Finite Group Databasebeta

← #10697181

Subgroups — #10697181

Order 768.

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

normal subgroups only

This group has 530 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
1#83317C211yesmaximal, characteristic
2#83314C211yesmaximal, characteristic
3C2×#4850C211yesmaximal, characteristic, complemented
4C3×#1480C211yesmaximal, characteristic
5#70786C211yesmaximal, characteristic, complemented
6#70780C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10181—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—21no—
11C2×#869C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5232—21no—
16C2×#818—21no—
17C2×#816—21no—
18#5093—21no—
19#5090—21no—
20#5076—21no—
21#5075—21no—
22#5054—22no—
23#5040—22no—
24#5019—22no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4850—22no—
28#4850—22no—
29C3×D4⋊C8—22no—
30#4482—22no—
31#4469—22no—
32#3206—31no—
33C2×#477—31no—
34#3037—31no—
35#1549—31no—
36#1480D311yescharacteristic
37#1461—31no—
38#1301—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C42C2311yescharacteristic
51#951—42no—
52#951—42no—
53C2×#216—22no—
54C2×#214—22no—
55C2×#214—24no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C2×#208—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D412yes—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C22⋊C8—21no—
74C3×C4×C8—21no—
75#835—22no—
76#832—22no—
77#830D412yes—
78#818—24no—
79#816—24no—
80#810—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C2×#137—61no—
86#532C2×D311yescharacteristic
87C2×C4⋊Q8—31no—
88C2×C4×Q8—31no—
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#431—31no—
99#431C2×D311yescharacteristic
100C2×#107—61no—
101#423—31no—
102C2×C4⋊C8C2×D311yescharacteristic
103#417—61no—
104C2×#103—61no—
105#332—62no—
106D4⋊C8—22no—
107#324—62no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C2×C3×Q8—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—22no—
130C2×Dic6—41no—
131C2×Dic6—82no—
132C2×Dic6SD1612yescomplemented
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
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#216—44no—
146#214—44no—
147#214—84no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—22no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158#208—41no—
159#208—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×#41—21no—
163C22×Q8—31no—
164C22×Q8—61no—
165#137—122no—
166#137—122no—
167C2×Q16—62no—
168C2×SD16—62no—
169C2×SD16—64no—
170C2×M16—61no—
171C22×C8—21no—
172C4⋊Q8—32no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8D1211yescharacteristic
177#127D1211yescharacteristic
178#124—21no—
179#121—21no—
180#121—21no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×D4—21no—
186C4×D4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C42—61no—
192C2×C42C22×D311yescharacteristic
193#107—64no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8#6712yes—
200#103—64no—
201#103—64no—
202#103—62no—
203#102—62no—
204C22⋊C8—21no—
205C22⋊C8—32no—
206C4×C8—21no—
207C4×C8—61no—
208C23×C3—41no—
209C23×C3#12111yescharacteristic
210C3×Q8—41no—
211C3×Q8—41no—
212C3×D4—42no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—42no—
216C3×D4—81no—
217C3×D4—81no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4#13711yescharacteristic
233C2×Dic3—22no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—42no—
238C2×Dic3—82no—
239C2×Dic3—81no—
240Dic6—22no—
241Dic6—22no—
242Dic6—41no—
243Dic6—41no—
244Dic6—44no—
245Dic6—82no—
246C3×C8—42no—
247C3×C8—81no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—32no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—62no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—121no—
263C2×Q8—242no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C22×C4—41no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—121no—
272C22×C4#23511yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275Q16—124no—
276SD16—124no—
277SD16—244no—
278M16—121no—
279M16—121no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—62no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—61no—
289C4⋊C4—62no—
290C4⋊C4—62no—
291C4⋊C4—61no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C42—121no—
299C42—121no—
300C42C2×D1211yescharacteristic
301C42D3×D411yescharacteristic
302C42D3×D411yescharacteristic
303C42C2×#6711yescharacteristic
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3#39211yescharacteristic
310C22×C3#46511yescharacteristic
311C22×C3#46811yescharacteristic
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—81no—
316C3×C4#44811yescharacteristic
317C3×C4#44911yescharacteristic
318C3×C4#45011yescharacteristic
319C3×C4#45211yescharacteristic
320C3×C4#47312yes—
321Dic3—42no—
322Dic3—82no—
323Dic3—81no—
324C23—41no—
325C23#88411yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—62no—
329Q8—62no—
330Q8—124no—
331Q8—121no—
332Q8—121no—
333Q8—242no—
334D4—42no—
335D4—42no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—62no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—122no—
350C2×C4—242no—
351C2×C4—241no—
352C2×C4#88211yescharacteristic
353C2×C4#88211yescharacteristic
354C2×C4#88411yescharacteristic
355C2×C4#91111yescharacteristic
356C2×C4#91211yescharacteristic
357C2×C4D3×SD1611yescharacteristic
358C2×C4#91511yescharacteristic
359C2×C4#93011yescharacteristic
360C2×C4#93711yescharacteristic
361C2×C4#94011yescharacteristic
362C8—42no—
363C8—81no—
364C8—122no—
365C8—122no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—161no—
369C2×C3#200011yescharacteristic
370C2×C3#200811yescharacteristic
371C2×C3#202011yescharacteristic
372C22—21no—
373C22—21no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22#477811yescharacteristic
378C22#478011yescharacteristic
379C22#496011yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—81no—
384C4—122no—
385C4—242no—
386C4—241no—
387C4#477611yescharacteristic
388C4#478311yescharacteristic
389C4#514312yes—
390C4#515811yescharacteristic
391C4#517011yescharacteristic
392C3#1018111yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—161no—
396C2#7000811yescharacteristic
397C2#7020511yescharacteristic
398C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1480C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1480D3
hypocenterC3#10181
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
3. C2×#4850C2semidirect
5. #70786C2semidirect
6. #70780C2semidirect
7. #67707C2semidirect
132. C2×Dic6SD16semidirect
392. C3#10181semidirect

Maximal subgroups

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

subgroupindex
#833172
#833142
C2×#48502
C3×#14802
#707862
#707802
#677072
#101813