Finite Group Databasebeta

← #10698889

Subgroups — #10698889

Order 768.

428 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 95 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#79067C211yesmaximal, characteristic
2#70528C211yesmaximal, characteristic, complemented
3#70498C211yesmaximal, characteristic
4#70490C211yesmaximal, characteristic, complemented
5#70432C211yesmaximal, characteristic
6C2×#4484C211yesmaximal, characteristic, complemented
7C3×#1292C211yesmaximal, characteristic, complemented
8#10459—31nomaximal, Sylow
9C2×#909—21no—
10#5477C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#824—21no—
16#5133—21no—
17#4924C2211yescharacteristic
18C2×C4×C3⋊C8C2211yescharacteristic
19#4849—21no—
20#4848—21no—
21#4845—22no—
22#4554—21no—
23#4486—21no—
24#4485—21no—
25#4484—21no—
26#4484—21no—
27#4483—21no—
28#4483—21no—
29#4480—22no—
30#3037—31no—
31#1801—31no—
32#1566—31no—
33#1547—31no—
34#1492—31no—
35C2×Q8⋊C8—31no—
36#1292D311yescharacteristic, complemented
37C2×#235—21no—
38C22×Dic6D411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×M16—21no—
41C22×C3×C8—21no—
42C2×C3×C42C2311yescharacteristic
43#930—21no—
44#930—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×C4×Dic3—21no—
48C2×#208—21no—
49C22×C3⋊C8—21no—
50#909—42no—
51#909—42no—
52C2×Dic12—22no—
53C2×#204—22no—
54#875—21no—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869D411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69#824—42no—
70#821—22no—
71#816—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D411yescharacteristic
76C4×C3⋊C8D411yescharacteristic
77C2×#137—61no—
78#532—31no—
79C2×C4×Q8—31no—
80#464—62no—
81#462—61no—
82#460—61no—
83#423—31no—
84#423C2×D311yescharacteristic
85#423C2×D311yescharacteristic
86C2×C4⋊C8C2×D311yescharacteristic
87#417—61no—
88C2×#101—61no—
89C2×C4×C8—31no—
90#343—61no—
91#334—62no—
92#332—61no—
93#331—61no—
94Q8⋊C8—61no—
95Q8⋊C8—61no—
96D4⋊C8—61no—
97D4⋊C8—61no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C2×#67—41no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104#235—41no—
105#235—41no—
106#235—41no—
107#235—41no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×C4×D3—41no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6SD1612yes—
119C3×M16—41no—
120C3×M16—41no—
121C3×M16—41no—
122C3×M16—41no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—81no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139C4×Dic3—41no—
140C4×Dic3—41no—
141#208—41no—
142#208—41no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—41no—
147Dic12—44no—
148#204—44no—
149C2×#41—61no—
150C22×Q8—31no—
151#137—122no—
152#137—122no—
153C2×Q16—62no—
154C2×SD16—62no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—61no—
158C22×C8—21no—
159C22×C8—61no—
160C4⋊Q8—31no—
161#127—31no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×Q8—61no—
166C4×Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C42—61no—
174C2×C42C22×D311yescharacteristic
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8D1212yes—
183#103—62no—
184#102—62no—
185#101—122no—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8—61no—
189C23×C3#12311yescharacteristic
190C22×D3—41no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12311yescharacteristic
199C2×C3×C4#12411yescharacteristic
200C2×C3×C4C2×SD1611yescharacteristic
201C2×C3×C4#13511yescharacteristic
202C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
203C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
204#67—81no—
205#67—81no—
206C2×Dic3—22no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—81no—
212D12—41no—
213D12—41no—
214D12—41no—
215D12—41no—
216D12—81no—
217C4×D3—81no—
218C4×D3—81no—
219Dic6—22no—
220Dic6—22no—
221Dic6—41no—
222Dic6—41no—
223Dic6—41no—
224Dic6—41no—
225C3×C8—42no—
226C3×C8—81no—
227C3×C8—81no—
228C3⋊C8—22no—
229C3⋊C8—41no—
230C3⋊C8—81no—
231#41—121no—
232#41—121no—
233#41—121no—
234#41—121no—
235C2×Q8—32no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—121no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—121no—
249C22×C4#23511yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4#23911yescharacteristic
252Q16—124no—
253SD16—124no—
254M16—41no—
255M16—41no—
256M16—41no—
257M16—41no—
258M16—121no—
259M16—121no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—81no—
266C2×C8—122no—
267C2×C8—121no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—121no—
277C42—121no—
278C42C2×D1211yescharacteristic
279C42#23511yescharacteristic
280C42D3×D411yescharacteristic
281C42D3×D411yescharacteristic
282C22×C3—21no—
283C22×C3—21no—
284C22×C3#39411yescharacteristic
285C22×C3#48111yescharacteristic
286C22×C3#48211yescharacteristic
287C2×D3—81no—
288C2×D3—81no—
289C2×D3—81no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4#44911yescharacteristic
294C3×C4#45011yescharacteristic
295C3×C4#46511yescharacteristic
296C3×C4#46911yescharacteristic
297C3×C4#47111yescharacteristic
298C3×C4#47311yescharacteristic
299Dic3—42no—
300Dic3—81no—
301Dic3—81no—
302C23—121no—
303C23#89411yescharacteristic
304Q8—62no—
305Q8—62no—
306Q8—121no—
307Q8—121no—
308Q8—121no—
309Q8—121no—
310D4—121no—
311D4—121no—
312D4—121no—
313D4—121no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—62no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4#87511yescharacteristic
329C2×C4#88211yescharacteristic
330C2×C4#88411yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#89411yescharacteristic
333C2×C4#89511yescharacteristic
334C2×C4#91111yescharacteristic
335C2×C4D3×SD1611yescharacteristic
336C2×C4#91411yescharacteristic
337C2×C4#91911yescharacteristic
338C8—42no—
339C8—62no—
340C8—81no—
341C8—81no—
342C8—121no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#203411yescharacteristic
347C2×C3#205011yescharacteristic
348C2×C3#205211yescharacteristic
349D3—161no—
350C22—21no—
351C22—21no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22#467211yescharacteristic
356C22#486311yescharacteristic
357C22#488511yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—122no—
362C4—241no—
363C4—241no—
364C4#474811yescharacteristic
365C4#475511yescharacteristic
366C4#477611yescharacteristic
367C4#478211yescharacteristic
368C4#480711yescharacteristic
369C4#481011yescharacteristic
370C3#1045911yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—481no—
374C2#6871911yescharacteristic
375C2#6882311yescharacteristic
376C2#6898711yescharacteristic
subgroupG/H
centerC22#4672
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#394
prime radical#5477C22
hypercenter#1292D3
hypocenterC3#10459
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #70528C2semidirect
4. #70490C2semidirect
6. C2×#4484C2semidirect
7. C3×#1292C2semidirect
36. #1292D3semidirect
370. C3#10459semidirect

Maximal subgroups

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

subgroupindex
#790672
#705282
#704982
#704902
#704322
C2×#44842
C3×#12922
#104593