Finite Group Databasebeta

← #10696656

Subgroups — #10696656

Order 768.

574 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 87 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#83213C211yesmaximal, characteristic, complemented
2#83203C211yesmaximal, characteristic, complemented
3#79064C211yesmaximal, characteristic
4C3×#1997C211yesmaximal, characteristic, complemented
5#69859C211yesmaximal, characteristic, complemented
6#69853C211yesmaximal, characteristic, complemented
7#67710C211yesmaximal, characteristic, complemented
8#10106—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#942—22no—
11#5477C2211yescharacteristic
12#5477C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5227—22no—
16#5135—22no—
17#5092—21no—
18#5086—22no—
19#5085—21no—
20#5085—21no—
21#5083—22no—
22#5079—21no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4843—21no—
26#4843—21no—
27#4842—21no—
28#4842—21no—
29C3×#343—22no—
30#4464—22no—
31#4459—22no—
32#3253—31no—
33#3206—31no—
34#3152—31no—
35#1997D311yescharacteristic, complemented
36#1525—31no—
37#1484—31no—
38#1299—31no—
39C22×C3×Q8D412yes—
40C2×#235—21no—
41C2×#235—21no—
42C2×C3×M16—21no—
43C2×C3×M16D412yes—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—24no—
52#942—48no—
53C2×#215—24no—
54C2×#215—24no—
55#930—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59C2×#208—21no—
60C2×D24—22no—
61#875—21no—
62#875—21no—
63#874D411yescharacteristic
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69#869D411yescharacteristic
70#869D411yescharacteristic
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×#97—21no—
74C3×C4×C8—21no—
75#835—24no—
76#830—21no—
77#821—22no—
78#821—22no—
79#817—22no—
80C4⋊(C3⋊C8)D412yes—
81#803D412yes—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84C2×#137—62no—
85#532—31no—
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88#497—62no—
89#494—61no—
90#492—61no—
91#492—61no—
92#487—61no—
93#476—61no—
94#476—61no—
95#469—62no—
96#461—62no—
97#459—61no—
98#459—61no—
99#448—62no—
100#431—31no—
101#423C2×D311yescharacteristic
102C4×M16—31no—
103C4×M16C2×D311yescharacteristic
104#343—22no—
105#329—62no—
106D4⋊C8—62no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—44no—
110C22×C3×C4—22no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C2×#67—41no—
115C2×#67—41no—
116#235—44no—
117#235—82no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C3×M16—22no—
129C3×M16—41no—
130C3×M16—42no—
131C2×C3×C8—22no—
132C2×C3×C8—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141#216—48no—
142#215—48no—
143#215—84no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152#208—24no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155D24—82no—
156C2×#41—61no—
157C2×#41—61no—
158C22×Q8D1212yes—
159#137—128no—
160C2×Q16—64no—
161C2×SD16—64no—
162C2×SD16—64no—
163C2×D8—62no—
164C2×M16—21no—
165C2×M16—61no—
166C2×M16#6712yes—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173#127—31no—
174#127—31no—
175#124—61no—
176#124—61no—
177#121—61no—
178#121—61no—
179#121—61no—
180#121—61no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C4×D4—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—32no—
192#103—64no—
193#102—62no—
194#102—62no—
195C22⋊C8—61no—
196#97—21no—
197#97—32no—
198C4×C8—21no—
199C4×C8—61no—
200C4×C8—61no—
201C23×C3#12111yescharacteristic
202C22×D3—41no—
203C22×D3—41no—
204C3×Q8—44no—
205C3×Q8—44no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—42no—
211C2×C3×C4—82no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218#67—82no—
219#67—82no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222D12—42no—
223D12—42no—
224D12—41no—
225D12—41no—
226D12—41no—
227D12—41no—
228D12—42no—
229D12—81no—
230C4×D3—82no—
231C4×D3—82no—
232Dic6—42no—
233Dic6—81no—
234C3×C8—42no—
235C3×C8—42no—
236C3⋊C8—24no—
237C3⋊C8—42no—
238#41—124no—
239#41—242no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—121no—
252C2×D4—121no—
253C22×C4—22no—
254C22×C4—121no—
255C22×C4—121no—
256C22×C4C2×D1211yescharacteristic
257C22×C4#23511yescharacteristic
258C22×C4C2×#6711yescharacteristic
259Q16—128no—
260SD16—128no—
261SD16—244no—
262D8—242no—
263M16—22no—
264M16—41no—
265M16—42no—
266M16—64no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—62no—
270C2×C8—62no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3#39211yescharacteristic
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—81no—
296C2×D3—81no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—82no—
301C3×C4#45312yes—
302C3×C4#45511yescharacteristic
303C3×C4#45611yescharacteristic
304C3×C4#47112yes—
305Dic3—81no—
306Dic3—81no—
307C23—121no—
308C23—121no—
309C23#93011yescharacteristic
310Q8—44no—
311Q8—44no—
312Q8—122no—
313Q8—241no—
314D4—122no—
315D4—122no—
316D4—122no—
317D4—121no—
318D4—121no—
319D4—121no—
320D4—121no—
321D4—241no—
322D4—242no—
323D4—242no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—242no—
333C2×C4—242no—
334C2×C4#88211yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4#93711yescharacteristic
339C2×C4#94011yescharacteristic
340C8—42no—
341C8—42no—
342C8—64no—
343C8—122no—
344C2×C3—41no—
345C2×C3#200211yescharacteristic
346C2×C3#200711yescharacteristic
347C2×C3#201911yescharacteristic
348D3—161no—
349D3—161no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—241no—
354C22—241no—
355C22—241no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22#496011yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—241no—
365C4—241no—
366C4#474712yes—
367C4#475412yes—
368C4#506111yescharacteristic
369C4#509511yescharacteristic
370C3#1010611yescharacteristic, complemented, Sylow
371C2—41no—
372C2—481no—
373C2—481no—
374C2#6969411yescharacteristic
375C2#7008211yescharacteristic
376C2#7011711yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#392
prime radical#79064C2
hypercenter#1997D3
hypocenterC3#10106
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
1. #83213C2semidirect
2. #83203C2semidirect
4. C3×#1997C2semidirect
5. #69859C2semidirect
6. #69853C2semidirect
7. #67710C2semidirect
35. #1997D3semidirect
370. C3#10106semidirect

Maximal subgroups

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

subgroupindex
#832132
#832032
#790642
C3×#19972
#698592
#698532
#677102
#101063