Finite Group Databasebeta

← #10697592

Subgroups — #10697592

Order 768.

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

normal subgroups only

This group has 554 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#79312C211yesmaximal, characteristic
2#79067C211yesmaximal, characteristic
3#73722C211yesmaximal, characteristic, complemented
4#70766C211yesmaximal, characteristic, complemented
5#70507C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic
7C3×#1303C211yesmaximal, characteristic, complemented
8#10244—31nomaximal, Sylow
9C2×#909—21no—
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5133—21no—
17#4973—22no—
18#4936—21no—
19#4929—21no—
20#4925C2211yescharacteristic
21#4849—21no—
22#4848—21no—
23#4845—22no—
24#4841—22no—
25#4738—22no—
26#4693—21no—
27#4681—21no—
28#4659—22no—
29#4484—22no—
30#4481—22no—
31#4451—22no—
32#3206—31no—
33#3037—31no—
34#2568—31no—
35#1548—31no—
36#1502—31no—
37#1479—31no—
38#1303D311yescharacteristic, complemented
39C2×#235—21no—
40C22×Dic6D411yescharacteristic
41C22×Dic6D412yes—
42C2×C3×M16—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46#930—21no—
47#930—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53#909—42no—
54#909—42no—
55C2×Dic12—22no—
56C2×#204—22no—
57C2×#204—24no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4⋊C8D412yes—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76C3×#95D412yes—
77C3⋊#95—24no—
78C3⋊#95—24no—
79#830—21no—
80#821—22no—
81#816—22no—
82#816—24no—
83C4⋊(C3⋊C8)—21no—
84C4×C3⋊C8—21no—
85C2×#137—61no—
86#532—31no—
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#423C2×D311yescharacteristic
99#417—61no—
100#400—62no—
101#391—61no—
102#391—62no—
103#382—61no—
104C4×C4⋊C4C2×D311yescharacteristic
105#333—62no—
106#327—62no—
107Q8⋊C8—62no—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—22no—
116#235—41no—
117#235—41no—
118#235—41no—
119#235—41no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—22no—
130C2×Dic6—21no—
131C2×Dic6—22no—
132C2×Dic6—44no—
133C2×Dic6SD1612yescomplemented
134C3×M16—41no—
135C3×M16—41no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C3×C4⋊C4—42no—
139C3×C42—41no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
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—
159C2×C3⋊C8—41no—
160Dic12—44no—
161#204—44no—
162#204—84no—
163C2×#41—61no—
164C22×Q8—31no—
165C22×Q8—32no—
166#137—122no—
167#137—122no—
168C2×Q16—62no—
169C2×SD16—62no—
170C2×SD16—64no—
171C2×M16—21no—
172C4⋊Q8—31no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177#127—31no—
178#124—61no—
179#121—61no—
180#121—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×D4—61no—
186C4×D4—61no—
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—61no—
197C4⋊C8D1212yes—
198C4⋊C8D1212yes—
199#103—62no—
200#103—64no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204C4×C8—21no—
205C4×C8—61no—
206#95—64no—
207#95—64no—
208#95D1212yes—
209C23×C3#12011yescharacteristic
210C22×D3—41no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×C3×C4C2×SD1611yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C4#13711yescharacteristic
226#67—81no—
227#67—81no—
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—44no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236D12—41no—
237D12—41no—
238D12—42no—
239D12—81no—
240C4×D3—81no—
241C4×D3—81no—
242Dic6—22no—
243Dic6—22no—
244Dic6—41no—
245Dic6—41no—
246Dic6—41no—
247Dic6—41no—
248Dic6—44no—
249Dic6—82no—
250C3×C8—42no—
251C3×C8—42no—
252C3⋊C8—81no—
253#41—121no—
254#41—121no—
255#41—121no—
256#41—121no—
257C2×Q8—32no—
258C2×Q8—61no—
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—61no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—121no—
271C22×C4—22no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—62no—
275C22×C4—121no—
276C22×C4C2×D1211yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279Q16—124no—
280SD16—124no—
281SD16—244no—
282M16—41no—
283M16—41no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—121no—
287C4⋊C4—42no—
288C4⋊C4—61no—
289C4⋊C4—61no—
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—121no—
299C22⋊C4—121no—
300C42—41no—
301C42—41no—
302C42—121no—
303C42—121no—
304C42C2×D1211yescharacteristic
305C42C2×D1211yescharacteristic
306C42D3×D411yescharacteristic
307C42#23911yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3#39211yescharacteristic
311C22×C3#44811yescharacteristic
312C22×C3#44911yescharacteristic
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—82no—
320C3×C4#45011yescharacteristic
321C3×C4#45211yescharacteristic
322C3×C4#45512yes—
323C3×C4#46511yescharacteristic
324C3×C4#46811yescharacteristic
325Dic3—42no—
326Dic3—81no—
327Dic3—82no—
328Dic3—81no—
329C23—121no—
330C23#88211yescharacteristic
331Q8—62no—
332Q8—62no—
333Q8—121no—
334Q8—121no—
335Q8—121no—
336Q8—121no—
337Q8—124no—
338Q8—242no—
339D4—121no—
340D4—121no—
341D4—122no—
342D4—241no—
343D4—241no—
344D4—241no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—62no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—124no—
354C2×C4—122no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—122no—
358C2×C4—241no—
359C2×C4—241no—
360C2×C4#87411yescharacteristic
361C2×C4#88211yescharacteristic
362C2×C4#88211yescharacteristic
363C2×C4#89511yescharacteristic
364C2×C4#89511yescharacteristic
365C2×C4#89511yescharacteristic
366C2×C4#91111yescharacteristic
367C2×C4D3×SD1611yescharacteristic
368C2×C4#91611yescharacteristic
369C2×C4#91811yescharacteristic
370C8—42no—
371C8—42no—
372C8—241no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3#200011yescharacteristic
376C2×C3#200411yescharacteristic
377C2×C3#200911yescharacteristic
378D3—161no—
379C22—21no—
380C22—21no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#467111yescharacteristic
385C22#478111yescharacteristic
386C22#480611yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—122no—
392C4—241no—
393C4—242no—
394C4—241no—
395C4#477611yescharacteristic
396C4#480811yescharacteristic
397C4#482512yes—
398C4#486411yescharacteristic
399C4#488611yescharacteristic
400C3#1024411yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—481no—
404C2#6876211yescharacteristic
405C2#6888711yescharacteristic
406C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1303D3
hypocenterC3#10244
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #73722C2semidirect
4. #70766C2semidirect
5. #70507C2semidirect
7. C3×#1303C2semidirect
38. #1303D3semidirect
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
#793122
#790672
#737222
#707662
#705072
#694692
C3×#13032
#102443