Finite Group Databasebeta

← #10696050

Subgroups — #10696050

Order 768.

560 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 95 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#83368C211yesmaximal, characteristic, complemented
2#83351C211yesmaximal, characteristic
3#73419C211yesmaximal, characteristic
4C3×#1995C211yesmaximal, characteristic, complemented
5#70500C211yesmaximal, characteristic, complemented
6#70496C211yesmaximal, characteristic
7#67614C211yesmaximal, characteristic, complemented
8#10016—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5091—21no—
15#5090—21no—
16#5084—21no—
17#5083—21no—
18#5050—21no—
19#5046—21no—
20#5044—21no—
21#5043—21no—
22C4×#216—22no—
23C4×#215—22no—
24C4×(C2×D3)⋊C4—22no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4674—21no—
28#4673—21no—
29#4672—21no—
30#4659—21no—
31C3×#339—22no—
32#4484—22no—
33#4483—22no—
34#3104—31no—
35#3101—31no—
36#2398—31no—
37#1995D311yescharacteristic, complemented
38#1487—31no—
39#1486—31no—
40#1274—31no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C42D412yes—
54C2×C3×C42C2311yescharacteristic
55C2×#216—22no—
56C2×#215—22no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—21no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×#95—21no—
75C3×#95—21no—
76#835—22no—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80#830—21no—
81#830—21no—
82#810—22no—
83#809—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4×C3⋊C8D412yes—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91C2×C4×Q8—31no—
92C2×C4×D4—31no—
93#484—61no—
94#483—61no—
95#477—61no—
96#476—61no—
97#462—61no—
98#461—61no—
99#451—61no—
100#450—61no—
101C4×Q16—62no—
102C4×SD16—62no—
103#431—31no—
104#431—31no—
105#423C2×D311yescharacteristic
106#400—61no—
107#397—61no—
108#394—61no—
109#393—61no—
110C4×C22⋊C4—62no—
111#376C2×D311yescharacteristic
112#339—22no—
113Q8⋊C8—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4—22no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—42no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—42no—
131C2×C4×D3—42no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—42no—
135C3×M16—42no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—22no—
145C3×C42—22no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#216—44no—
152#215—44no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—44no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C22×Q8—31no—
168C22×D4—31no—
169C23×C4—61no—
170C2×Q16—62no—
171C2×SD16—62no—
172C2×M16—21no—
173C4⋊Q8D1211yescharacteristic
174C4⋊Q8D1211yescharacteristic
175#125—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188#117—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—61no—
195C2×C42—61no—
196C2×C42#6712yes—
197C2×C42C22×D311yescharacteristic
198#107—62no—
199#106—62no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#103—62no—
205#103—62no—
206#103—62no—
207#102—62no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C4×C8—32no—
211C4×C8—32no—
212#95—21no—
213#95—21no—
214#95—62no—
215#95—62no—
216C23×C3#12011yescharacteristic
217C22×D3—22no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—42no—
222C3×Q8—42no—
223C3×Q8—42no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12711yescharacteristic
237C2×C3×C4#13512yes—
238C2×C3×C4#13712yes—
239C2×Dic3—22no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246D12—22no—
247D12—22no—
248D12—41no—
249D12—41no—
250C4×D3—82no—
251C4×D3—81no—
252C4×D3—81no—
253Dic6—22no—
254Dic6—22no—
255Dic6—41no—
256Dic6—41no—
257C3×C8—81no—
258C3⋊C8—42no—
259C3⋊C8—42no—
260C24—61no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—122no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—122no—
269C22×C4—22no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—122no—
276C22×C4—122no—
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4C2×#6711yescharacteristic
280Q16—124no—
281SD16—124no—
282M16—42no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—62no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—41no—
299C22⋊C4—124no—
300C22⋊C4—124no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C42—22no—
304C42—22no—
305C42—41no—
306C42—121no—
307C42—121no—
308C42C2×D1211yescharacteristic
309C42#23511yescharacteristic
310C42D3×D411yescharacteristic
311C42D3×D411yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3#39211yescharacteristic
315C22×C3#45212yes—
316C2×D3—44no—
317C2×D3—42no—
318C2×D3—82no—
319C2×D3—81no—
320C2×D3—81no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—82no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4#46412yes—
329C3×C4#49511yescharacteristic
330C3×C4#49711yescharacteristic
331Dic3—42no—
332Dic3—81no—
333Dic3—81no—
334C23—62no—
335C23—122no—
336C23—121no—
337C23—121no—
338C23—122no—
339C23#93711yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—62no—
343Q8—62no—
344Q8—121no—
345Q8—121no—
346D4—62no—
347D4—62no—
348D4—121no—
349D4—121no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—62no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—242no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4#88211yescharacteristic
368C2×C4#88211yescharacteristic
369C2×C4#88411yescharacteristic
370C2×C4#88411yescharacteristic
371C2×C4#93011yescharacteristic
372C2×C4#94011yescharacteristic
373C2×C4#94212yes—
374C2×C4#95012yes—
375C8—81no—
376C8—122no—
377C8—122no—
378C2×C3—22no—
379C2×C3#201011yescharacteristic
380C2×C3#201011yescharacteristic
381C2×C3#201511yescharacteristic
382D3—82no—
383C22—21no—
384C22—21no—
385C22—124no—
386C22—122no—
387C22—242no—
388C22—241no—
389C22—241no—
390C22#496011yescharacteristic
391C22#504712yes—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—82no—
397C4—81no—
398C4—81no—
399C4—122no—
400C4—241no—
401C4—241no—
402C4#502812yes—
403C4#505911yescharacteristic
404C4#508611yescharacteristic
405C3#1001611yescharacteristic, complemented, Sylow
406C2—22no—
407C2—242no—
408C2#6968911yescharacteristic
409C2#7040611yescharacteristic
410C2#7040611yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1995C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1995D3
hypocenterC3#10016
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
1. #83368C2semidirect
4. C3×#1995C2semidirect
5. #70500C2semidirect
7. #67614C2semidirect
37. #1995D3semidirect
405. C3#10016semidirect

Maximal subgroups

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

subgroupindex
#833682
#833512
#734192
C3×#19952
#705002
#704962
#676142
#100163