Finite Group Databasebeta

← #10697675

Subgroups — #10697675

Order 768.

528 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 101 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#83367C211yesmaximal, characteristic, complemented
2#83363C211yesmaximal, characteristic
3#79067C211yesmaximal, characteristic
4C3×#1493C211yesmaximal, characteristic, complemented
5#69788C211yesmaximal, characteristic, complemented
6#69735C211yesmaximal, characteristic
7#67711C211yesmaximal, characteristic, complemented
8#10257—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#909—21no—
11#5477C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5133—21no—
16#5094—21no—
17#5091—21no—
18#5084—21no—
19#5083—21no—
20#5063—22no—
21#5048—22no—
22#5027—21no—
23#5020—21no—
24#5004—21no—
25#4995—21no—
26#4924C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4849—21no—
29#4848—21no—
30#4845—22no—
31C3×Q8⋊C8—22no—
32#4550—22no—
33#4459—22no—
34#3246—31no—
35#3078—31no—
36#3037—31no—
37#1984—31no—
38#1493D311yescharacteristic, complemented
39#1492—31no—
40#1292—31no—
41C2×#235—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C22×C3×C8—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×C42C2311yescharacteristic
54C2×#215—22no—
55#930—21no—
56#930—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×C4×Dic3—21no—
60C2×#208—21no—
61C22×C3⋊C8D412yes—
62#909—42no—
63#909—42no—
64C2×Dic12—22no—
65C2×#204—22no—
66#875—21no—
67#874D411yescharacteristic
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C4×Dic6—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×C4×C8—21no—
78C3×C4×C8—21no—
79#835—22no—
80#821—22no—
81#816—22no—
82#810—22no—
83#810—22no—
84#809—22no—
85#808—22no—
86#808—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)D412yes—
92C2×#137—61no—
93#533C2×D311yescharacteristic
94#532—31no—
95C2×C4×Q8—31no—
96#479—62no—
97#477—61no—
98#476—61no—
99#475—61no—
100#475—61no—
101#465—62no—
102#464—62no—
103#462—61no—
104#461—61no—
105#460—61no—
106#460—61no—
107#428—61no—
108#423—31no—
109#423C2×D311yescharacteristic
110C2×C4⋊C8—31no—
111#419—61no—
112#417—61no—
113C2×C4×C8C2×D311yescharacteristic
114#336—62no—
115Q8⋊C8—22no—
116D4⋊C8—62no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—41no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125#235—41no—
126#235—41no—
127#235—41no—
128#235—41no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6SD1612yescomplemented
140C3×M16—41no—
141C3×M16—41no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158#215—82no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C4×Dic3—41no—
168C4×Dic3—41no—
169#208—41no—
170#208—41no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—42no—
174Dic12—44no—
175#204—44no—
176C2×#41—61no—
177C22×Q8—31no—
178#137—122no—
179#137—122no—
180C2×Q16—62no—
181C2×SD16—62no—
182C2×SD16—62no—
183C2×M16—21no—
184C2×M16—61no—
185C22×C8—21no—
186C22×C8—32no—
187C4⋊Q8—31no—
188C4⋊Q8D1211yescharacteristic
189C4⋊Q8D1211yescharacteristic
190#127—31no—
191#125—21no—
192#124—61no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195#121—61no—
196#121—61no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—61no—
200C4×Q8—61no—
201C4×Q8—61no—
202C4×Q8—61no—
203C4×D4—61no—
204C4×D4—61no—
205#117—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210#107—62no—
211#106—62no—
212#106—62no—
213C4⋊C8—21no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218C4⋊C8#6712yes—
219#103—62no—
220#103—62no—
221#103—62no—
222#103—62no—
223#102—62no—
224#102—62no—
225C4×C8—21no—
226C4×C8—21no—
227C23×C3#12111yescharacteristic
228C22×D3—41no—
229C3×Q8—42no—
230C3×Q8—41no—
231C3×Q8—41no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12711yescharacteristic
243C2×C3×C4C2×SD1611yescharacteristic
244C2×C3×C4#13511yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
246C2×C3×C4#13711yescharacteristic
247#67—81no—
248#67—81no—
249C2×Dic3—22no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255D12—42no—
256D12—41no—
257D12—41no—
258D12—41no—
259D12—41no—
260C4×D3—81no—
261C4×D3—81no—
262Dic6—22no—
263Dic6—22no—
264Dic6—41no—
265Dic6—41no—
266Dic6—41no—
267Dic6—41no—
268C3×C8—42no—
269C3×C8—42no—
270C3⋊C8—42no—
271C3⋊C8—42no—
272#41—121no—
273#41—121no—
274#41—121no—
275#41—121no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—32no—
279C2×Q8—61no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—121no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—121no—
292C22×C4#23511yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4D3×D411yescharacteristic
295Q16—124no—
296SD16—124no—
297SD16—242no—
298M16—41no—
299M16—41no—
300M16—121no—
301M16—121no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—62no—
307C2×C8—62no—
308C2×C8—122no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C22⋊C4—41no—
322C22⋊C4—121no—
323C22⋊C4—121no—
324C42—41no—
325C42—121no—
326C42—121no—
327C42C2×D1211yescharacteristic
328C42D3×D411yescharacteristic
329C42D3×D411yescharacteristic
330C42C2×#6711yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3#39211yescharacteristic
334C22×C3#46511yescharacteristic
335C22×C3#46811yescharacteristic
336C2×D3—81no—
337C2×D3—81no—
338C2×D3—81no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—81no—
343C3×C4—81no—
344C3×C4#44811yescharacteristic
345C3×C4#44911yescharacteristic
346C3×C4#45011yescharacteristic
347C3×C4#45211yescharacteristic
348C3×C4#45511yescharacteristic
349C3×C4#45611yescharacteristic
350Dic3—42no—
351Dic3—81no—
352Dic3—81no—
353C23—121no—
354C23#88411yescharacteristic
355Q8—42no—
356Q8—41no—
357Q8—41no—
358Q8—62no—
359Q8—62no—
360Q8—121no—
361Q8—121no—
362Q8—121no—
363Q8—121no—
364D4—122no—
365D4—121no—
366D4—121no—
367D4—121no—
368D4—121no—
369D4—241no—
370D4—241no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—62no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4#88211yescharacteristic
385C2×C4#88211yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4D3×SD1611yescharacteristic
388C2×C4#91411yescharacteristic
389C2×C4#91811yescharacteristic
390C2×C4#91911yescharacteristic
391C2×C4#93011yescharacteristic
392C2×C4#93711yescharacteristic
393C2×C4#94011yescharacteristic
394C8—42no—
395C8—42no—
396C8—122no—
397C8—122no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3#200011yescharacteristic
401C2×C3#200711yescharacteristic
402C2×C3#201711yescharacteristic
403D3—161no—
404C22—21no—
405C22—21no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22#480711yescharacteristic
410C22#481211yescharacteristic
411C22#496011yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—81no—
416C4—81no—
417C4—122no—
418C4—241no—
419C4—241no—
420C4#480611yescharacteristic
421C4#481011yescharacteristic
422C4#482511yescharacteristic
423C4#482611yescharacteristic
424C4#516911yescharacteristic
425C4#518711yescharacteristic
426C3#1025711yescharacteristic, complemented, Sylow
427C2—21no—
428C2—21no—
429C2—481no—
430C2#7001311yescharacteristic
431C2#7027211yescharacteristic
432C2#7039311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1493D3
hypocenterC3#10257
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
1. #83367C2semidirect
4. C3×#1493C2semidirect
5. #69788C2semidirect
7. #67711C2semidirect
38. #1493D3semidirect
139. C2×Dic6SD16semidirect
426. C3#10257semidirect

Maximal subgroups

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

subgroupindex
#833672
#833632
#790672
C3×#14932
#697882
#697352
#677112
#102573