemented
12C3×#490C412yescomplemented
13C3×#481—21no—
14C3×#480—21no—
15C3×#478—22no—
16C3×#432C2211yescharacteristic
17C2×#837—21no—
18C2×#837—21no—
19#4984—21no—
20#4980—21no—
21#4980—21no—
22#4980—21no—
23#4980—21no—
24Dic3×C4⋊C4—21no—
25#4937C2211yescharacteristic
26C4×C4⋊Dic3C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4543—22no—
30#4537—21no—
31#4537—21no—
32#3234D311yescharacteristic
33#2336—31no—
34#1980—31no—
35#1528—31no—
36#1514—31no—
37#1288—31no—
38#1285—31no—
39C2×C3×#41—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#125—21no—
43C3×#125—21no—
44C3×#125C2×C412yes—
45C3×#125D412yes—
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D411yescharacteristic
62C2×C4×Dic3D411yescharacteristic
63C2×#208—21no—
64C3×#107—22no—
65C3×#106—22no—
66C3×C4⋊C8—21no—
67C3×#102—22no—
68C3×#102—22no—
69C3×C22⋊C8—21no—
70C3×#97C2×C412yes—
71#837—42no—
72#837—42no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D411yescharacteristic
81C3⋊#95D411yescharacteristic
82#830—21no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803—21no—
86C4×C3⋊C8—21no—
87#532C2×D311yescharacteristic
88C2×#125C2×D311yescharacteristic
89#501Dic312yescomplemented
90#490Dic312yescomplemented
91#481—21no—
92#480—21no—
93#478—22no—
94#432—31no—
95#432C2×D311yescharacteristic
96C2×#104—61no—
97C2×#104—61no—
98C4×M16—31no—
99#389—61no—
100#387—61no—
101#387—61no—
102#387—61no—
103#387—61no—
104#383—31no—
105C4×C4⋊C4—31no—
106C4×C4⋊C4—61no—
107#330—62no—
108#327—61no—
109#327—61no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C2×C3×Q8—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C2×C3×C8—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4C22⋊C412yes—
139C3×C4⋊C4C22⋊C412yes—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146C4⋊Dic3—42no—
147C4⋊Dic3—42no—
148C4⋊Dic3—81no—
149C4×Dic3—42no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—81no—
153#208—22no—
154#208—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×#41—21no—
158C2×M16—61no—
159C4⋊Q8C2×Dic311yescharacteristic
160#127C2×Dic311yescharacteristic
161#125—21no—
162#125—21no—
163#125C2×Dic312yes—
164#125#6712yes—
165#124—21no—
166#121—21no—
167#121—21no—
168C4×D4—21no—
169C4×D4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4#6711yescharacteristic
176C2×C4⋊C4#6711yescharacteristic
177C2×C42—31no—
178C2×C42—31no—
179C2×C42—61no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—22no—
183#106—22no—
184C4⋊C8—21no—
185C4⋊C8—61no—
186#104—122no—
187#104—122no—
188#102—22no—
189#102—22no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192#97—61no—
193#97—61no—
194#97C2×Dic312yes—
195C4×C8—61no—
196#95—31no—
197#95—31no—
198#95—61no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205C23×C3—41no—
206C23×C3#12011yescharacteristic
207C3×Q8—41no—
208C3×Q8—41no—
209C3×D4—42no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—81no—
213C3×D4—81no—
214C3×D4—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×Dic3—22no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C3×C8—42no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4C2×#6711yescharacteristic
268M16—62no—
269M16—121no—
270C2×C8—22no—
271C2×C8—121no—
272C2×C8—121no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—122no—
282C4⋊C4—122no—
283C4⋊C4—241no—
284C4⋊C4#21712yes—
285C4⋊C4#21712yes—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42—121no—
289C42—121no—
290C42—122no—
291C42—241no—
292C42#23711yescharacteristic
293C42#23711yescharacteristic
294C42C22×Dic311yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3#38611yescharacteristic
302C22×C3#45311yescharacteristic
303C22×C3#45411yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—42no—
311C3×C4—42no—
312C3×C4—81no—
313Dic3—42no—
314Dic3—81no—
315Dic3—81no—
316Dic3—81no—
317C23—41no—
318C23#93711yescharacteristic
319Q8—41no—
320Q8—41no—
321D4—41no—
322D4—41no—
323D4—42no—
324D4—81no—
325D4—81no—
326D4—81no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—62no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4D4×Dic311yescharacteristic
353C2×C4D4×Dic311yescharacteristic
354C2×C4#93511yescharacteristic
355C2×C4#93911yescharacteristic
356C2×C4#93911yescharacteristic
357C2×C4C2×#21711yescharacteristic
358C8—42no—
359C8—122no—
360C8—241no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—161no—
364C2×C3#188311yescharacteristic
365C2×C3#190611yescharacteristic
366C2×C3#191211yescharacteristic
367C22—21no—
368C22—21no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#519811yescharacteristic
373C22#520111yescharacteristic
374C22#522011yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—42no—
383C4—81no—
384C4—122no—
385C4—241no—
386C4—241no—
387C4—241no—
388C3#938211yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—161no—
392C2#7186411yescharacteristic
393C2#7208811yescharacteristic
394C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3234C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3234D3
hypocenterC3#9382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3234

Complements

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

NH ≅ G/Ntype
2. #83024C2semidirect
5. #71735C2semidirect
6. #67752C2semidirect
7. #67749C2semidirect
11. C3×#501C4semidirect
12. C3×#490C4semidirect
89. #501Dic3semidirect
90. #490Dic3semidirect
388. C3#9382semidirect

Maximal subgroups

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

subgroupindex
C3×#32342
#830242
#718182
#718132
#717352
#677522
#677492
#93823