Finite Group Databasebeta

← #10695939

Subgroups — #10695939

Order 768.

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

normal subgroups only

This group has 554 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#82480C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic
3#79067C211yesmaximal, characteristic
4C3×#1487C211yesmaximal, characteristic, complemented
5#70790C211yesmaximal, characteristic
6#70766C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9999—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#909—21no—
11#5477C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5230—21no—
16#5227—21no—
17#5133—21no—
18#5027—21no—
19C4×#215—22no—
20#5020—21no—
21#4985—21no—
22#4978—22no—
23#4968—21no—
24#4936C2211yescharacteristic
25#4929—21no—
26#4928—21no—
27#4925C2211yescharacteristic
28#4849—21no—
29#4848—21no—
30#4845—22no—
31C3×Q8⋊C8—22no—
32#4484—22no—
33#4470—22no—
34#3103—31no—
35#3037—31no—
36#2581—31no—
37#1517—31no—
38#1487D311yescharacteristic, complemented
39#1479—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#215—22no—
54#930—21no—
55#930—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—22no—
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61#909—42no—
62#909—42no—
63C2×Dic12—22no—
64C2×#204—22no—
65#875—21no—
66#874D411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69#869D411yescharacteristic
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×C8—21no—
79#835—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95D412yes—
85#830—21no—
86#821—22no—
87#816—22no—
88#809—22no—
89#808—22no—
90C4⋊(C3⋊C8)—21no—
91C4×C3⋊C8D412yes—
92C2×#137—61no—
93#532—31no—
94C2×C4×Q8—31no—
95C2×C4×Q8C2×D311yescharacteristic
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#382—31no—
114#327—62no—
115Q8⋊C8—22no—
116Q8⋊C8—62no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C2×#67—41no—
126C22×Dic3—21no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129#235—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×Dic6—21no—
143C2×Dic6SD1612yescomplemented
144C3×M16—41no—
145C3×M16—41no—
146C2×C3×C8—22no—
147C2×C3×C8—41no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158#215—44no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—41no—
166C4⋊Dic3—41no—
167Dic3⋊C4—44no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C4×Dic3—41no—
171C4×Dic3—41no—
172C2×C3⋊C8—22no—
173Dic12—44no—
174#204—44no—
175C2×#41—61no—
176C22×Q8—31no—
177C22×Q8D1211yescharacteristic
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—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
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—62no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4D1211yescharacteristic
205C2×C42—21no—
206C2×C42—61no—
207C2×C42C22×D311yescharacteristic
208#106—62no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—61no—
212C4⋊C8#6712yes—
213#103—62no—
214#103—62no—
215#102—62no—
216#102—62no—
217C22⋊C8—21no—
218C22⋊C8—61no—
219C4×C8—21no—
220C4×C8—32no—
221#95—32no—
222#95—62no—
223#95—62no—
224#95—62no—
225#95—61no—
226C23×C3#12011yescharacteristic
227C22×D3—41no—
228C3×Q8—22no—
229C3×Q8—22no—
230C3×Q8—41no—
231C3×Q8—41no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12711yescharacteristic
245C2×C3×C4C2×SD1611yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4#13711yescharacteristic
249#67—81no—
250#67—81no—
251C2×Dic3—22no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—42no—
260D12—42no—
261D12—41no—
262D12—41no—
263D12—81no—
264C4×D3—81no—
265C4×D3—81no—
266Dic6—22no—
267Dic6—22no—
268Dic6—41no—
269Dic6—41no—
270Dic6—41no—
271Dic6—41no—
272C3×C8—42no—
273C3×C8—81no—
274C3⋊C8—42no—
275#41—121no—
276#41—121no—
277#41—121no—
278#41—121no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—32no—
282C2×Q8—42no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—121no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—61no—
296C22×C4—62no—
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—41no—
309C2×C8—62no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
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—41no—
326C42—41no—
327C42—121no—
328C42—121no—
329C42#23511yescharacteristic
330C42D3×D411yescharacteristic
331C42D3×D411yescharacteristic
332C42C2×#6711yescharacteristic
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—42no—
346C3×C4—81no—
347C3×C4#45011yescharacteristic
348C3×C4#45211yescharacteristic
349C3×C4#46511yescharacteristic
350C3×C4#46811yescharacteristic
351Dic3—42no—
352Dic3—82no—
353Dic3—81no—
354Dic3—81no—
355C23—121no—
356C23#88211yescharacteristic
357Q8—22no—
358Q8—22no—
359Q8—41no—
360Q8—41no—
361Q8—62no—
362Q8—62no—
363Q8—121no—
364Q8—121no—
365Q8—121no—
366Q8—121no—
367D4—122no—
368D4—121no—
369D4—121no—
370D4—241no—
371D4—241no—
372D4—241no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—62no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—122no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4#88211yescharacteristic
392C2×C4#88411yescharacteristic
393C2×C4#88411yescharacteristic
394C2×C4D3×SD1611yescharacteristic
395C2×C4#91411yescharacteristic
396C2×C4#91811yescharacteristic
397C2×C4#91911yescharacteristic
398C2×C4#93011yescharacteristic
399C2×C4#93711yescharacteristic
400C2×C4#94011yescharacteristic
401C8—42no—
402C8—81no—
403C8—122no—
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#480611yescharacteristic
416C22#481011yescharacteristic
417C22#496011yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—81no—
424C4—122no—
425C4—242no—
426C4—241no—
427C4—241no—
428C4#480711yescharacteristic
429C4#481211yescharacteristic
430C4#516911yescharacteristic
431C4#518711yescharacteristic
432C3#999911yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—481no—
436C2#6999311yescharacteristic
437C2#7027211yescharacteristic
438C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1487D3
hypocenterC3#9999
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
1. #82480C2semidirect
4. C3×#1487C2semidirect
6. #70766C2semidirect
7. #67713C2semidirect
38. #1487D3semidirect
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
#824802
#824422
#790672
C3×#14872
#707902
#707662
#677132
#99993