Finite Group Databasebeta

← #10697300

Subgroups — #10697300

Order 768.

508 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 99 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#83370C211yesmaximal, characteristic, complemented
2#83363C211yesmaximal, characteristic
3#78528C211yesmaximal, characteristic
4C3×#1986C211yesmaximal, characteristic, complemented
5#69489C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic
7#67668C211yesmaximal, characteristic, complemented
8#10198—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5133—22no—
15#5093—21no—
16#5092—21no—
17#5084—21no—
18#5083—21no—
19#5050—22no—
20#5048—22no—
21#5031—22no—
22#5027—21no—
23#5020—21no—
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4714—22no—
27#4694—21no—
28#4681—21no—
29C3×#339—22no—
30#4529—22no—
31#4451—22no—
32#3246—31no—
33#3209—31no—
34#3141—31no—
35#1987—31no—
36#1986D311yescharacteristic, complemented
37#1502—31no—
38#1283—31no—
39C2×#235—21no—
40C22×Dic6—21no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#216—24no—
54C2×#215—22no—
55#930—21no—
56#930—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C22×C3⋊C8D412yes—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D411yescharacteristic
68#869D412yes—
69C3×#97—21no—
70C3×C4×C8—21no—
71C3×#95—21no—
72#835—22no—
73#830D412yes—
74#821—24no—
75#816—24no—
76#810—24no—
77#809—22no—
78#808—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#532—31no—
85C2×C4⋊Q8—31no—
86#494—61no—
87#489—62no—
88#487—61no—
89#487—61no—
90#486—61no—
91#476—61no—
92#475—61no—
93#465—62no—
94#462—62no—
95#461—61no—
96#460—61no—
97#451—62no—
98#431—31no—
99C2×C4⋊C8—31no—
100#417—62no—
101C4×M16C2×D311yescharacteristic
102C4×C4⋊C4C2×D311yescharacteristic
103#340—62no—
104#339—22no—
105#327—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113C22×Dic3—41no—
114#235—42no—
115#235—42no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×C4×D3—41no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C3×M16—22no—
125C3×M16—22no—
126C2×C3×C8—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—41no—
137C3×C42—42no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#216—84no—
143#215—82no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155C2×#41—61no—
156C22×Q8—61no—
157C2×Q16—64no—
158C2×SD16—62no—
159C2×M16#6712yes—
160C22×C8—32no—
161C4⋊Q8—32no—
162C4⋊Q8—31no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8D1211yescharacteristic
167C4⋊Q8D1211yescharacteristic
168#127—31no—
169#125—21no—
170#124—61no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173#121—61no—
174#121—61no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×D4—61no—
178C4×D4—61no—
179#117—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C42—21no—
184C2×C42C22×D311yescharacteristic
185#106—62no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#103—64no—
190#103—64no—
191#103—62no—
192#102—62no—
193#102—64no—
194C22⋊C8—32no—
195#97—21no—
196C4×C8—21no—
197C4×C8—61no—
198#95—21no—
199C23×C3#12711yescharacteristic
200C22×D3—41no—
201C3×Q8—42no—
202C3×Q8—41no—
203C3×Q8—41no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#13512yes—
219C2×C3×C4#13712yes—
220#67—81no—
221#67—81no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—82no—
225D12—42no—
226D12—41no—
227D12—41no—
228D12—81no—
229C4×D3—82no—
230Dic6—44no—
231Dic6—42no—
232Dic6—82no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236#41—122no—
237#41—122no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—62no—
241C2×Q8—61no—
242C2×Q8—121no—
243C2×Q8—242no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—121no—
248C22×C4—41no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254Q16—244no—
255SD16—242no—
256M16—22no—
257M16—22no—
258C2×C8—22no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—122no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—61no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—62no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C42—41no—
280C42—42no—
281C42C2×D1211yescharacteristic
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#39211yescharacteristic
288C22×C3#49511yescharacteristic
289C22×C3#49711yescharacteristic
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—82no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4#45212yes—
300C3×C4#45511yescharacteristic
301C3×C4#45611yescharacteristic
302C3×C4#46412yes—
303Dic3—82no—
304Dic3—81no—
305C23—121no—
306C23#94011yescharacteristic
307Q8—42no—
308Q8—41no—
309Q8—41no—
310Q8—124no—
311Q8—122no—
312Q8—242no—
313D4—122no—
314D4—121no—
315D4—121no—
316D4—241no—
317D4—241no—
318D4—241no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—242no—
330C2×C4—242no—
331C2×C4#88211yescharacteristic
332C2×C4#88211yescharacteristic
333C2×C4#88411yescharacteristic
334C2×C4#88411yescharacteristic
335C2×C4#93011yescharacteristic
336C2×C4#93711yescharacteristic
337C2×C4#94212yes—
338C2×C4#95012yes—
339C8—42no—
340C8—122no—
341C8—122no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3#200711yescharacteristic
345C2×C3#200811yescharacteristic
346C2×C3#201511yescharacteristic
347D3—161no—
348C22—21no—
349C22—21no—
350C22—241no—
351C22—241no—
352C22—241no—
353C22#496011yescharacteristic
354C22#505911yescharacteristic
355C22#508611yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—82no—
360C4—81no—
361C4—81no—
362C4—242no—
363C4—241no—
364C4#482511yescharacteristic
365C4#482611yescharacteristic
366C4#502812yes—
367C4#504712yes—
368C3#1019811yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—481no—
372C2#6968911yescharacteristic
373C2#7039311yescharacteristic
374C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1986D3
hypocenterC3#10198
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
1. #83370C2semidirect
4. C3×#1986C2semidirect
5. #69489C2semidirect
7. #67668C2semidirect
36. #1986D3semidirect
368. C3#10198semidirect

Maximal subgroups

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

subgroupindex
#833702
#833632
#785282
C3×#19862
#694892
#694692
#676682
#101983