Finite Group Databasebeta

← #10696212

Subgroups — #10696212

Order 768.

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

normal subgroups only

This group has 498 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×#3110C211yesmaximal, characteristic, complemented
2#83361C211yesmaximal, characteristic, complemented
3#82308C211yesmaximal, characteristic
4#71827C211yesmaximal, characteristic, complemented
5#71603C211yesmaximal, characteristic, complemented
6#70811C211yesmaximal, characteristic, complemented
7#67756C211yesmaximal, characteristic, complemented
8#10042—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#484—21no—
13C3×#474—21no—
14C3×#461—21no—
15C3×#450—21no—
16C3×#440—22no—
17C3×#432C2211yescharacteristic
18#5217—21no—
19#5087—21no—
20#5084—21no—
21#5067—21no—
22#5064—21no—
23#5015—21no—
24C4×#213—22no—
25#5007—21no—
26#5003—22no—
27#4967—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30#4925C2211yescharacteristic
31#4535—22no—
32#4525—22no—
33#4464—22no—
34#3235—31no—
35#3110D311yescharacteristic, complemented
36#3100—31no—
37#1536—31no—
38#1497—31no—
39#1466—31no—
40#1287—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C2×C3×SD16—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×#117—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#213—22no—
61#930—21no—
62#930—21no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869D411yescharacteristic
68C3×#107—22no—
69C3×C4⋊C8—21no—
70C3×#103—22no—
71C3×#102—22no—
72C3×C22⋊C8—21no—
73C3×#97D412yes—
74#832—22no—
75#830—21no—
76#830—21no—
77#830D412yes—
78#809—22no—
79#809—22no—
80#809—22no—
81#808—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91#532—31no—
92C2×C4×D4C2×D311yescharacteristic
93#501—61no—
94#490—61no—
95#484—21no—
96#482—62no—
97#480—61no—
98#476—61no—
99#474—21no—
100#474—61no—
101#474—61no—
102#461—21no—
103#459—61no—
104#450—21no—
105#447—61no—
106#440—22no—
107C4×D8—62no—
108#432—31no—
109#432—31no—
110#432C2×D311yescharacteristic
111#431—31no—
112#329—62no—
113Q8⋊C8—62no—
114#323—62no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—82no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—41no—
129#235—82no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×C4×D3—41no—
134C2×Dic6—21no—
135C3×SD16—44no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153#213—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×#41—61no—
162C22×D4#6711yescharacteristic
163C23×C4—21no—
164C2×SD16—22no—
165C2×D8—62no—
166C4⋊Q8—31no—
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—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—61no—
183C4×D4—61no—
184#117—21no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C22⋊C4—21no—
187C2×C42C22×D311yescharacteristic
188#107—22no—
189#107—62no—
190#107—62no—
191#106—62no—
192C4⋊C8—21no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196#103—22no—
197#102—22no—
198#102—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206#97—61no—
207#97—61no—
208#97#6712yes—
209C4×C8—32no—
210C23×C3—22no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—81no—
214C23×C3—81no—
215C23×C3#12111yescharacteristic
216C22×D3—41no—
217C3×Q8—42no—
218C3×Q8—42no—
219C3×D4—22no—
220C3×D4—22no—
221C3×D4—41no—
222C3×D4—41no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—82no—
230C2×C3×C4—82no—
231C2×C3×C4—81no—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12711yescharacteristic
238#67—82no—
239C2×Dic3—41no—
240D12—42no—
241D12—42no—
242D12—81no—
243C4×D3—82no—
244Dic6—81no—
245C3×C8—42no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249C24—21no—
250#41—242no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—61no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—42no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—121no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—82no—
265C22×C4—121no—
266C22×C4D3×D411yescharacteristic
267C22×C4#23711yescharacteristic
268C22×C4C2×#6711yescharacteristic
269SD16—44no—
270D8—124no—
271C2×C8—22no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8—121no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C42—41no—
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C42C2×#6711yescharacteristic
294C42C2×#6711yescharacteristic
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—42no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3#39211yescharacteristic
304C2×D3—81no—
305C2×D3—81no—
306C2×D3—81no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4#47212yes—
316Dic3—81no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—81no—
321C23—81no—
322C23—121no—
323C23#93911yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—241no—
327D4—22no—
328D4—22no—
329D4—41no—
330D4—41no—
331D4—122no—
332D4—122no—
333D4—241no—
334D4—242no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—121no—
345C2×C4—242no—
346C2×C4#93711yescharacteristic
347C2×C4#93711yescharacteristic
348C2×C4#93811yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4#94011yescharacteristic
351C2×C4#95311yescharacteristic
352C8—42no—
353C8—122no—
354C8—122no—
355C8—241no—
356C2×C3—41no—
357C2×C3—82no—
358C2×C3#200911yescharacteristic
359C2×C3#201011yescharacteristic
360C2×C3#201811yescharacteristic
361D3—161no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—82no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22—241no—
371C22—241no—
372C22—241no—
373C22#522311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—241no—
383C4#519912yes—
384C3#1004211yescharacteristic, complemented, Sylow
385C2—41no—
386C2—82no—
387C2—481no—
388C2#7171511yescharacteristic
389C2#7209511yescharacteristic
390C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3110C2
socleC22×C3#392
prime radical#82308C2
hypercenter#3110D3
hypocenterC3#10042
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. C3×#3110C2semidirect
2. #83361C2semidirect
4. #71827C2semidirect
5. #71603C2semidirect
6. #70811C2semidirect
7. #67756C2semidirect
35. #3110D3semidirect
384. C3#10042semidirect

Maximal subgroups

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

subgroupindex
C3×#31102
#833612
#823082
#718272
#716032
#708112
#677562
#100423