Finite Group Databasebeta

← #10697588

Subgroups — #10697588

Order 768.

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

normal subgroups only

This group has 554 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
3#73722C211yesmaximal, characteristic, complemented
4C3×#1502C211yesmaximal, characteristic
5#70779C211yesmaximal, characteristic, complemented
6#70767C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10244—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—21no—
11C2×#869C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5232—21no—
16#5093—21no—
17#5090—21no—
18#5076—21no—
19#5075—21no—
20#5054—22no—
21#5040—22no—
22#5019—22no—
23#4973—22no—
24#4936—21no—
25#4929—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4659—22no—
29C3×#327—22no—
30#4481—22no—
31#4456—22no—
32#3206—31no—
33#3037—31no—
34#2568—31no—
35#1548—31no—
36#1502D311yescharacteristic
37#1479—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×Dic6D411yescharacteristic
41C22×Dic6D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#951—42no—
53#951—42no—
54C2×#216—22no—
55C2×#214—22no—
56C2×#214—24no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×Dic3⋊C4—21no—
61C2×C4×Dic3—21no—
62C2×#208—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74C3×#95D412yes—
75#835—22no—
76#832—22no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79#830—21no—
80#810—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
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#464—62no—
92#462—61no—
93#461—62no—
94#460—61no—
95#448—62no—
96#431—31no—
97#431C2×D311yescharacteristic
98#423—31no—
99#417—61no—
100#400—62no—
101#391—61no—
102#391—62no—
103#382—61no—
104C4×C4⋊C4C2×D311yescharacteristic
105#333—62no—
106#327—22no—
107Q8⋊C8—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—22no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—22no—
131C2×Dic6—22no—
132C2×Dic6—44no—
133C2×Dic6SD1612yescomplemented
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—42no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#216—44no—
147#214—44no—
148#214—84no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×#41—21no—
164C22×Q8—31no—
165C22×Q8—32no—
166#137—122no—
167#137—122no—
168C2×Q16—62no—
169C2×SD16—62no—
170C2×SD16—64no—
171C2×M16—61no—
172C4⋊Q8—61no—
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—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C42—21no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—32no—
198C4⋊C8—61no—
199#103—64no—
200#103—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204C4×C8—21no—
205C4×C8—61no—
206#95—64no—
207#95—64no—
208#95#6712yes—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C3×Q8—41no—
212C3×Q8—41no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—42no—
216C3×D4—81no—
217C3×D4—81no—
218C3×D4—81no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×SD1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×Dic3—22no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—44no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244Dic6—22no—
245Dic6—22no—
246Dic6—41no—
247Dic6—41no—
248Dic6—44no—
249Dic6—82no—
250C3×C8—81no—
251C3⋊C8—42no—
252C3⋊C8—42no—
253#41—41no—
254#41—41no—
255#41—41no—
256#41—41no—
257C2×Q8—21no—
258C2×Q8—32no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—62no—
262C2×Q8—61no—
263C2×Q8—62no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—124no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4C2×#6711yescharacteristic
279Q16—124no—
280SD16—124no—
281SD16—244no—
282M16—121no—
283M16—121no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—62no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—42no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—41no—
301C42—41no—
302C42—121no—
303C42—121no—
304C42C2×D1211yescharacteristic
305C42#23511yescharacteristic
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#39211yescharacteristic
314C22×C3#44811yescharacteristic
315C22×C3#44911yescharacteristic
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—82no—
320C3×C4—81no—
321C3×C4#45011yescharacteristic
322C3×C4#45211yescharacteristic
323C3×C4#45512yes—
324C3×C4#46511yescharacteristic
325C3×C4#46811yescharacteristic
326Dic3—42no—
327Dic3—82no—
328Dic3—81no—
329C23—41no—
330C23#93711yescharacteristic
331Q8—41no—
332Q8—41no—
333Q8—62no—
334Q8—62no—
335Q8—121no—
336Q8—121no—
337Q8—124no—
338Q8—242no—
339D4—41no—
340D4—41no—
341D4—42no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—62no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—124no—
357C2×C4—122no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4#88211yescharacteristic
361C2×C4#88211yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4#91111yescharacteristic
365C2×C4#91211yescharacteristic
366C2×C4D3×SD1611yescharacteristic
367C2×C4#91511yescharacteristic
368C2×C4#93011yescharacteristic
369C2×C4#94011yescharacteristic
370C8—81no—
371C8—122no—
372C8—122no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—161no—
376C2×C3#200011yescharacteristic
377C2×C3#200411yescharacteristic
378C2×C3#200911yescharacteristic
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#496011yescharacteristic
385C22#515811yescharacteristic
386C22#517011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—81no—
392C4—122no—
393C4—241no—
394C4—242no—
395C4#477611yescharacteristic
396C4#477811yescharacteristic
397C4#478011yescharacteristic
398C4#478311yescharacteristic
399C4#482512yes—
400C3#1024411yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—161no—
404C2#7005311yescharacteristic
405C2#7020511yescharacteristic
406C2#7038611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1502D3
hypocenterC3#10244
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
3. #73722C2semidirect
5. #70779C2semidirect
6. #70767C2semidirect
7. #67620C2semidirect
133. C2×Dic6SD16semidirect
400. C3#10244semidirect

Maximal subgroups

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

subgroupindex
#833172
#833142
#737222
C3×#15022
#707792
#707672
#676202
#102443