Finite Group Databasebeta

← #10696133

Subgroups — #10696133

Order 768.

490 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3113C211yesmaximal, characteristic, complemented
2#83363C211yesmaximal, characteristic
3#82444C211yesmaximal, characteristic
4#71822C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#70806C211yesmaximal, characteristic
7#67755C211yesmaximal, characteristic, complemented
8#10029—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#483—21no—
13C3×#475—21no—
14C3×#462—21no—
15C3×#451—21no—
16C3×#441—22no—
17C3×#432C2211yescharacteristic
18#5227—21no—
19#5084—21no—
20#5083—21no—
21#5048—22no—
22#5027—21no—
23#5027—21no—
24#5026—22no—
25#5021—21no—
26#5020—21no—
27#4967—21no—
28#4926C2211yescharacteristic
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30C2×#803C2211yescharacteristic
31#4543—22no—
32#4524—22no—
33#4464—22no—
34#3246—31no—
35#3113D311yescharacteristic, complemented
36#3112—31no—
37#1985—31no—
38#1531—31no—
39#1530—31no—
40#1290—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C2×C3×Q16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#215—22no—
61C2×#215—22no—
62#930—21no—
63#930—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D412yes—
66#875—21no—
67#874D411yescharacteristic
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71C3×#106—22no—
72C3×C4⋊C8—21no—
73C3×#103—22no—
74C3×#103—22no—
75C3×C22⋊C8—21no—
76C3×#97D412yes—
77#835—22no—
78#835—22no—
79#830—21no—
80#809—22no—
81#809—22no—
82#808—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89#803D412yes—
90#533C2×D311yescharacteristic
91#532—31no—
92C2×C4×Q8C2×D311yescharacteristic
93#483—21no—
94#480—61no—
95#477—61no—
96#476—61no—
97#475—21no—
98#475—61no—
99#465—62no—
100#462—21no—
101#461—61no—
102#460—61no—
103#460—61no—
104#451—21no—
105#448—61no—
106#441—22no—
107#440—62no—
108#432—31no—
109#432C2×D311yescharacteristic
110C2×C4⋊C8—31no—
111C2×#97—31no—
112#336—62no—
113#330—62no—
114#329—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126#235—82no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×C4×D3—41no—
131C2×Dic6—21no—
132C3×Q16—44no—
133C2×C3×C8—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152#215—44no—
153#215—82no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—42no—
164C2×#41—61no—
165C22×Q8#6711yescharacteristic
166C2×Q16—22no—
167C2×SD16—62no—
168C2×SD16—62no—
169C22×C8—32no—
170C22×C8—61no—
171C4⋊Q8—31no—
172C4⋊Q8#6711yescharacteristic
173C4⋊Q8#6711yescharacteristic
174#127—31no—
175#125—21no—
176#124—61no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179#121—61no—
180#121—61no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×D4—61no—
188C4×D4—61no—
189#117—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C42—21no—
193C2×C42C22×D311yescharacteristic
194#107—62no—
195#106—22no—
196#106—62no—
197C4⋊C8—21no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201#103—22no—
202#103—22no—
203#103—62no—
204#103—62no—
205#102—62no—
206#102—62no—
207C22⋊C8—21no—
208C22⋊C8—61no—
209#97—32no—
210#97—61no—
211#97—61no—
212#97#6712yes—
213C23×C3#12111yescharacteristic
214C22×D3—41no—
215C3×Q8—22no—
216C3×Q8—22no—
217C3×Q8—41no—
218C3×Q8—41no—
219C3×Q8—42no—
220C3×Q8—41no—
221C3×Q8—41no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12711yescharacteristic
239#67—82no—
240C2×Dic3—41no—
241D12—42no—
242D12—41no—
243D12—41no—
244D12—81no—
245C4×D3—82no—
246Dic6—81no—
247C3×C8—42no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C3⋊C8—42no—
251#41—242no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—42no—
257C2×Q8—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—121no—
265C22×C4D3×D411yescharacteristic
266C22×C4#23711yescharacteristic
267C22×C4C2×#6711yescharacteristic
268Q16—44no—
269SD16—124no—
270SD16—242no—
271C2×C8—22no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8—62no—
275C2×C8—121no—
276C2×C8—121no—
277C2×C8—122no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C22⋊C4—41no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42D3×D411yescharacteristic
297C42D3×D411yescharacteristic
298C42C2×#6711yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3#39211yescharacteristic
303C22×C3#47112yes—
304C2×D3—81no—
305C2×D3—81no—
306C2×D3—81no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—42no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4—81no—
316C3×C4#45511yescharacteristic
317C3×C4#45611yescharacteristic
318Dic3—81no—
319C23—121no—
320C23#93911yescharacteristic
321Q8—22no—
322Q8—22no—
323Q8—41no—
324Q8—41no—
325Q8—42no—
326Q8—41no—
327Q8—41no—
328Q8—241no—
329D4—122no—
330D4—121no—
331D4—121no—
332D4—241no—
333D4—242no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—121no—
346C2×C4—242no—
347C2×C4#93711yescharacteristic
348C2×C4#93711yescharacteristic
349C2×C4#93811yescharacteristic
350C2×C4#93911yescharacteristic
351C2×C4#94011yescharacteristic
352C2×C4#95311yescharacteristic
353C8—42no—
354C8—122no—
355C8—122no—
356C8—122no—
357C2×C3—22no—
358C2×C3#200711yescharacteristic
359C2×C3#201011yescharacteristic
360C2×C3#201911yescharacteristic
361D3—161no—
362C22—21no—
363C22—21no—
364C22—241no—
365C22—241no—
366C22—241no—
367C22#520312yes—
368C22#522311yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—241no—
379C4#520211yescharacteristic
380C4#520511yescharacteristic
381C3#1002911yescharacteristic, complemented, Sylow
382C2—22no—
383C2—481no—
384C2#7171511yescharacteristic
385C2#7210711yescharacteristic
386C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3113C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3113D3
hypocenterC3#10029
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
1. C3×#3113C2semidirect
4. #71822C2semidirect
5. #71635C2semidirect
7. #67755C2semidirect
35. #3113D3semidirect
381. C3#10029semidirect

Maximal subgroups

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

subgroupindex
C3×#31132
#833632
#824442
#718222
#716352
#708062
#677552
#100293