Finite Group Databasebeta

← #10696217

Subgroups — #10696217

Order 768.

498 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 79 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#83298C211yesmaximal, characteristic
2#83274C211yesmaximal, characteristic, complemented
3#79344C211yesmaximal, characteristic, complemented
4C3×#1536C211yesmaximal, characteristic, complemented
5#70827C211yesmaximal, characteristic, complemented
6#70803C211yesmaximal, characteristic, complemented
7#67756C211yesmaximal, characteristic, complemented
8#10042—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5087—21no—
15#5081—21no—
16#5072—21no—
17#5071—21no—
18#5036—21no—
19#5035—22no—
20#5033—21no—
21C4×#213—22no—
22#5008—21no—
23#5007—21no—
24#4926C2211yescharacteristic
25#4925C2211yescharacteristic
26#4846—21no—
27#4841—21no—
28#4731—21no—
29#4722—21no—
30#4710—22no—
31C3×#329—22no—
32#4484—22no—
33#4468—22no—
34#3235—31no—
35#3110—31no—
36#3100—31no—
37#1536D311yescharacteristic, complemented
38#1497—31no—
39#1466—31no—
40#1287—31no—
41C2×C3×#41—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#213—22no—
53C2×(C2×D3)⋊C4—21no—
54#926—21no—
55C2×C4⋊Dic3D411yescharacteristic
56#924—21no—
57#924—21no—
58C2×#204—22no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#97—21no—
73C3×#97D412yes—
74#832—22no—
75#832—22no—
76#832—22no—
77#830—21no—
78#830D412yes—
79#821—22no—
80#818—22no—
81#816—22no—
82#809—22no—
83#808—22no—
84#807—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#532C2×D311yescharacteristic
92C2×C4×D4—31no—
93#501—61no—
94#490—61no—
95#484—61no—
96#482—62no—
97#480—61no—
98#476—61no—
99#474—61no—
100#474—61no—
101#474—61no—
102#461—61no—
103#459—61no—
104#450—61no—
105#447—61no—
106#440—62no—
107C4×D8—62no—
108#432—31no—
109#432C2×D311yescharacteristic
110#432C2×D311yescharacteristic
111#431—31no—
112#329—22no—
113Q8⋊C8—62no—
114#323—62no—
115C23×D3—21no—
116C3×#41—82no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—42no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×C4×D3—82no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145#217—41no—
146#213—44no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160#204—44no—
161C2×#41—21no—
162C22×D4—31no—
163C23×C4—61no—
164C2×SD16—62no—
165C2×D8—62no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8D1211yescharacteristic
169#127D1211yescharacteristic
170#125—61no—
171#124—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174#121—21no—
175#121—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184#117—61no—
185C2×C4⋊C4—31no—
186C2×C22⋊C4—61no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#107—62no—
190#107—62no—
191#106—62no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196#103—62no—
197#102—62no—
198#102—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—32no—
205C22⋊C8—61no—
206#97—21no—
207#97—61no—
208#97#6712yes—
209C4×C8—32no—
210C23×C3—41no—
211C23×C3#12111yescharacteristic
212C22×D3—22no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—81no—
216C22×D3—81no—
217C3×Q8—81no—
218C3×D4—42no—
219C3×D4—42no—
220C3×D4—81no—
221C3×D4—82no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—82no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—81no—
237D12—22no—
238D12—22no—
239D12—41no—
240D12—41no—
241C4×D3—82no—
242C4×D3—82no—
243Dic6—42no—
244Dic6—42no—
245C3×C8—42no—
246C3×C8—81no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249C24—61no—
250#41—82no—
251C2×Q8—21no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—122no—
261C22×C4—41no—
262C22×C4—61no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4—242no—
266C22×C4#23511yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269SD16—124no—
270D8—124no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—62no—
274C2×C8—62no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C42—121no—
291C42C2×D1211yescharacteristic
292C42D3×D411yescharacteristic
293C42D3×D411yescharacteristic
294C42C2×#6711yescharacteristic
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3#39211yescharacteristic
302C2×D3—42no—
303C2×D3—82no—
304C2×D3—81no—
305C2×D3—81no—
306C2×D3—82no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—81no—
313C3×C4#47212yes—
314Dic3—81no—
315Dic3—81no—
316Dic3—81no—
317C23—41no—
318C23—62no—
319C23—121no—
320C23—121no—
321C23—241no—
322C23—241no—
323C23#88411yescharacteristic
324Q8—81no—
325Q8—122no—
326Q8—122no—
327D4—42no—
328D4—42no—
329D4—62no—
330D4—62no—
331D4—81no—
332D4—82no—
333D4—121no—
334D4—121no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—82no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—242no—
344C2×C4—242no—
345C2×C4—241no—
346C2×C4#88211yescharacteristic
347C2×C4#88211yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#93011yescharacteristic
350C2×C4#93711yescharacteristic
351C2×C4#94011yescharacteristic
352C8—42no—
353C8—81no—
354C8—122no—
355C8—122no—
356C2×C3—41no—
357C2×C3—161no—
358C2×C3#200911yescharacteristic
359C2×C3#201011yescharacteristic
360C2×C3#201811yescharacteristic
361D3—82no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—122no—
369C22—242no—
370C22—242no—
371C22—241no—
372C22—241no—
373C22#496011yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—81no—
380C4—241no—
381C4—241no—
382C4—241no—
383C4#474912yes—
384C3#1004211yescharacteristic, complemented, Sylow
385C2—41no—
386C2—161no—
387C2—242no—
388C2#7009011yescharacteristic
389C2#7040611yescharacteristic
390C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1536C2
socleC22×C3#392
prime radical#83298C2
hypercenter#1536D3
hypocenterC3#10042
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1536

Complements

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

NH ≅ G/Ntype
2. #83274C2semidirect
3. #79344C2semidirect
4. C3×#1536C2semidirect
5. #70827C2semidirect
6. #70803C2semidirect
7. #67756C2semidirect
37. #1536D3semidirect
384. C3#10042semidirect

Maximal subgroups

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

subgroupindex
#832982
#832742
#793442
C3×#15362
#708272
#708032
#677562
#100423