Finite Group Databasebeta

← #10697319

Subgroups — #10697319

Order 768.

528 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#83216C211yesmaximal, characteristic
2#83207C211yesmaximal, characteristic, complemented
3#79027C211yesmaximal, characteristic
4C3×#1981C211yesmaximal, characteristic, complemented
5#69492C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#10201—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#103—22no—
14C3×C4×C4⋊C4C2211yescharacteristic
15#5114—22no—
16#5088—21no—
17#5084—22no—
18#5081—22no—
19#5080—21no—
20#5080—21no—
21#5078—21no—
22#4968—22no—
23#4964—22no—
24C2×#805—22no—
25#4925C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4851—21no—
28#4848—21no—
29#4842—21no—
30#4841—21no—
31#4452—22no—
32#3246—31no—
33#3227—31no—
34#3165—31no—
35#1981D311yescharacteristic, complemented
36#1980—31no—
37#1503—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×C3×C8D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53#926—21no—
54#924—21no—
55#924—21no—
56C2×#208D412yes—
57C2×#204—22no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#103—44no—
71C3×#95—21no—
72#830D412yes—
73#821—22no—
74#817—22no—
75#816—22no—
76#809—24no—
77#808—24no—
78#808—24no—
79#807—24no—
80#805—44no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88#501—62no—
89#500—61no—
90#498—61no—
91#498—61no—
92#486—61no—
93#481—62no—
94#476—61no—
95#476—62no—
96#475—61no—
97#465—62no—
98#461—61no—
99#460—61no—
100#431—31no—
101#428—62no—
102C2×C4⋊C8C2×D311yescharacteristic
103C2×#104—62no—
104C2×#103—22no—
105C4×M16—31no—
106C4×C4⋊C4C2×D311yescharacteristic
107#328—62no—
108C2×C3×Q8—22no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C2×C3×Q8—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118#235—42no—
119#235—42no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×C3×C8—22no—
128C2×C3×C8—42no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C42—42no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—41no—
152#208—22no—
153#208SD1614yescomplemented
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156#204—82no—
157C2×#41—61no—
158C22×Q8—21no—
159C2×SD16—62no—
160C2×M16—32no—
161C22×C8#6712yes—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8D1212yes—
169#127—31no—
170#125—61no—
171#124—61no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174#121—61no—
175#121—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—61no—
179C4×D4—61no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C42—21no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187#106—64no—
188#106—64no—
189#106—62no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—61no—
193#104—124no—
194#103—44no—
195#103—62no—
196#102—64no—
197#102—62no—
198C22⋊C8—32no—
199#97—61no—
200C4×C8—61no—
201C4×C8—61no—
202#95—21no—
203C23×C3#12711yescharacteristic
204C22×D3—41no—
205C3×Q8—44no—
206C3×Q8—81no—
207C3×Q8—81no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—81no—
214C2×C3×C4—82no—
215C2×C3×C4—81no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4C2×SD1612yes—
223C2×C3×C4#13712yes—
224#67—81no—
225#67—81no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229D12—42no—
230D12—41no—
231D12—41no—
232D12—42no—
233C4×D3—82no—
234Dic6—41no—
235Dic6—41no—
236Dic6—42no—
237Dic6—81no—
238C3×C8—42no—
239C3⋊C8—24no—
240C3⋊C8—42no—
241#41—122no—
242#41—122no—
243C2×Q8—22no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—121no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—121no—
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4C2×#6711yescharacteristic
260SD16—242no—
261M16—34no—
262M16—62no—
263C2×C8—22no—
264C2×C8—42no—
265C2×C8—62no—
266C2×C8—62no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C42—42no—
285C42—121no—
286C42C2×D1211yescharacteristic
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#39211yescharacteristic
293C22×C3#49211yescharacteristic
294C22×C3#49711yescharacteristic
295C2×D3—81no—
296C2×D3—81no—
297C2×D3—81no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—82no—
303C3×C4#44812yes—
304C3×C4#45311yescharacteristic
305C3×C4#45411yescharacteristic
306C3×C4#46112yes—
307Dic3—81no—
308Dic3—81no—
309Dic3—81no—
310C23—121no—
311C23#94011yescharacteristic
312Q8—44no—
313Q8—81no—
314Q8—81no—
315Q8—121no—
316Q8—122no—
317Q8—121no—
318Q8—241no—
319D4—122no—
320D4—121no—
321D4—122no—
322D4—121no—
323D4—241no—
324D4—241no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—242no—
337C2×C4#88211yescharacteristic
338C2×C4#88211yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4#88411yescharacteristic
341C2×C4D3×SD1612yes—
342C2×C4#91812yes—
343C2×C4#93011yescharacteristic
344C2×C4#93711yescharacteristic
345C8—42no—
346C8—64no—
347C8—122no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#200611yescharacteristic
351C2×C3#200911yescharacteristic
352C2×C3#201311yescharacteristic
353D3—161no—
354C22—21no—
355C22—21no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22#496011yescharacteristic
360C22#517511yescharacteristic
361C22#519011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—82no—
367C4—241no—
368C4—241no—
369C4—241no—
370C4#480612yes—
371C4#481812yes—
372C4#506211yescharacteristic
373C4#507711yescharacteristic
374C3#1020111yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—481no—
378C2#6966411yescharacteristic
379C2#7028211yescharacteristic
380C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1981C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1981D3
hypocenterC3#10201
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. #83207C2semidirect
4. C3×#1981C2semidirect
6. #69471C2semidirect
7. #67706C2semidirect
35. #1981D3semidirect
153. #208SD16semidirect
374. C3#10201semidirect

Maximal subgroups

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

subgroupindex
#832162
#832072
#790272
C3×#19812
#694922
#694712
#677062
#102013