Finite Group Databasebeta

← #10696061

Subgroups — #10696061

Order 768.

662 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#3153C211yesmaximal, characteristic, complemented
2#83300C211yesmaximal, characteristic, complemented
3#83002C211yesmaximal, characteristic, complemented
4#71813C211yesmaximal, characteristic, complemented
5#71719C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#10018—31nomaximal, Sylow
9C2×C3×#137—21no—
10C2×C3×AGL(1,ℤ/8)—21no—
11C3×#532C2211yescharacteristic
12C2×C3×#124C2211yescharacteristic
13C2×C4×D12C2211yescharacteristic
14C3×#497—21no—
15C3×#496—21no—
16C3×#495—22no—
17C3×C4×M16C2211yescharacteristic
18#5231—21no—
19#5223—21no—
20#5084—21no—
21#5072—21no—
22#5036—21no—
23#5034—21no—
24#5014—22no—
25#4960—21no—
26#4956—22no—
27#4953—21no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31#4540—22no—
32#4537—22no—
33#4459—22no—
34#3153D311yescharacteristic, complemented
35#3109—31no—
36#2631—31no—
37#1533—31no—
38#1514—31no—
39#1490—31no—
40#1288—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C22×#67—22no—
45C22×D12D411yescharacteristic
46C22×C4×D3—21no—
47C3×#137—44no—
48C3×AGL(1,ℤ/8)—44no—
49C2×C3×Q16—22no—
50C2×C3×SD16—22no—
51C2×C3×SD16—22no—
52C2×C3×D8—22no—
53C2×C3×M16—21no—
54C3×C4⋊Q8D411yescharacteristic, complemented
55C3×#127D411yescharacteristic, complemented
56C3×#124—21no—
57C3×#124—21no—
58C3×#124—21no—
59C3×#124—21no—
60C3×#124—21no—
61C3×#121—21no—
62C3×#121—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C42C2311yescharacteristic
68C2×#217—22no—
69C2×#213—22no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3D411yescharacteristic
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77C3×#97—21no—
78C3×C4×C8D412yescomplemented
79#832—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86#809—22no—
87#808—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90#803—21no—
91#803D412yescomplemented
92C2×#137—21no—
93C2×AGL(1,ℤ/8)—21no—
94#532C2×D311yescharacteristic
95C2×#124C2×D311yescharacteristic
96C2×C4×D4—31no—
97#497—21no—
98#496—21no—
99#495—22no—
100#481—61no—
101#476—61no—
102#459—61no—
103#447—61no—
104#442—62no—
105#432—31no—
106#432—31no—
107C4×M16C2×D311yescharacteristic
108#397—61no—
109#393—61no—
110#392—61no—
111#392—61no—
112#386—62no—
113#383—31no—
114#329—62no—
115#327—62no—
116D4⋊C8—62no—
117C24×C3—21no—
118C23×D3—21no—
119C3×#41—42no—
120C3×#41—42no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—42no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—42no—
131C2×C3×D4—41no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C2×#67—44no—
138C2×#67—82no—
139C2×#67—82no—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12—42no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3—82no—
148C3×Q16—44no—
149C3×SD16—44no—
150C3×SD16—44no—
151C3×D8—44no—
152C3×M16—24no—
153C2×C3×C8—22no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42#4111yescharacteristic
166#217—82no—
167#213—44no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—82no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C2×#41—21no—
176C22×Q8#6711yescharacteristic
177C22×D4—31no—
178C22×D4—62no—
179C22×D4#6711yescharacteristic
180C23×C4—61no—
181#137—44no—
182AGL(1,ℤ/8)—44no—
183C2×Q16—22no—
184C2×SD16—22no—
185C2×SD16—22no—
186C2×D8—22no—
187C2×D8—62no—
188C2×M16—21no—
189C4⋊Q8#6711yescharacteristic, complemented
190#127#6711yescharacteristic, complemented
191#124—21no—
192#124—21no—
193#124—21no—
194#124—21no—
195#124—21no—
196#121—21no—
197#121—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C4×D4—61no—
204C2×C4⋊C4—31no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C42C22×D311yescharacteristic
211#106—62no—
212C4⋊C8—61no—
213C4⋊C8—61no—
214#102—62no—
215#102—62no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218#97—21no—
219#97—32no—
220#97—61no—
221C4×C8#6712yescomplemented
222#95—32no—
223#95—62no—
224#95—61no—
225#95—61no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—42no—
232C23×C3#12011yescharacteristic
233C22×D3—22no—
234C22×D3—42no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—42no—
238C3×Q8—41no—
239C3×Q8—42no—
240C3×Q8—41no—
241C3×Q8—81no—
242C3×D4—42no—
243C3×D4—41no—
244C3×D4—42no—
245C3×D4—41no—
246C3×D4—81no—
247C3×D4—81no—
248C3×D4—82no—
249C2×C3×C4—22no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—82no—
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12711yescharacteristic
262#67—88no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—41no—
267C2×Dic3—42no—
268D12—22no—
269D12—22no—
270D12—41no—
271D12—41no—
272C4×D3—82no—
273C4×D3—82no—
274C3×C8—24no—
275C3⋊C8—42no—
276C3⋊C8—81no—
277C24—21no—
278C24—61no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—42no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—122no—
295C2×D4—124no—
296C2×D4—242no—
297C2×D4—242no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—62no—
301C22×C4—61no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—242no—
305C22×C4D3×D411yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4C2×#6711yescharacteristic
308Q16—44no—
309SD16—44no—
310SD16—44no—
311D8—44no—
312D8—124no—
313M16—24no—
314C2×C8—22no—
315C2×C8—62no—
316C2×C8—121no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C22⋊C4—242no—
330C22⋊C4—242no—
331C42D3×D411yescharacteristic
332C42#23711yescharacteristic
333C42C2×#6711yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—82no—
338C22×C3—82no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3#39211yescharacteristic
346C22×C3#45312yes—
347C2×D3—44no—
348C2×D3—42no—
349C2×D3—82no—
350C2×D3—81no—
351C2×D3—81no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—81no—
359C3×C4—81no—
360Dic3—82no—
361Dic3—81no—
362C23—41no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—62no—
369C23—122no—
370C23—121no—
371C23—121no—
372C23—122no—
373C23#93711yescharacteristic
374Q8—41no—
375Q8—42no—
376Q8—41no—
377Q8—81no—
378D4—41no—
379D4—42no—
380D4—42no—
381D4—41no—
382D4—62no—
383D4—62no—
384D4—81no—
385D4—81no—
386D4—82no—
387D4—121no—
388D4—121no—
389D4—248no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—122no—
402C2×C4—242no—
403C2×C4—242no—
404C2×C4#93711yescharacteristic
405C2×C4#93811yescharacteristic
406C2×C4#93911yescharacteristic
407C2×C4#93911yescharacteristic
408C2×C4#94011yescharacteristic
409C2×C4#95311yescharacteristic
410C8—24no—
411C8—122no—
412C8—241no—
413C2×C3—22no—
414C2×C3—161no—
415C2×C3—161no—
416C2×C3#200211yescharacteristic
417C2×C3#200811yescharacteristic
418C2×C3#201011yescharacteristic
419D3—82no—
420C22—21no—
421C22—21no—
422C22—82no—
423C22—82no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—124no—
431C22—122no—
432C22—242no—
433C22—241no—
434C22—241no—
435C22#507712yes—
436C22#522311yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—81no—
444C4—81no—
445C4—242no—
446C4—241no—
447C3#1001811yescharacteristic, complemented, Sylow
448C2—22no—
449C2—161no—
450C2—161no—
451C2—242no—
452C2#7176911yescharacteristic
453C2#7211811yescharacteristic
454C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3153C2
socleC22×C3#392
hypercenter#3153D3
hypocenterC3#10018
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3153

Complements

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

NH ≅ G/Ntype
1. C3×#3153C2semidirect
2. #83300C2semidirect
3. #83002C2semidirect
4. #71813C2semidirect
5. #71719C2semidirect
6. #69855C2semidirect
7. #67666C2semidirect
34. #3153D3semidirect
54. C3×C4⋊Q8D4semidirect
55. C3×#127D4semidirect
78. C3×C4×C8D4semidirect
91. #803D4semidirect
189. C4⋊Q8#67semidirect
190. #127#67semidirect
221. C4×C8#67semidirect
447. C3#10018semidirect

Maximal subgroups

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

subgroupindex
C3×#31532
#833002
#830022
#718132
#717192
#698552
#676662
#100183