Finite Group Databasebeta

← #10696037

Subgroups — #10696037

Order 768.

598 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 97 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#79309C211yesmaximal, characteristic, complemented
2#79297C211yesmaximal, characteristic
3#73419C211yesmaximal, characteristic, complemented
4#70770C211yesmaximal, characteristic, complemented
5#70762C211yesmaximal, characteristic, complemented
6#69440C211yesmaximal, characteristic, complemented
7C3×#1274C211yesmaximal, characteristic, complemented
8#10013—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#376C2211yescharacteristic
15C4×(C2×D3)⋊C4—22no—
16C2×#805—22no—
17#4924C2211yescharacteristic
18#4851—21no—
19#4848—21no—
20#4846—21no—
21#4843—21no—
22#4738—21no—
23#4736—21no—
24#4735—21no—
25#4732—21no—
26C4×D24—22no—
27C4×#204—22no—
28#4674—21no—
29#4673—21no—
30#4672—21no—
31#4659—21no—
32#4459—22no—
33#4456—22no—
34#3103—31no—
35#3100—31no—
36#2398—31no—
37#1963—31no—
38#1487—31no—
39#1486—31no—
40#1274D311yescharacteristic, complemented
41C2×#235—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C22×Dic6D411yescharacteristic
45C2×C3×C42D412yes—
46C2×C3×C42C2311yescharacteristic
47#930—21no—
48#930—21no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3—21no—
56C2×#208—21no—
57C2×D24—22no—
58C2×#204—22no—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8D412yes—
77C3×C4×C8D412yes—
78C3×#95—21no—
79C3×#95—21no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82#821—22no—
83#821—22no—
84#821—22no—
85#818—22no—
86#817—22no—
87#816—22no—
88#805—44no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91#532—31no—
92C2×C4×Q8—31no—
93C2×C4×D4—31no—
94#481—61no—
95#480—61no—
96#475—61no—
97#474—61no—
98#460—61no—
99#459—61no—
100#448—61no—
101#447—61no—
102C4×SD16—62no—
103C4×D8—62no—
104#431C2×D311yescharacteristic
105#431C2×D311yescharacteristic
106#423—31no—
107C2×#104—62no—
108#400—61no—
109#397—61no—
110#394—61no—
111#393—61no—
112C4×C22⋊C4—62no—
113#376C2×D311yescharacteristic
114Q8⋊C8—62no—
115D4⋊C8—62no—
116C23×D3—21no—
117C22×C3×C4—22no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—41no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125#235—42no—
126#235—42no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×C4×D3—41no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—42no—
137C2×C4×D3—42no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—42no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C3×C42—22no—
145C3×C42—22no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—44no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164C4×Dic3—41no—
165C4×Dic3—41no—
166#208—42no—
167C2×C3⋊C8—41no—
168D24—44no—
169#204—44no—
170C2×#41—61no—
171C22×Q8—31no—
172C22×D4—31no—
173C23×C4—61no—
174C2×SD16—62no—
175C2×D8—62no—
176C2×M16—61no—
177C4⋊Q8—31no—
178#127—31no—
179#124—61no—
180#121—61no—
181#121—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C4×D4—61no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—61no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C42—61no—
199C2×C42D1212yes—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207#104—124no—
208#103—62no—
209#102—62no—
210#102—62no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C4×C8D1212yes—
215C4×C8D1212yes—
216#95—21no—
217#95—21no—
218#95—62no—
219#95—62no—
220C23×C3#12011yescharacteristic
221C22×D3—22no—
222C22×D3—42no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—42no—
226C22×D3—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12711yescharacteristic
238C2×C3×C4#13512yes—
239C2×C3×C4AGL(1,ℤ/8)12yes—
240#67—82no—
241C2×Dic3—22no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—42no—
249D12—22no—
250D12—22no—
251D12—41no—
252D12—41no—
253D12—42no—
254D12—42no—
255D12—42no—
256C4×D3—82no—
257C4×D3—81no—
258C4×D3—81no—
259C4×D3—81no—
260C4×D3—81no—
261Dic6—22no—
262Dic6—22no—
263Dic6—41no—
264Dic6—41no—
265Dic6—42no—
266C3×C8—42no—
267C3×C8—42no—
268C3⋊C8—81no—
269C24—61no—
270#41—122no—
271#41—122no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—122no—
282C2×D4—121no—
283C22×C4—22no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4—121no—
290C22×C4—122no—
291C22×C4—122no—
292C22×C4C2×D1211yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4D3×D411yescharacteristic
295SD16—124no—
296D8—124no—
297M16—122no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—121no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—124no—
310C22⋊C4—124no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C42—22no—
316C42—22no—
317C42—121no—
318C42—121no—
319C42C2×D1211yescharacteristic
320C42C2×D1211yescharacteristic
321C42D3×D411yescharacteristic
322C42#23911yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3#39211yescharacteristic
326C22×C3#44912yes—
327C2×D3—44no—
328C2×D3—42no—
329C2×D3—82no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C2×D3—81no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4#46312yes—
341C3×C4#49511yescharacteristic
342C3×C4#49611yescharacteristic
343Dic3—42no—
344Dic3—81no—
345Dic3—81no—
346Dic3—81no—
347C23—62no—
348C23—122no—
349C23—121no—
350C23—121no—
351C23—122no—
352C23—121no—
353C23#88211yescharacteristic
354Q8—62no—
355Q8—62no—
356Q8—121no—
357Q8—121no—
358Q8—122no—
359D4—62no—
360D4—62no—
361D4—121no—
362D4—121no—
363D4—122no—
364D4—122no—
365D4—122no—
366D4—242no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—62no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4—242no—
385C2×C4#87411yescharacteristic
386C2×C4#88211yescharacteristic
387C2×C4#88211yescharacteristic
388C2×C4#89511yescharacteristic
389C2×C4#89511yescharacteristic
390C2×C4#89511yescharacteristic
391C2×C4#90412yes—
392C2×C4#90812yes—
393C8—42no—
394C8—42no—
395C8—241no—
396C2×C3—22no—
397C2×C3#201011yescharacteristic
398C2×C3#201011yescharacteristic
399C2×C3#201211yescharacteristic
400D3—82no—
401D3—161no—
402C22—21no—
403C22—21no—
404C22—124no—
405C22—122no—
406C22—242no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22—241no—
412C22#467111yescharacteristic
413C22#473412yes—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—82no—
419C4—122no—
420C4—241no—
421C4—241no—
422C4—241no—
423C4#469611yescharacteristic
424C4#471211yescharacteristic
425C4#484412yes—
426C3#1001311yescharacteristic, complemented, Sylow
427C2—22no—
428C2—242no—
429C2—481no—
430C2#6867511yescharacteristic
431C2#6870311yescharacteristic
432C2#6870311yescharacteristic
subgroupG/H
centerC2×C4#874
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#392
prime radical#79297C2
hypercenter#1274D3
hypocenterC3#10013
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
1. #79309C2semidirect
3. #73419C2semidirect
4. #70770C2semidirect
5. #70762C2semidirect
6. #69440C2semidirect
7. C3×#1274C2semidirect
40. #1274D3semidirect
426. C3#10013semidirect

Maximal subgroups

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

subgroupindex
#793092
#792972
#734192
#707702
#707622
#694402
C3×#12742
#100133