Finite Group Databasebeta

← #10696624

Subgroups — #10696624

Order 768.

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

normal subgroups only

This group has 568 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×#3165C211yesmaximal, characteristic, complemented
2#83357C211yesmaximal, characteristic
3#83264C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic, complemented
5#71776C211yesmaximal, characteristic, complemented
6#69860C211yesmaximal, characteristic
7#67665C211yesmaximal, characteristic, complemented
8#10102—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#947—21no—
12C2×#944—21no—
13C2×#875C2211yescharacteristic
14C3×#501—22no—
15C3×#500—21no—
16C3×#498—21no—
17C3×#428—22no—
18C3×C4×M16C2211yescharacteristic
19#5089—21no—
20#5086—21no—
21#5050—21no—
22#5046—21no—
23#5028—22no—
24#4993—22no—
25#4993—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4547—22no—
30#4538—22no—
31#4464—22no—
32#3215—31no—
33#3165D311yescharacteristic, complemented
34#2658—31no—
35#1556—31no—
36#1527—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D411yescharacteristic
48C3×C4⋊Q8D411yescharacteristic
49C3×#125—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C42C2311yescharacteristic
58#947—44no—
59#944—44no—
60C2×#216—22no—
61C2×#215—22no—
62C2×(C2×D3)⋊C4—24no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×Dic3⋊C4—24no—
66C2×C4×Dic3—21no—
67#875—21no—
68#875—21no—
69#875—21no—
70#875—21no—
71C3×#107—24no—
72C3×#106—24no—
73C3×#97D412yes—
74C3×C4×C8—21no—
75C3⋊#95—21no—
76C3⋊#95D412yes—
77#830—21no—
78#830—21no—
79#810—22no—
80#809—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83#803—21no—
84#803—21no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#124—31no—
89C2×#124—61no—
90#501—22no—
91#500—21no—
92#498—21no—
93#497—61no—
94#489—61no—
95#464—62no—
96#451—61no—
97#450—61no—
98#432—31no—
99#432—31no—
100#428—22no—
101C4×M16C2×D311yescharacteristic
102#399—62no—
103#399—62no—
104#376—31no—
105#332—62no—
106#329—62no—
107#328—62no—
108C23×D3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×Q8—42no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C22×C3×C4—22no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—22no—
120C22×Dic3—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—42no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—42no—
127C3×M16—24no—
128C2×C3×C8—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#216—82no—
146#215—82no—
147(C2×D3)⋊C4—48no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152Dic3⋊C4—48no—
153C4×Dic3—24no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C22×Q8—21no—
157C22×Q8—31no—
158C22×D4—31no—
159C2×Q16—62no—
160C2×SD16—62no—
161C2×M16—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—124no—
167C4⋊Q8#6711yescharacteristic
168C4⋊Q8#6711yescharacteristic
169#125—21no—
170#124—61no—
171#124—61no—
172#124—61no—
173#124—61no—
174#124—124no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179#117—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—64no—
182C2×C4⋊C4#6712yes—
183C2×C22⋊C4—64no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—24no—
189#106—24no—
190C4⋊C8—32no—
191C4⋊C8—61no—
192#103—62no—
193#102—62no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196#97—61no—
197#97—61no—
198#97#6712yes—
199C4×C8—21no—
200#95—32no—
201#95—61no—
202C23×C3#12011yescharacteristic
203C22×D3—42no—
204C22×D3—42no—
205C22×D3—41no—
206C22×D3—42no—
207C3×Q8—42no—
208C3×Q8—81no—
209C3×Q8—82no—
210C3×Q8—82no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
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
225C2×Dic3—24no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—41no—
230D12—42no—
231D12—81no—
232Dic6—42no—
233Dic6—81no—
234C3×C8—24no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C24—61no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—122no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—122no—
250C22×C4—22no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256Q16—242no—
257SD16—242no—
258M16—24no—
259C2×C8—22no—
260C2×C8—62no—
261C2×C8—121no—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—128no—
273C22⋊C4—41no—
274C22⋊C4—128no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C42—41no—
280C42—64no—
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C42C2×#6711yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#39211yescharacteristic
288C22×C3#45612yes—
289C2×D3—84no—
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—81no—
301C3×C4#45312yes—
302Dic3—44no—
303Dic3—81no—
304C23—122no—
305C23—122no—
306C23—121no—
307C23—122no—
308C23#93711yescharacteristic
309Q8—42no—
310Q8—81no—
311Q8—82no—
312Q8—82no—
313Q8—122no—
314Q8—241no—
315D4—122no—
316D4—241no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—64no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4#93711yescharacteristic
331C2×C4#93811yescharacteristic
332C2×C4#93911yescharacteristic
333C2×C4#93911yescharacteristic
334C2×C4#94011yescharacteristic
335C2×C4#95311yescharacteristic
336C8—24no—
337C8—122no—
338C8—241no—
339C2×C3—22no—
340C2×C3#200211yescharacteristic
341C2×C3#200511yescharacteristic
342C2×C3#200911yescharacteristic
343D3—161no—
344C22—21no—
345C22—21no—
346C22—244no—
347C22—241no—
348C22—241no—
349C22—241no—
350C22#509512yes—
351C22#522311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—82no—
358C4—81no—
359C4—81no—
360C4—124no—
361C4—241no—
362C4#519812yes—
363C3#1010211yescharacteristic, complemented, Sylow
364C2—22no—
365C2—481no—
366C2#7179511yescharacteristic
367C2#7209411yescharacteristic
368C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3165D3
hypocenterC3#10102
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. C3×#3165C2semidirect
4. #71836C2semidirect
5. #71776C2semidirect
7. #67665C2semidirect
33. #3165D3semidirect
363. C3#10102semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833572
#832642
#718362
#717762
#698602
#676652
#101023