Finite Group Databasebeta

← #10698887

Subgroups — #10698887

Order 768.

428 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 99 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#82441C211yesmaximal, characteristic
2C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
3#71637C211yesmaximal, characteristic, complemented
4#70432C211yesmaximal, characteristic
5#69735C211yesmaximal, characteristic
6#69730C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#10459—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#942—21no—
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#836—21no—
15#5031—21no—
16#5028—22no—
17#5027—21no—
18#4966—21no—
19#4924C2211yescharacteristic
20#4924C2211yescharacteristic
21#4924C2211yescharacteristic
22C3×Q8⋊C8—21no—
23C3×Q8⋊C8—21no—
24#4530—21no—
25#4486—21no—
26#4485—21no—
27#4459—21no—
28#4459—21no—
29#4457—22no—
30#3037—31no—
31#1801—31no—
32#1566—31no—
33#1547—31no—
34#1492—31no—
35C2×Q8⋊C8D311yescharacteristic, complemented
36#1292—31no—
37C22×C3×Q8D411yescharacteristic
38C2×#235—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#216—22no—
50#942—42no—
51#942—42no—
52C2×#215—22no—
53#930—21no—
54#930—21no—
55C2×#208—21no—
56C2×#208—21no—
57C2×#208—21no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4×C8—21no—
66C3×C4×C8D411yescharacteristic
67C3×C4×C8D411yescharacteristic
68#836—42no—
69#810—22no—
70#809—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C2×#137—61no—
78#532—31no—
79C2×C4×Q8C2×D311yescharacteristic
80#464—62no—
81#462—61no—
82#460—61no—
83#423—31no—
84#423—31no—
85#423—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87#417—61no—
88C2×#101—61no—
89C2×C4×C8C2×D311yescharacteristic
90#343—61no—
91#334—62no—
92#332—61no—
93#331—61no—
94Q8⋊C8—21no—
95Q8⋊C8—21no—
96D4⋊C8—61no—
97D4⋊C8—61no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8SD1612yes—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—41no—
110#235—41no—
111#235—41no—
112#235—41no—
113#235—41no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×C4×D3—41no—
118C2×Dic6—21no—
119C2×C3×C8—21no—
120C2×C3×C8—21no—
121C2×C3×C8—42no—
122C2×C3×C8—41no—
123C2×C3×C8—81no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C42—41no—
129C3×C42—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134#216—44no—
135#215—44no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140#208—41no—
141#208—41no—
142#208—41no—
143#208—41no—
144#208—41no—
145#208—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×#41—61no—
150C22×Q8D1211yescharacteristic
151#137—122no—
152#137—122no—
153C2×Q16—62no—
154C2×SD16—62no—
155C2×M16—61no—
156C2×M16—61no—
157C2×M16—61no—
158C22×C8—21no—
159C22×C8—21no—
160C4⋊Q8—31no—
161#127—31no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4D1211yescharacteristic
173C2×C42—21no—
174C2×C42C22×D311yescharacteristic
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—32no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183#103—62no—
184#102—62no—
185#101—122no—
186C4×C8—21no—
187C4×C8#6711yescharacteristic
188C4×C8#6711yescharacteristic
189C23×C3#12311yescharacteristic
190C22×D3—41no—
191C3×Q8—22no—
192C3×Q8—22no—
193C3×Q8—41no—
194C3×Q8—41no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—81no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12311yescharacteristic
208C2×C3×C4#12411yescharacteristic
209C2×C3×C4C2×SD1611yescharacteristic
210C2×C3×C4#13511yescharacteristic
211C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213#67—81no—
214#67—81no—
215C2×Dic3—41no—
216D12—41no—
217D12—41no—
218D12—41no—
219D12—41no—
220D12—81no—
221C4×D3—81no—
222C4×D3—81no—
223Dic6—41no—
224Dic6—41no—
225C3×C8—22no—
226C3×C8—41no—
227C3×C8—81no—
228C3⋊C8—42no—
229C3⋊C8—81no—
230C3⋊C8—81no—
231#41—121no—
232#41—121no—
233#41—121no—
234#41—121no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—61no—
241C2×Q8#20412yes—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—121no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—121no—
249C22×C4C2×D1211yescharacteristic
250C22×C4#23711yescharacteristic
251C22×C4#23711yescharacteristic
252Q16—124no—
253SD16—124no—
254M16—121no—
255M16—121no—
256M16—121no—
257M16—121no—
258M16—121no—
259M16—121no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—62no—
265C2×C8—81no—
266C2×C8—121no—
267C2×C8—121no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—41no—
277C42—41no—
278C42#23511yescharacteristic
279C42D3×D411yescharacteristic
280C42D3×D411yescharacteristic
281C42C2×#6711yescharacteristic
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—42no—
294C3×C4—81no—
295C3×C4#44911yescharacteristic
296C3×C4#45011yescharacteristic
297C3×C4#46511yescharacteristic
298C3×C4#46911yescharacteristic
299C3×C4#47111yescharacteristic
300C3×C4#47311yescharacteristic
301Dic3—81no—
302C23—121no—
303C23#88611yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—41no—
307Q8—41no—
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—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—121no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4#88211yescharacteristic
329C2×C4#88311yescharacteristic
330C2×C4#88511yescharacteristic
331C2×C4C2×#20411yescharacteristic
332C2×C4#90411yescharacteristic
333C2×C4#90811yescharacteristic
334C2×C4#90811yescharacteristic
335C2×C4#93011yescharacteristic
336C2×C4#93811yescharacteristic
337C2×C4#93911yescharacteristic
338C8—22no—
339C8—41no—
340C8—81no—
341C8—122no—
342C8—241no—
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#473511yescharacteristic
356C22#473711yescharacteristic
357C22#496111yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—81no—
363C4—241no—
364C4#473111yescharacteristic
365C4#473411yescharacteristic
366C4#511411yescharacteristic
367C4#513511yescharacteristic
368C4#517711yescharacteristic
369C4#519211yescharacteristic
370C3#1045911yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—481no—
374C2#6993511yescharacteristic
375C2#7026411yescharacteristic
376C2#7037911yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC22×C3#394
prime radical#5477C22
hypercenterC2×Q8⋊C8D3
hypocenterC3#10459
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8⋊C8C2semidirect
3. #71637C2semidirect
6. #69730C2semidirect
7. #67705C2semidirect
35. C2×Q8⋊C8D3semidirect
370. C3#10459semidirect

Maximal subgroups

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

subgroupindex
#824412
C2×C3×Q8⋊C82
#716372
#704322
#697352
#697302
#677052
#104593