Finite Group Databasebeta

← #10697290

Subgroups — #10697290

Order 768.

524 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 109 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#83363C211yesmaximal, characteristic
3#83213C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic, complemented
5#71779C211yesmaximal, characteristic, complemented
6#69867C211yesmaximal, characteristic
7#67668C211yesmaximal, characteristic, complemented
8#10197—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#428—22no—
16C3×C4×M16C2211yescharacteristic
17#5227—22no—
18C2×#825—22no—
19#5085—21no—
20#5084—21no—
21#5083—22no—
22#5083—21no—
23#5079—21no—
24#5048—22no—
25#5027—21no—
26#5020—21no—
27C4×C4⋊Dic3C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30#4538—22no—
31#4529—22no—
32#3246—31no—
33#3206—31no—
34#3165D311yescharacteristic, complemented
35#1987—31no—
36#1980—31no—
37#1503—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#215—24no—
58C2×#215—22no—
59#930—21no—
60#930—21no—
61C2×C4⋊Dic3—21no—
62C2×C4×Dic3—21no—
63C22×C3⋊C8D412yes—
64#875—21no—
65#874D411yescharacteristic
66C4×D12—21no—
67C4×D12—21no—
68#869D411yescharacteristic
69C3×#107—24no—
70C3×#106—24no—
71C3×#97—21no—
72C3×C4×C8—21no—
73#835—24no—
74#835—22no—
75C3⋊#95—21no—
76#830D412yes—
77#825—44no—
78#809—22no—
79#808—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4×C3⋊C8—21no—
84#533C2×D311yescharacteristic
85#532—31no—
86C2×C4⋊Q8C2×D311yescharacteristic
87#501—22no—
88#500—21no—
89#498—21no—
90#492—61no—
91#487—61no—
92#476—61no—
93#475—61no—
94#465—62no—
95#461—62no—
96#461—61no—
97#460—61no—
98#448—62no—
99#431—31no—
100#428—22no—
101C2×C4⋊C8—31no—
102C2×#104—62no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4—31no—
105#340—62no—
106#328—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—41no—
111C2×C3×Q8—82no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×#67—41no—
117C22×Dic3—41no—
118#235—42no—
119#235—42no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C3×M16—22no—
126C3×M16SD1614yescomplemented
127C2×C3×C8—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144#215—84no—
145#215—82no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—41no—
152C4×Dic3—42no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—42no—
156C2×#41—61no—
157C22×Q8—21no—
158C2×SD16—64no—
159C2×SD16—62no—
160C2×M16#6712yes—
161C22×C8—32no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—31no—
166C4⋊Q8#6712yes—
167C4⋊Q8#6711yescharacteristic
168C4⋊Q8#6711yescharacteristic
169#127—31no—
170#125—21no—
171#124—61no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174#121—61no—
175#121—61no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—61no—
179C4×D4—61no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—61no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186#107—24no—
187#106—24no—
188#106—62no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192#104—124no—
193#103—64no—
194#103—62no—
195#102—62no—
196C22⋊C8—32no—
197#97—21no—
198C4×C8—21no—
199C4×C8—61no—
200#95—61no—
201C23×C3#12711yescharacteristic
202C22×D3—41no—
203C3×Q8—44no—
204C3×Q8—41no—
205C3×Q8—41no—
206C3×Q8—81no—
207C3×Q8—82no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—82no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4C2×SD1612yes—
222C2×C3×C4#13712yes—
223#67—81no—
224#67—81no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229D12—42no—
230D12—41no—
231D12—41no—
232D12—42no—
233C4×D3—82no—
234Dic6—42no—
235C3×C8—24no—
236C3⋊C8—42no—
237C3⋊C8—42no—
238#41—122no—
239#41—122no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—61no—
245C2×Q8—82no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—121no—
250C22×C4—41no—
251C22×C4—121no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256SD16—244no—
257SD16—242no—
258M16—22no—
259M16#21514yescomplemented
260C2×C8—22no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—122no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—62no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—121no—
278C22⋊C4—41no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C42—41no—
282C42—122no—
283C42D3×D411yescharacteristic
284C42#23711yescharacteristic
285C42C2×#6711yescharacteristic
286C42C2×#6711yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#39211yescharacteristic
290C22×C3#49211yescharacteristic
291C22×C3#49711yescharacteristic
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—81no—
299C3×C4—82no—
300C3×C4—81no—
301C3×C4#44812yes—
302C3×C4#45511yescharacteristic
303C3×C4#45611yescharacteristic
304C3×C4#46112yes—
305Dic3—82no—
306Dic3—81no—
307C23—121no—
308C23#94011yescharacteristic
309Q8—44no—
310Q8—41no—
311Q8—41no—
312Q8—81no—
313Q8—82no—
314Q8—122no—
315D4—122no—
316D4—121no—
317D4—121no—
318D4—122no—
319D4—241no—
320D4—241no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4—242no—
333C2×C4#93711yescharacteristic
334C2×C4#93711yescharacteristic
335C2×C4#93811yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4C2×#21512yes—
339C2×C4#94212yes—
340C2×C4#95311yescharacteristic
341C8—24no—
342C8—122no—
343C8—122no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#200711yescharacteristic
347C2×C3#200811yescharacteristic
348C2×C3#201311yescharacteristic
349D3—161no—
350C22—21no—
351C22—21no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22#508511yescharacteristic
356C22#508611yescharacteristic
357C22#522311yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—242no—
365C4—241no—
366C4#508312yes—
367C4#520211yescharacteristic
368C4#520511yescharacteristic
369C4#522712yes—
370C3#1019711yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—481no—
374C2#7179111yescharacteristic
375C2#7209911yescharacteristic
376C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3165D3
hypocenterC3#10197
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. #71835C2semidirect
5. #71779C2semidirect
7. #67668C2semidirect
34. #3165D3semidirect
126. C3×M16SD16semidirect
259. M16#215semidirect
370. C3#10197semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833632
#832132
#718352
#717792
#698672
#676682
#101973