Finite Group Databasebeta

← #10697604

Subgroups — #10697604

Order 768.

740 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 103 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#82378C211yesmaximal, characteristic
2#82296C211yesmaximal, characteristic, complemented
3#79305C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic, complemented
5#70792C211yesmaximal, characteristic, complemented
6#70774C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10246—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#940—21no—
11C2×#930—22no—
12C2×#874C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#431C2211yescharacteristic
16#5215—21no—
17#5009—22no—
18#5008—21no—
19#5007—21no—
20#4988—21no—
21#4964—21no—
22#4960—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4924C2211yescharacteristic
25#4848—22no—
26#4731—22no—
27#4694—21no—
28#4693—21no—
29C3×D4⋊C8—22no—
30#4485—22no—
31#4467—22no—
32#3208—31no—
33#3077—31no—
34#2541—31no—
35#1545—31no—
36#1503—31no—
37#1478D311yescharacteristic, complemented
38#1303—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×#67—24no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12D412yes—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C42C2311yescharacteristic
52#940—42no—
53#940—42no—
54#930—48no—
55C2×(C2×D3)⋊C4—24no—
56#926—21no—
57C2×C4⋊Dic3—21no—
58#924—21no—
59#924—21no—
60C2×C4×Dic3—21no—
61C2×#208—21no—
62C2×#204—24no—
63#874—21no—
64#874—21no—
65#874—21no—
66#874—21no—
67#870—21no—
68#869D411yescharacteristic, complemented
69#869D411yescharacteristic, complemented
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yescomplemented
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8—21no—
77C3×C4×C8—21no—
78#832—22no—
79#832—22no—
80C3⋊#95D412yes—
81#821—24no—
82#808—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yescomplemented
86#533—31no—
87C2×#127—31no—
88C2×#127—61no—
89C2×#121—62no—
90C2×C4×D4C2×D311yescharacteristic
91#492—61no—
92#486—61no—
93#478—62no—
94#476—61no—
95#474—61no—
96#460—62no—
97#450—62no—
98#431C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100#428—61no—
101#423—31no—
102#418—61no—
103#392—62no—
104#390—61no—
105C4×C4⋊C4—31no—
106#331—62no—
107#328—62no—
108D4⋊C8—22no—
109C24×C3—21no—
110C23×D3—22no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4SD1612yescomplemented
117C22×C3×C4—21no—
118C22×C3×C4—44no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C2×#67—48no—
125C2×#67—48no—
126C2×#67—84no—
127C22×Dic3—22no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—44no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—41no—
136C2×D12—41no—
137C2×D12—41no—
138C2×D12—41no—
139C2×D12—82no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#217—41no—
153(C2×D3)⋊C4—48no—
154C4⋊Dic3—24no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—22no—
167#204—84no—
168C22×D4—32no—
169C22×D4—64no—
170C22×D4—61no—
171C22×D4—61no—
172C22×D4D1211yescharacteristic
173C23×C4—21no—
174C2×SD16—64no—
175C2×M16—61no—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178#127—61no—
179#127—61no—
180#127—61no—
181#127—61no—
182#127—122no—
183#127—122no—
184#125—61no—
185C22⋊Q8—61no—
186C22⋊Q8—61no—
187#121—128no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194#117—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4D1211yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—64no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8#6712yescomplemented
207#102—64no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214#95—32no—
215C23×C3—22no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3#12011yescharacteristic
221C22×D3—44no—
222C22×D3—44no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—42no—
226C3×D4—22no—
227C3×D4—22no—
228C3×D4—41no—
229C3×D4—41no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—82no—
235C2×C3×C4—82no—
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12711yescharacteristic
242C2×C3×C4C2×SD1611yescharacteristic
243C2×C3×C4#13511yescharacteristic
244C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
245C2×C3×C4#13711yescharacteristic
246#67—816no—
247C2×Dic3—24no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252D12—44no—
253D12—82no—
254D12—82no—
255D12—82no—
256D12—81no—
257D12—81no—
258D12—81no—
259D12—81no—
260Dic6—42no—
261Dic6—81no—
262C3×C8—42no—
263C3×C8—81no—
264C3⋊C8—42no—
265C24—21no—
266C24—62no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—62no—
275C2×D4—62no—
276C2×D4—124no—
277C2×D4—128no—
278C2×D4—128no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—242no—
288C2×D4—244no—
289C2×D4#20412yescomplemented
290C22×C4—21no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—62no—
295C22×C4C2×D1211yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298SD16—244no—
299M16—121no—
300M16—121no—
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—62no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—64no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—128no—
316C22⋊C4—121no—
317C42—121no—
318C42—121no—
319C42—121no—
320C42#23511yescharacteristic
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C42C2×#6711yescharacteristic
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—44no—
327C22×C3—42no—
328C22×C3—82no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3#39211yescharacteristic
332C22×C3#45011yescharacteristic
333C22×C3#45211yescharacteristic
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—81no—
343C3×C4#44811yescharacteristic
344C3×C4#44911yescharacteristic
345C3×C4#45312yes—
346C3×C4#46511yescharacteristic
347C3×C4#46811yescharacteristic
348Dic3—44no—
349Dic3—81no—
350Dic3—81no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—124no—
357C23—124no—
358C23—122no—
359C23—122no—
360C23—122no—
361C23#88211yescharacteristic
362Q8—122no—
363Q8—241no—
364D4—22no—
365D4—22no—
366D4—41no—
367D4—41no—
368D4—124no—
369D4—242no—
370D4—242no—
371D4—242no—
372D4—2416no—
373D4—241no—
374D4—241no—
375D4—241no—
376D4—241no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—64no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4#88211yescharacteristic
389C2×C4#88411yescharacteristic
390C2×C4#88411yescharacteristic
391C2×C4C2×#20411yescharacteristic
392C2×C4#90411yescharacteristic
393C2×C4#90811yescharacteristic
394C2×C4#90911yescharacteristic
395C2×C4#93011yescharacteristic
396C2×C4#93711yescharacteristic
397C2×C4#94011yescharacteristic
398C8—42no—
399C8—81no—
400C8—122no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—82no—
404C2×C3#200011yescharacteristic
405C2×C3#200211yescharacteristic
406C2×C3#200811yescharacteristic
407D3—162no—
408C22—21no—
409C22—21no—
410C22—44no—
411C22—42no—
412C22—82no—
413C22—81no—
414C22—81no—
415C22—244no—
416C22—244no—
417C22—242no—
418C22—242no—
419C22—242no—
420C22#473111yescharacteristic
421C22#473311yescharacteristic
422C22#496011yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—81no—
427C4—124no—
428C4—241no—
429C4—241no—
430C4#473411yescharacteristic
431C4#473811yescharacteristic
432C4#511411yescharacteristic
433C4#513411yescharacteristic
434C4#519812yes—
435C3#1024611yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—82no—
439C2—482no—
440C2#6996211yescharacteristic
441C2#7032411yescharacteristic
442C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
prime radical#82378C2
hypercenter#1478D3
hypocenterC3#10246
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
2. #82296C2semidirect
3. #79305C2semidirect
4. C3×#1478C2semidirect
5. #70792C2semidirect
6. #70774C2semidirect
7. #67746C2semidirect
37. #1478D3semidirect
68. #869D4semidirect
69. #869D4semidirect
73. C3×C4⋊C8D4semidirect
85. C4⋊(C3⋊C8)D4semidirect
116. C2×C3×D4SD16semidirect
206. C4⋊C8#67semidirect
289. C2×D4#204semidirect
435. C3#10246semidirect

Maximal subgroups

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

subgroupindex
#823782
#822962
#793052
C3×#14782
#707922
#707742
#677462
#102463