Finite Group Databasebeta

← #10696557

Subgroups — #10696557

Order 768.

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

normal subgroups only

This group has 534 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×#2640C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic, complemented
3#82947C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10092—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#397—22no—
13C3×#393—22no—
14C3×#391—21no—
15C3×#390—21no—
16C3×#376C2211yescharacteristic
17#5227—21no—
18#5217—21no—
19#5093—21no—
20#5089—22no—
21#5088—21no—
22#5058—21no—
23#5057—22no—
24#5055—21no—
25#4966—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4547—22no—
30#4540—22no—
31#4451—22no—
32#3213—31no—
33#3141—31no—
34#2640D311yescharacteristic, complemented
35#1556—31no—
36#1527—31no—
37#1505—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C2×#235—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#215—22no—
59C2×#213—22no—
60#930—21no—
61#930—21no—
62C2×#208—21no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869D411yescharacteristic
68C3×#95—24no—
69C3×#95—21no—
70C3×#95D412yes—
71#835—22no—
72#832—22no—
73#830—21no—
74#830—21no—
75#810—24no—
76#809—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#532—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124C2×D311yescharacteristic
86#496—61no—
87#489—62no—
88#489—61no—
89#487—61no—
90#486—61no—
91#463—62no—
92#448—61no—
93#447—61no—
94#432—31no—
95#432—31no—
96#417—62no—
97C4×M16—31no—
98#397—22no—
99#393—22no—
100#391—21no—
101#390—21no—
102#376C2×D311yescharacteristic
103#332—62no—
104#329—62no—
105#327—62no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—42no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—82no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—42no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—41no—
123#235—44no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×C4×D3—41no—
128C2×Dic6—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—84no—
138C3×C42—42no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#215—82no—
144#213—82no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149#208—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×#41—61no—
154C22×Q8—21no—
155C22×Q8#6711yescharacteristic
156C22×D4#6711yescharacteristic
157C2×SD16—62no—
158C2×D8—62no—
159C2×M16—61no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—31no—
165#127—31no—
166#124—21no—
167#124—21no—
168#124—21no—
169#124—21no—
170#124—61no—
171#121—61no—
172#121—61no—
173C4×D4—61no—
174C4×D4—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4#6712yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—24no—
180C2×C42—21no—
181C2×C42C22×D311yescharacteristic
182C4⋊C8—32no—
183C4⋊C8—61no—
184#103—64no—
185#103—62no—
186#102—64no—
187#102—62no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190#97—32no—
191#97—61no—
192#97—61no—
193C4×C8—61no—
194#95—24no—
195#95—21no—
196#95#6712yes—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3#12011yescharacteristic
202C22×D3—41no—
203C3×Q8—42no—
204C3×Q8—81no—
205C3×Q8—82no—
206C3×Q8—82no—
207C3×D4—42no—
208C3×D4—81no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—44no—
215C2×C3×C4—42no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225#67—82no—
226C2×Dic3—41no—
227D12—42no—
228D12—42no—
229D12—81no—
230C4×D3—82no—
231Dic6—42no—
232C3⋊C8—42no—
233C3⋊C8—42no—
234C3⋊C8—81no—
235C24—21no—
236#41—124no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—61no—
243C2×Q8—82no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—42no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—121no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—121no—
255C22×C4C2×#6711yescharacteristic
256C22×C4C2×#6711yescharacteristic
257C22×C4C2×#6711yescharacteristic
258SD16—242no—
259D8—242no—
260M16—122no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—121no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—84no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C42—42no—
278C42D3×D411yescharacteristic
279C42D3×D411yescharacteristic
280C42D3×D411yescharacteristic
281C42#23711yescharacteristic
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—84no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3#39211yescharacteristic
289C22×C3#45412yes—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—82no—
299C3×C4—81no—
300C3×C4—82no—
301C3×C4#45512yes—
302Dic3—81no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—42no—
307C23—121no—
308C23#95311yescharacteristic
309Q8—42no—
310Q8—81no—
311Q8—82no—
312Q8—82no—
313Q8—122no—
314D4—42no—
315D4—81no—
316D4—122no—
317D4—122no—
318D4—241no—
319D4—242no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—121no—
331C2×C4—242no—
332C2×C4#93711yescharacteristic
333C2×C4#93711yescharacteristic
334C2×C4#93811yescharacteristic
335C2×C4#93911yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4#94011yescharacteristic
338C8—122no—
339C8—122no—
340C8—241no—
341C2×C3—22no—
342C2×C3—161no—
343C2×C3#200311yescharacteristic
344C2×C3#200411yescharacteristic
345C2×C3#200811yescharacteristic
346D3—161no—
347C22—21no—
348C22—21no—
349C22—84no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—241no—
354C22—241no—
355C22—241no—
356C22#522311yescharacteristic
357C22#524512yes—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—81no—
365C4—82no—
366C4—241no—
367C4#506112yes—
368C3#1009211yescharacteristic, complemented, Sylow
369C2—22no—
370C2—161no—
371C2—481no—
372C2#7173111yescharacteristic
373C2#7185811yescharacteristic
374C2#7186711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2640C2
socleC22×C3#392
prime radical#82947C2
hypercenter#2640D3
hypocenterC3#10092
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2640

Complements

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

NH ≅ G/Ntype
1. C3×#2640C2semidirect
2. #83199C2semidirect
4. #71776C2semidirect
5. #71718C2semidirect
6. #69432C2semidirect
7. #67615C2semidirect
34. #2640D3semidirect
368. C3#10092semidirect

Maximal subgroups

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

subgroupindex
C3×#26402
#831992
#829472
#717762
#717182
#694322
#676152
#100923