Finite Group Databasebeta

← #10697602

Subgroups — #10697602

Order 768.

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

normal subgroups only

This group has 740 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#83355C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic, complemented
3#73799C211yesmaximal, characteristic
4C3×#1503C211yesmaximal, characteristic, complemented
5#70774C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10246—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#895—22no—
11C2×#874—21no—
12C2×#874C2211yescharacteristic
13C2×C4×D12C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5088—21no—
17#5087—21no—
18#5085—21no—
19#5084—21no—
20#5066—22no—
21#5046—22no—
22#5027—22no—
23#5001—21no—
24#4995—21no—
25#4939—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4671—22no—
29C3×#328—22no—
30#4485—22no—
31#4459—22no—
32#3208—31no—
33#3077—31no—
34#2541—31no—
35#1545—31no—
36#1503D311yescharacteristic, complemented
37#1478—31no—
38#1303—31no—
39C22×D12—21no—
40C22×D12—24no—
41C22×D12—21no—
42C22×D12D411yescharacteristic
43C22×D12D412yes—
44C22×C4×D3—21no—
45C3×C4⋊Q8D411yescharacteristic, complemented
46C3×C4⋊Q8D411yescharacteristic, complemented
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#215—24no—
57C2×(C2×D3)⋊C4—24no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×#208—21no—
61#895—48no—
62#874—21no—
63#874—21no—
64#874—21no—
65#874—21no—
66#874—42no—
67#874—42no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C3×C4⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C4×C8—21no—
75C3×#95D412yes—
76#830—21no—
77#809—22no—
78#809—22no—
79#809—24no—
80#808—22no—
81#807—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yescomplemented
84C4⋊(C3⋊C8)D412yescomplemented
85C4×C3⋊C8—21no—
86#533C2×D311yescharacteristic
87C2×#127—31no—
88C2×#127—61no—
89C2×#121—62no—
90C2×C4×D4—31no—
91#492—61no—
92#486—61no—
93#478—62no—
94#476—61no—
95#474—61no—
96#460—62no—
97#450—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#428—61no—
101#423—31no—
102#418—61no—
103#392—62no—
104#390—61no—
105C4×C4⋊C4C2×D311yescharacteristic
106#331—62no—
107#328—22no—
108D4⋊C8—62no—
109C23×D3—21no—
110C23×D3—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C22×C3×C4—22no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—22no—
123C2×D12—21no—
124C2×D12—22no—
125C2×D12—44no—
126C2×D12—48no—
127C2×D12—48no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—41no—
136C2×D12—82no—
137C2×D12—84no—
138C2×D12SD1612yescomplemented
139C2×C4×D3—44no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C3×C8—41no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158#215—84no—
159(C2×D3)⋊C4—48no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C22×D4—31no—
169C22×D4—32no—
170C22×D4—64no—
171C22×D4—61no—
172C22×D4—61no—
173C23×C4—61no—
174C2×SD16—64no—
175C2×M16—61no—
176C4⋊Q8D1211yescharacteristic, complemented
177C4⋊Q8D1211yescharacteristic, complemented
178#127—61no—
179#127—61no—
180#127—61no—
181#127—61no—
182#127—122no—
183#127—122no—
184#125—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187#121—128no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×D4—61no—
191C4×D4—61no—
192C4×D4—61no—
193C4×D4—61no—
194#117—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—31no—
197C2×C22⋊C4—64no—
198C2×C22⋊C4—61no—
199C2×C42—21no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—32no—
206C4⋊C8—61no—
207#102—62no—
208#102—62no—
209#102—64no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C4×C8—21no—
213C4×C8—61no—
214#95#6712yes—
215C23×C3#12011yescharacteristic
216C22×D3—22no—
217C22×D3—42no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—44no—
222C22×D3—44no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—42no—
226C3×Q8—42no—
227C3×Q8—81no—
228C2×C3×C4—22no—
229C2×C3×C4—24no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×C3×C4C2×SD1611yescharacteristic
242C2×C3×C4#13511yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4#13711yescharacteristic
245C2×Dic3—42no—
246C2×Dic3—41no—
247D12—22no—
248D12—22no—
249D12—41no—
250D12—44no—
251D12—41no—
252D12—82no—
253D12—82no—
254D12—82no—
255D12—816no—
256D12—81no—
257D12—81no—
258D12—81no—
259D12—81no—
260C4×D3—82no—
261C4×D3—82no—
262C3×C8—81no—
263C3⋊C8—42no—
264C3⋊C8—42no—
265C24—61no—
266C24—62no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×D4—32no—
270C2×D4—61no—
271C2×D4—62no—
272C2×D4—61no—
273C2×D4—62no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—124no—
278C2×D4—128no—
279C2×D4—128no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—242no—
289C2×D4—244no—
290C22×C4—22no—
291C22×C4—61no—
292C22×C4—124no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—244no—
299M16—121no—
300M16—121no—
301C2×C8—41no—
302C2×C8—62no—
303C2×C8—62no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C22⋊C4—41no—
314C22⋊C4—128no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42C2×D1211yescharacteristic
321C42#23511yescharacteristic
322C42D3×D411yescharacteristic
323C42D3×D411yescharacteristic
324C22×C3—21no—
325C22×C3—21no—
326C22×C3#39211yescharacteristic
327C22×C3#45011yescharacteristic
328C22×C3#45211yescharacteristic
329C2×D3—44no—
330C2×D3—42no—
331C2×D3—82no—
332C2×D3—81no—
333C2×D3—81no—
334C2×D3—84no—
335C2×D3—84no—
336C2×D3—82no—
337C2×D3—82no—
338C2×D3—82no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—44no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4#44811yescharacteristic
346C3×C4#44911yescharacteristic
347C3×C4#45312yes—
348C3×C4#46511yescharacteristic
349C3×C4#46811yescharacteristic
350Dic3—81no—
351C23—62no—
352C23—122no—
353C23—122no—
354C23—124no—
355C23—124no—
356C23—121no—
357C23—121no—
358C23—122no—
359C23—122no—
360C23—122no—
361C23#93711yescharacteristic
362Q8—42no—
363Q8—81no—
364D4—62no—
365D4—62no—
366D4—121no—
367D4—121no—
368D4—124no—
369D4—242no—
370D4—242no—
371D4—242no—
372D4—2416no—
373D4—241no—
374D4—241no—
375D4—241no—
376D4—241no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—122no—
385C2×C4—121no—
386C2×C4—242no—
387C2×C4—242no—
388C2×C4#88211yescharacteristic
389C2×C4#88211yescharacteristic
390C2×C4#88411yescharacteristic
391C2×C4#88411yescharacteristic
392C2×C4D3×SD1611yescharacteristic
393C2×C4#91411yescharacteristic
394C2×C4#91811yescharacteristic
395C2×C4#91911yescharacteristic
396C2×C4#93011yescharacteristic
397C2×C4#94011yescharacteristic
398C8—81no—
399C8—122no—
400C8—122no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3#200011yescharacteristic
404C2×C3#200211yescharacteristic
405C2×C3#200811yescharacteristic
406D3—82no—
407D3—162no—
408C22—21no—
409C22—21no—
410C22—124no—
411C22—122no—
412C22—242no—
413C22—244no—
414C22—244no—
415C22—241no—
416C22—241no—
417C22—242no—
418C22—242no—
419C22—242no—
420C22#496011yescharacteristic
421C22#516911yescharacteristic
422C22#518711yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—44no—
427C4—81no—
428C4—81no—
429C4—241no—
430C4#480611yescharacteristic
431C4#480711yescharacteristic
432C4#481011yescharacteristic
433C4#481211yescharacteristic
434C4#482212yes—
435C3#1024611yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—242no—
439C2—482no—
440C2#7005011yescharacteristic
441C2#7027211yescharacteristic
442C2#7038511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#392
prime radical#73799C2
hypercenter#1503D3
hypocenterC3#10246
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
1. #83355C2semidirect
2. #83350C2semidirect
4. C3×#1503C2semidirect
5. #70774C2semidirect
6. #70757C2semidirect
7. #67620C2semidirect
36. #1503D3semidirect
45. C3×C4⋊Q8D4semidirect
46. C3×C4⋊Q8D4semidirect
83. C4⋊(C3⋊C8)D4semidirect
84. C4⋊(C3⋊C8)D4semidirect
138. C2×D12SD16semidirect
176. C4⋊Q8D12semidirect
177. C4⋊Q8D12semidirect
435. C3#10246semidirect

Maximal subgroups

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

subgroupindex
#833552
#833502
#737992
C3×#15032
#707742
#707572
#676202
#102463