Finite Group Databasebeta

← #10696096

Subgroups — #10696096

Order 768.

518 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3166C211yesmaximal, characteristic, complemented
2#83353C211yesmaximal, characteristic
3#83066C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic, complemented
5#71736C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic
7#67666C211yesmaximal, characteristic, complemented
8#10023—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#501—21no—
13C3×#501—21no—
14C3×#499—22no—
15C3×#428—21no—
16C3×#428—21no—
17C3×C4×M16C2211yescharacteristic
18#5087—21no—
19#5083—21no—
20#5046—21no—
21#5044—21no—
22#5026—22no—
23#4993—21no—
24#4992—21no—
25#4991—21no—
26#4989—22no—
27#4981—21no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31#4543—22no—
32#4538—22no—
33#4459—22no—
34#3166D311yescharacteristic, complemented
35#3110—31no—
36#2633—31no—
37#1534—31no—
38#1515—31no—
39#1490—31no—
40#1288—31no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C3×#125—21no—
50C3×#125—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42C2311yescharacteristic
61C2×#215—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×Dic3⋊C4—22no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C3×#107—22no—
72C3×#107—22no—
73C3×#106—22no—
74C3×#106—22no—
75C3×#97—21no—
76C3×C4×C8D412yes—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95D412yes—
82#830—21no—
83#830—21no—
84#809—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803D412yes—
90#533C2×D311yescharacteristic
91C2×#125C2×D311yescharacteristic
92C2×C4×D4—31no—
93#501—21no—
94#501—21no—
95#499—22no—
96#484—61no—
97#474—61no—
98#461—61no—
99#450—61no—
100#440—62no—
101#432—31no—
102#432—31no—
103#428—21no—
104#428—21no—
105C4×M16C2×D311yescharacteristic
106#399—61no—
107#398—61no—
108#397—61no—
109#394—61no—
110#385—62no—
111#383—31no—
112#330—62no—
113#328—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—22no—
124C22×Dic3—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—42no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—82no—
131C3×M16—24no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
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—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#215—44no—
154(C2×D3)⋊C4—44no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—82no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—82no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C22×D4—31no—
164C23×C4—61no—
165C2×SD16—62no—
166C2×M16—21no—
167C4⋊Q8#6711yescharacteristic
168C4⋊Q8#6711yescharacteristic
169#125—21no—
170#125—21no—
171#125—21no—
172#125—21no—
173#125—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182#117—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4#6711yescharacteristic
188C2×C4⋊C4#6711yescharacteristic
189C2×C22⋊C4—62no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C42C22×D311yescharacteristic
193#107—22no—
194#107—22no—
195#107—62no—
196#106—22no—
197#106—22no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#103—62no—
201#102—62no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204#97—21no—
205#97—32no—
206#97—61no—
207C4×C8#6712yes—
208#95—32no—
209#95—62no—
210#95—61no—
211#95—61no—
212C23×C3#12011yescharacteristic
213C22×D3—22no—
214C22×D3—42no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—42no—
218C3×Q8—42no—
219C3×Q8—81no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241D12—22no—
242D12—22no—
243D12—41no—
244D12—41no—
245C4×D3—82no—
246C4×D3—82no—
247C3×C8—24no—
248C3⋊C8—42no—
249C3⋊C8—81no—
250C24—61no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—122no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—62no—
259C22×C4—61no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4—242no—
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4C2×#6711yescharacteristic
266SD16—124no—
267M16—24no—
268C2×C8—22no—
269C2×C8—62no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—242no—
288C22⋊C4—41no—
289C22⋊C4—124no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—242no—
293C42—41no—
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
296C42C2×#6711yescharacteristic
297C42C2×#6711yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3#39211yescharacteristic
301C22×C3#45612yes—
302C2×D3—44no—
303C2×D3—42no—
304C2×D3—82no—
305C2×D3—81no—
306C2×D3—81no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4—81no—
316C3×C4—81no—
317Dic3—82no—
318Dic3—81no—
319C23—62no—
320C23—122no—
321C23—121no—
322C23—121no—
323C23—122no—
324C23#93711yescharacteristic
325Q8—42no—
326Q8—81no—
327D4—62no—
328D4—62no—
329D4—121no—
330D4—121no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4—242no—
347C2×C4—242no—
348C2×C4#93711yescharacteristic
349C2×C4#93811yescharacteristic
350C2×C4#93911yescharacteristic
351C2×C4#93911yescharacteristic
352C2×C4#94011yescharacteristic
353C2×C4#95311yescharacteristic
354C8—24no—
355C8—122no—
356C8—241no—
357C2×C3—22no—
358C2×C3#200511yescharacteristic
359C2×C3#200811yescharacteristic
360C2×C3#201011yescharacteristic
361D3—82no—
362C22—21no—
363C22—21no—
364C22—124no—
365C22—122no—
366C22—242no—
367C22—241no—
368C22—241no—
369C22#509512yes—
370C22#522311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—242no—
382C4—241no—
383C3#1002311yescharacteristic, complemented, Sylow
384C2—22no—
385C2—242no—
386C2#7179511yescharacteristic
387C2#7211811yescharacteristic
388C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3166C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3166D3
hypocenterC3#10023
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
1. C3×#3166C2semidirect
4. #71837C2semidirect
5. #71736C2semidirect
7. #67666C2semidirect
34. #3166D3semidirect
383. C3#10023semidirect

Maximal subgroups

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

subgroupindex
C3×#31662
#833532
#830662
#718372
#717362
#698552
#676662
#100233