Finite Group Databasebeta

← #10695941

Subgroups — #10695941

Order 768.

554 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 99 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
2#79067C211yesmaximal, characteristic
3#74191C211yesmaximal, characteristic, complemented
4#70767C211yesmaximal, characteristic, complemented
5#70500C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic
7C3×#1286C211yesmaximal, characteristic, complemented
8#9999—31nomaximal, Sylow
9C2×#909—21no—
10#5477C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5133—21no—
17#4984—21no—
18#4973—21no—
19#4925C2211yescharacteristic
20#4851—21no—
21#4849—21no—
22#4848—21no—
23#4848—21no—
24#4845—22no—
25#4738—21no—
26#4735—21no—
27C4×#204—22no—
28#4659—21no—
29#4656—21no—
30#4653—22no—
31#4484—22no—
32#4456—22no—
33#4451—22no—
34#3103—31no—
35#3037—31no—
36#2581—31no—
37#1517—31no—
38#1487—31no—
39#1479—31no—
40#1286D311yescharacteristic, complemented
41C2×#235—21no—
42C22×Dic6D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47#930—21no—
48#930—21no—
49C2×C4⋊Dic3—22no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56#909—42no—
57#909—42no—
58C2×Dic12—22no—
59C2×#204—22no—
60C2×#204—22no—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×C22⋊C8—21no—
78C3×C4×C8D412yes—
79C3×#95—21no—
80C3×#95D412yes—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84#830—21no—
85#821—22no—
86#821—22no—
87#817—22no—
88#816—22no—
89#816—22no—
90C4⋊(C3⋊C8)—21no—
91C4×C3⋊C8—21no—
92C2×#137—61no—
93#532—31no—
94C2×C4×Q8—31no—
95C2×C4×Q8—31no—
96#481—61no—
97#475—61no—
98#464—62no—
99#462—61no—
100#460—61no—
101#460—61no—
102#448—61no—
103C4×SD16—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423C2×D311yescharacteristic
107#417—61no—
108#400—61no—
109#395—61no—
110#391—61no—
111#389—61no—
112#387—62no—
113#382C2×D311yescharacteristic
114#327—62no—
115Q8⋊C8—62no—
116Q8⋊C8—62no—
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—
125C22×Dic3—21no—
126#235—41no—
127#235—41no—
128#235—41no—
129#235—41no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×C4×D3—41no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×Dic6—42no—
143C2×Dic6SD1612yescomplemented
144C3×M16—41no—
145C3×M16—41no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
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—
162C4⋊Dic3—44no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C4×Dic3—41no—
168C4×Dic3—41no—
169C4×Dic3—41no—
170C4×Dic3—41no—
171C2×C3⋊C8—41no—
172Dic12—44no—
173#204—44no—
174#204—44no—
175C2×#41—61no—
176C22×Q8—31no—
177C22×Q8—31no—
178#137—122no—
179#137—122no—
180C2×Q16—62no—
181C2×SD16—62no—
182C2×SD16—62no—
183C2×M16—21no—
184C4⋊Q8—31no—
185#127—31no—
186#124—61no—
187#121—61no—
188#121—61no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×Q8—61no—
192C4×Q8—61no—
193C4×Q8—61no—
194C4×Q8—61no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×D4—61no—
198C4×D4—61no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C42—61no—
206C2×C42—61no—
207C2×C42C22×D311yescharacteristic
208#106—62no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—61no—
212C4⋊C8D1212yes—
213#103—62no—
214#103—62no—
215#102—62no—
216#102—62no—
217C22⋊C8—21no—
218C22⋊C8—61no—
219C4×C8—61no—
220C4×C8D1212yes—
221#95—21no—
222#95—62no—
223#95—62no—
224#95—62no—
225#95D1212yes—
226C23×C3#12011yescharacteristic
227C22×D3—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12711yescharacteristic
239C2×C3×C4C2×SD1611yescharacteristic
240C2×C3×C4#13511yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C3×C4#13711yescharacteristic
243#67—81no—
244#67—81no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256D12—42no—
257D12—41no—
258D12—41no—
259D12—81no—
260C4×D3—81no—
261C4×D3—81no—
262Dic6—22no—
263Dic6—22no—
264Dic6—22no—
265Dic6—22no—
266Dic6—41no—
267Dic6—41no—
268Dic6—41no—
269Dic6—41no—
270Dic6—41no—
271Dic6—41no—
272C3×C8—42no—
273C3×C8—42no—
274C3⋊C8—81no—
275#41—121no—
276#41—121no—
277#41—121no—
278#41—121no—
279C2×Q8—32no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—122no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—121no—
293C22×C4—22no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—121no—
299C22×C4C2×D1211yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302Q16—124no—
303SD16—124no—
304SD16—124no—
305M16—41no—
306M16—41no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—121no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—124no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C4⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C42—121no—
326C42—121no—
327C42—121no—
328C42—121no—
329C42C2×D1211yescharacteristic
330C42C2×D1211yescharacteristic
331C42D3×D411yescharacteristic
332C42#23911yescharacteristic
333C22×C3—21no—
334C22×C3—21no—
335C22×C3#39211yescharacteristic
336C22×C3#44811yescharacteristic
337C22×C3#44911yescharacteristic
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—81no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—82no—
346C3×C4#45011yescharacteristic
347C3×C4#45211yescharacteristic
348C3×C4#46511yescharacteristic
349C3×C4#46811yescharacteristic
350Dic3—42no—
351Dic3—42no—
352Dic3—81no—
353Dic3—81no—
354Dic3—81no—
355C23—121no—
356C23#88211yescharacteristic
357Q8—62no—
358Q8—62no—
359Q8—62no—
360Q8—62no—
361Q8—121no—
362Q8—121no—
363Q8—121no—
364Q8—121no—
365Q8—121no—
366Q8—121no—
367D4—122no—
368D4—121no—
369D4—121no—
370D4—241no—
371D4—241no—
372D4—241no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4#87411yescharacteristic
392C2×C4#88211yescharacteristic
393C2×C4#88211yescharacteristic
394C2×C4#89511yescharacteristic
395C2×C4#89511yescharacteristic
396C2×C4#89511yescharacteristic
397C2×C4#91111yescharacteristic
398C2×C4D3×SD1611yescharacteristic
399C2×C4#91611yescharacteristic
400C2×C4#91811yescharacteristic
401C8—42no—
402C8—42no—
403C8—241no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3#200011yescharacteristic
407C2×C3#200911yescharacteristic
408C2×C3#201011yescharacteristic
409D3—161no—
410C22—21no—
411C22—21no—
412C22—241no—
413C22—241no—
414C22—241no—
415C22#467111yescharacteristic
416C22#478111yescharacteristic
417C22#480611yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—82no—
423C4—122no—
424C4—122no—
425C4—241no—
426C4—241no—
427C4—241no—
428C4#477611yescharacteristic
429C4#480811yescharacteristic
430C4#486411yescharacteristic
431C4#488611yescharacteristic
432C3#999911yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—481no—
436C2#6870311yescharacteristic
437C2#6888711yescharacteristic
438C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1286D3
hypocenterC3#9999
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
3. #74191C2semidirect
4. #70767C2semidirect
5. #70500C2semidirect
7. C3×#1286C2semidirect
40. #1286D3semidirect
143. C2×Dic6SD16semidirect
432. C3#9999semidirect

Maximal subgroups

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

subgroupindex
#793092
#790672
#741912
#707672
#705002
#695452
C3×#12862
#99993