Finite Group Databasebeta

← #10697312

Subgroups — #10697312

Order 768.

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

normal subgroups only

This group has 550 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#83216C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic, complemented
3#79026C211yesmaximal, characteristic
4C3×#1981C211yesmaximal, characteristic, complemented
5#69492C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#10200—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#103—22no—
14C3×C4×C4⋊C4C2211yescharacteristic
15#5115—22no—
16#5093—21no—
17#5089—22no—
18#5088—21no—
19#5088—21no—
20#5084—22no—
21#5080—21no—
22#4968—22no—
23#4966—22no—
24C2×#805—22no—
25#4925C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4846—21no—
28#4846—21no—
29#4843—21no—
30#4843—21no—
31#4451—22no—
32#3245—31no—
33#3227—31no—
34#3141—31no—
35#1981D311yescharacteristic, complemented
36#1980—31no—
37#1502—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C2×#235—21no—
42C22×C3×C8D412yes—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#930—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56C2×#208D412yes—
57C2×D24—22no—
58#875—21no—
59#875—21no—
60#874D411yescharacteristic
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#103—44no—
71C3×#95—21no—
72#830D412yes—
73#821—22no—
74#821—22no—
75#818—22no—
76#810—24no—
77#809—24no—
78#809—24no—
79#808—24no—
80#805—44no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85#532—31no—
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88#498—61no—
89#489—62no—
90#487—61no—
91#486—61no—
92#486—61no—
93#481—62no—
94#476—62no—
95#474—61no—
96#474—61no—
97#466—62no—
98#459—61no—
99#459—61no—
100#431—31no—
101C2×C4⋊C8C2×D311yescharacteristic
102C2×#104—62no—
103#417—62no—
104C2×#103—22no—
105C4×M16—31no—
106C4×C4⋊C4C2×D311yescharacteristic
107#327—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—
118C2×#67—41no—
119#235—42no—
120#235—42no—
121#235—42no—
122#235—42no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×C3×C8—22no—
134C2×C3×C8—42no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C42—42no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154#208—22no—
155#208—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158D24—82no—
159C2×#41—61no—
160C2×#41—61no—
161C22×Q8—21no—
162C2×D8—62no—
163C2×M16—32no—
164C22×C8#6712yes—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170C4⋊Q8D1212yes—
171#127—31no—
172#127—31no—
173#124—61no—
174#124—61no—
175#121—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#106—64no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—61no—
193#104—124no—
194#103—44no—
195#103—64no—
196#102—64no—
197#102—64no—
198#102—62no—
199#102—62no—
200C22⋊C8—32no—
201#97—61no—
202C4×C8—61no—
203C4×C8—61no—
204#95—21no—
205C23×C3#12711yescharacteristic
206C22×D3—41no—
207C22×D3—41no—
208C3×Q8—44no—
209C3×Q8—81no—
210C3×Q8—81no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—81no—
217C2×C3×C4—82no—
218C2×C3×C4—81no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#13512yes—
226C2×C3×C4AGL(1,ℤ/8)12yes—
227#67—81no—
228#67—81no—
229#67—81no—
230#67—81no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233D12—42no—
234D12—42no—
235D12—41no—
236D12—41no—
237D12—41no—
238D12—41no—
239D12—42no—
240D12—81no—
241C4×D3—82no—
242C4×D3—82no—
243Dic6—42no—
244Dic6—42no—
245C3×C8—42no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248#41—122no—
249#41—122no—
250#41—122no—
251#41—122no—
252C2×Q8—22no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—121no—
265C2×D4—121no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4D3×D411yescharacteristic
271C22×C4D3×D411yescharacteristic
272C22×C4C2×#6711yescharacteristic
273D8—242no—
274M16—62no—
275M16—62no—
276C2×C8—22no—
277C2×C8—42no—
278C2×C8—62no—
279C2×C8—62no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—42no—
295C42C2×D1211yescharacteristic
296C42#23511yescharacteristic
297C42D3×D411yescharacteristic
298C42D3×D411yescharacteristic
299C22×C3—21no—
300C22×C3—21no—
301C22×C3#39211yescharacteristic
302C22×C3#49511yescharacteristic
303C22×C3#49611yescharacteristic
304C2×D3—81no—
305C2×D3—81no—
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—82no—
314C3×C4—82no—
315C3×C4#44912yes—
316C3×C4#45311yescharacteristic
317C3×C4#45411yescharacteristic
318C3×C4#46312yes—
319Dic3—81no—
320Dic3—81no—
321C23—121no—
322C23—121no—
323C23#94011yescharacteristic
324Q8—44no—
325Q8—81no—
326Q8—81no—
327Q8—122no—
328Q8—122no—
329D4—122no—
330D4—121no—
331D4—121no—
332D4—122no—
333D4—122no—
334D4—121no—
335D4—121no—
336D4—241no—
337D4—241no—
338D4—241no—
339D4—241no—
340D4—241no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—242no—
352C2×C4—242no—
353C2×C4#88211yescharacteristic
354C2×C4#88211yescharacteristic
355C2×C4#88411yescharacteristic
356C2×C4#88411yescharacteristic
357C2×C4#91412yes—
358C2×C4#91912yes—
359C2×C4#93011yescharacteristic
360C2×C4#93711yescharacteristic
361C8—42no—
362C8—122no—
363C8—122no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3#200611yescharacteristic
367C2×C3#200911yescharacteristic
368C2×C3#201211yescharacteristic
369D3—161no—
370D3—161no—
371C22—21no—
372C22—21no—
373C22—241no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22—241no—
378C22—241no—
379C22#496011yescharacteristic
380C22#517611yescharacteristic
381C22#519111yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—82no—
387C4—241no—
388C4—241no—
389C4#481012yes—
390C4#481912yes—
391C4#506211yescharacteristic
392C4#507711yescharacteristic
393C3#1020011yescharacteristic, complemented, Sylow
394C2—21no—
395C2—21no—
396C2—481no—
397C2—481no—
398C2#6966411yescharacteristic
399C2#7027711yescharacteristic
400C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1981C2
socleC22×C3#392
prime radical#79026C2
hypercenter#1981D3
hypocenterC3#10200
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
1. #83216C2semidirect
2. #83199C2semidirect
4. C3×#1981C2semidirect
5. #69492C2semidirect
6. #69469C2semidirect
7. #67706C2semidirect
35. #1981D3semidirect
393. C3#10200semidirect

Maximal subgroups

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

subgroupindex
#832162
#831992
#790262
C3×#19812
#694922
#694692
#677062
#102003