Finite Group Databasebeta

← #10697182

Subgroups — #10697182

Order 768.

530 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 103 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#82441C211yesmaximal, characteristic
2C2×#5021C211yesmaximal, characteristic, complemented
3#79312C211yesmaximal, characteristic
4C3×#1461C211yesmaximal, characteristic, complemented
5#70780C211yesmaximal, characteristic, complemented
6#70758C211yesmaximal, characteristic
7#67745C211yesmaximal, characteristic, complemented
8#10181—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#942—21no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C2×#835—21no—
16#5031—21no—
17#5028—22no—
18#5027—21no—
19#5021—22no—
20#5021—22no—
21#4966—21no—
22C2×#807—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4841—22no—
26#4738—22no—
27#4693—21no—
28#4681—21no—
29C3×#324—22no—
30#4482—22no—
31#4459—22no—
32#3206—31no—
33C2×#477—31no—
34#3037—31no—
35#1549—31no—
36#1480—31no—
37#1461D311yescharacteristic, complemented
38#1301—31no—
39C22×C3×Q8D411yescharacteristic
40C2×#235—21no—
41C22×Dic6—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#216—22no—
51#942—42no—
52#942—42no—
53C2×#215—22no—
54#930—21no—
55#930—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×#208—21no—
59C22×C3⋊C8—21no—
60C2×#204—24no—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869D411yescharacteristic
69#869D412yes—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8D412yes—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76#835—24no—
77#816—24no—
78#810—22no—
79#809—22no—
80#807—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4⋊(C3⋊C8)D412yes—
85C2×#137—61no—
86#532—31no—
87C2×C4⋊Q8—31no—
88C2×C4×Q8C2×D311yescharacteristic
89#492—61no—
90#487—61no—
91#477—62no—
92#477—62no—
93#464—62no—
94#462—61no—
95#461—62no—
96#460—61no—
97#448—62no—
98#431C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100C2×#107—61no—
101#423—31no—
102C2×C4⋊C8—31no—
103#417—61no—
104C2×#103—61no—
105#332—62no—
106D4⋊C8—62no—
107#324—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8SD1612yescomplemented
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—41no—
120C22×Dic3—41no—
121#235—41no—
122#235—41no—
123#235—41no—
124#235—41no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C2×Dic6—22no—
131C2×Dic6—41no—
132C2×Dic6—82no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C42—41no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145#216—44no—
146#215—44no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—22no—
150C4⋊Dic3—21no—
151C4⋊Dic3—22no—
152C4⋊Dic3—21no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161#204—84no—
162C2×#41—61no—
163C22×Q8—61no—
164C22×Q8D1211yescharacteristic
165#137—122no—
166#137—122no—
167C2×Q16—62no—
168C2×SD16—64no—
169C2×SD16—62no—
170C2×M16—61no—
171C22×C8—61no—
172C4⋊Q8—32no—
173C4⋊Q8—31no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177#127—31no—
178#124—61no—
179#121—61no—
180#121—61no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×D4—61no—
186C4×D4—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4D1211yescharacteristic
191C2×C42—21no—
192C2×C42C22×D311yescharacteristic
193#107—64no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—32no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#103—62no—
201#103—64no—
202#103—64no—
203#102—62no—
204C22⋊C8—21no—
205C22⋊C8#6712yes—
206C4×C8—21no—
207C4×C8—21no—
208C23×C3#12111yescharacteristic
209C22×D3—41no—
210C3×Q8—22no—
211C3×Q8—22no—
212C3×Q8—41no—
213C3×Q8—41no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—81no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4#13711yescharacteristic
231#67—81no—
232#67—81no—
233C2×Dic3—42no—
234C2×Dic3—41no—
235C2×Dic3—82no—
236D12—42no—
237D12—41no—
238D12—41no—
239D12—42no—
240C4×D3—81no—
241C4×D3—81no—
242Dic6—44no—
243Dic6—41no—
244Dic6—41no—
245Dic6—82no—
246C3×C8—42no—
247C3×C8—81no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250#41—121no—
251#41—121no—
252#41—121no—
253#41—121no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—61no—
260C2×Q8—62no—
261C2×Q8—121no—
262C2×Q8—242no—
263C2×Q8#20412yescomplemented
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—121no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4C2×D1211yescharacteristic
273C22×C4#23511yescharacteristic
274C22×C4C2×#6711yescharacteristic
275Q16—124no—
276SD16—124no—
277SD16—244no—
278M16—121no—
279M16—121no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—121no—
285C2×C8—121no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C42—41no—
299C42—41no—
300C42D3×D411yescharacteristic
301C42D3×D411yescharacteristic
302C42D3×D411yescharacteristic
303C42D3×D411yescharacteristic
304C22×C3—21no—
305C22×C3—21no—
306C22×C3#39211yescharacteristic
307C22×C3#46511yescharacteristic
308C22×C3#46811yescharacteristic
309C2×D3—81no—
310C2×D3—81no—
311C2×D3—81no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—42no—
316C3×C4—81no—
317C3×C4#44811yescharacteristic
318C3×C4#44911yescharacteristic
319C3×C4#45011yescharacteristic
320C3×C4#45211yescharacteristic
321C3×C4#47312yes—
322Dic3—81no—
323Dic3—82no—
324C23—121no—
325C23#93011yescharacteristic
326Q8—22no—
327Q8—22no—
328Q8—41no—
329Q8—41no—
330Q8—124no—
331Q8—121no—
332Q8—121no—
333Q8—242no—
334D4—122no—
335D4—121no—
336D4—121no—
337D4—122no—
338D4—241no—
339D4—241no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—242no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4#88211yescharacteristic
353C2×C4#88211yescharacteristic
354C2×C4#88411yescharacteristic
355C2×C4#88411yescharacteristic
356C2×C4C2×#20411yescharacteristic
357C2×C4#90411yescharacteristic
358C2×C4#90811yescharacteristic
359C2×C4#90911yescharacteristic
360C2×C4#93711yescharacteristic
361C2×C4#94011yescharacteristic
362C8—42no—
363C8—81no—
364C8—122no—
365C8—122no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3#200011yescharacteristic
369C2×C3#200811yescharacteristic
370C2×C3#202011yescharacteristic
371D3—161no—
372C22—21no—
373C22—21no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22#496011yescharacteristic
378C22#511411yescharacteristic
379C22#513411yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—241no—
386C4—242no—
387C4#473111yescharacteristic
388C4#473311yescharacteristic
389C4#473411yescharacteristic
390C4#473811yescharacteristic
391C4#475512yes—
392C3#1018111yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—481no—
396C2#6996211yescharacteristic
397C2#7005011yescharacteristic
398C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1461C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1461D3
hypocenterC3#10181
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. C2×#5021C2semidirect
4. C3×#1461C2semidirect
5. #70780C2semidirect
7. #67745C2semidirect
37. #1461D3semidirect
113. C2×C3×Q8SD16semidirect
263. C2×Q8#204semidirect
392. C3#10181semidirect

Maximal subgroups

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

subgroupindex
#824412
C2×#50212
#793122
C3×#14612
#707802
#707582
#677452
#101813