Finite Group Databasebeta

← #10697296

Subgroups — #10697296

Order 768.

508 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3209C211yesmaximal, characteristic, complemented
2#83363C211yesmaximal, characteristic
3#83199C211yesmaximal, characteristic
4#71841C211yesmaximal, characteristic, complemented
5#71779C211yesmaximal, characteristic, complemented
6#70789C211yesmaximal, characteristic
7#67744C211yesmaximal, characteristic, complemented
8#10198—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#494—21no—
13C3×#487—21no—
14C3×#462—22no—
15C3×#451—22no—
16C3×#431C2211yescharacteristic
17#5093—21no—
18#5089—22no—
19#5088—21no—
20#5084—21no—
21#5083—21no—
22#5048—22no—
23#5027—21no—
24#5020—21no—
25#4966—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4532—22no—
30#4529—22no—
31#4470—22no—
32#3246—31no—
33#3209D311yescharacteristic, complemented
34#3141—31no—
35#1987—31no—
36#1986—31no—
37#1502—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C2×C3×Q16—24no—
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—22no—
58#930—21no—
59#930—21no—
60C2×C4⋊Dic3—21no—
61C2×C4×Dic3—21no—
62C2×#208D412yes—
63C22×C3⋊C8D412yes—
64#875—21no—
65#874D411yescharacteristic
66C4×D12—21no—
67C4×D12—21no—
68#869D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×#103—24no—
71C3×C22⋊C8D412yes—
72C3×C4×C8—21no—
73#835—22no—
74C3⋊#95—21no—
75#810—24no—
76#809—24no—
77#809—22no—
78#808—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#532—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86#494—21no—
87#489—62no—
88#487—21no—
89#487—61no—
90#486—61no—
91#476—61no—
92#475—61no—
93#465—62no—
94#462—22no—
95#461—61no—
96#460—61no—
97#451—22no—
98#431C2×D311yescharacteristic
99C2×C4⋊C8—31no—
100#417—62no—
101C4×M16—31no—
102C4×C4⋊C4—31no—
103#340—62no—
104#339—62no—
105#327—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×#67—41no—
116C22×Dic3—41no—
117#235—42no—
118#235—42no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×C4×D3—41no—
123C2×Dic6—21no—
124C3×Q16—84no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#215—82no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—41no—
149C4×Dic3—42no—
150#208—22no—
151#208—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155C2×#41—61no—
156C22×Q8—21no—
157C2×Q16—24no—
158C2×SD16—62no—
159C2×M16—32no—
160C22×C8—32no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—31no—
165C4⋊Q8#6712yes—
166C4⋊Q8#6711yescharacteristic
167C4⋊Q8#6711yescharacteristic
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—21no—
182C2×C4⋊C4—61no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185#106—62no—
186C4⋊C8—21no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#103—24no—
190#103—64no—
191#103—62no—
192#102—64no—
193#102—62no—
194C22⋊C8#6712yes—
195#97—61no—
196C4×C8—21no—
197C4×C8—61no—
198#95—61no—
199C23×C3#12711yescharacteristic
200C22×D3—41no—
201C3×Q8—42no—
202C3×Q8—44no—
203C3×Q8—41no—
204C3×Q8—41no—
205C3×Q8—82no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—82no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#13512yes—
220C2×C3×C4#13712yes—
221#67—81no—
222#67—81no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227D12—42no—
228D12—41no—
229D12—41no—
230D12—81no—
231C4×D3—82no—
232Dic6—42no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236#41—122no—
237#41—122no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—41no—
242C2×Q8—61no—
243C2×Q8—82no—
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—84no—
255SD16—242no—
256M16—62no—
257M16—62no—
258C2×C8—22no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—122no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—62no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C42—41no—
280C42—122no—
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C42C2×#6711yescharacteristic
284C42C2×#6711yescharacteristic
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—81no—
297C3×C4—82no—
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—44no—
309Q8—41no—
310Q8—41no—
311Q8—82no—
312Q8—122no—
313D4—122no—
314D4—121no—
315D4—121no—
316D4—241no—
317D4—241no—
318D4—241no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—122no—
327C2×C4—121no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4—242no—
331C2×C4#91812yes—
332C2×C4#91912yes—
333C2×C4#93711yescharacteristic
334C2×C4#93711yescharacteristic
335C2×C4#93811yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4#95311yescharacteristic
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#519011yescharacteristic
354C22#519111yescharacteristic
355C22#522311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—81no—
360C4—82no—
361C4—81no—
362C4—242no—
363C4—241no—
364C4#518512yes—
365C4#518712yes—
366C4#520211yescharacteristic
367C4#520511yescharacteristic
368C3#1019811yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—481no—
372C2#7185811yescharacteristic
373C2#7207211yescharacteristic
374C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3209C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3209D3
hypocenterC3#10198
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
1. C3×#3209C2semidirect
4. #71841C2semidirect
5. #71779C2semidirect
7. #67744C2semidirect
33. #3209D3semidirect
368. C3#10198semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833632
#831992
#718412
#717792
#707892
#677442
#101983