Finite Group Databasebeta

← #10696490

Subgroups — #10696490

Order 768.

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

normal subgroups only

This group has 590 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
1C2×#5081C211yesmaximal, characteristic, complemented
2C2×#5086C211yesmaximal, characteristic
3#74255C211yesmaximal, characteristic
4C3×#1993C211yesmaximal, characteristic, complemented
5C2×#4465C211yesmaximal, characteristic, complemented
6C2×#4464C211yesmaximal, characteristic
7#67630C211yesmaximal, characteristic, complemented
8#10082—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—21no—
11C22×#215—21no—
12C2×#875C2211yescharacteristic
13C2×#870C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5086—22no—
17#5086—22no—
18#5081—22no—
19#5081—22no—
20#4994—22no—
21C2×#808—21no—
22C2×#807—21no—
23C2×#803C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4677—21no—
26#4674—21no—
27#4672—21no—
28#4656—21no—
29C3×#340—22no—
30#4465—22no—
31#4464—22no—
32C2×#501—31no—
33C2×#497—31no—
34#2820—31no—
35#1993D311yescharacteristic, complemented
36C2×#330—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D412yes—
50C2×C3×C42C2311yescharacteristic
51C2×#216—24no—
52C2×#216—44no—
53C2×#215—24no—
54C2×#215—44no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×C4⋊Dic3—22no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8—21no—
66#875—21no—
67#875—21no—
68#875—21no—
69#875—21no—
70#870—21no—
71#870—21no—
72#870—21no—
73#870—21no—
74C3×#97—21no—
75C3×#97—21no—
76C3×#95—21no—
77C3×#95—21no—
78C3⋊#95—22no—
79#808—24no—
80#807—24no—
81#803—21no—
82#803—21no—
83#803D412yes—
84#803D412yes—
85C22×Q16—61no—
86C22×SD16—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#125—31no—
89C2×#124—31no—
90#501—62no—
91#501—62no—
92#497—62no—
93#497—62no—
94C2×#107—61no—
95C2×#106—61no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97C2×D311yescharacteristic
99#399—61no—
100#397—61no—
101#396—62no—
102#395—61no—
103#394—61no—
104#382C2×D311yescharacteristic
105#340—22no—
106#330—62no—
107#329—62no—
108C23×D3—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C2×C3×Q8—41no—
113C22×C3×C4—22no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—42no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—42no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—42no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#216—48no—
140#215—48no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—82no—
146(C2×D3)⋊C4—82no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—82no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C2×C3⋊C8—24no—
157C2×C3⋊C8—24no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C22×Q8—21no—
161C22×Q8—31no—
162C22×D4—31no—
163C2×Q16—64no—
164C2×Q16—124no—
165C2×SD16—64no—
166C2×SD16—124no—
167C22×C8—21no—
168C22×C8—61no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8D1212yes—
174#125—61no—
175#125—61no—
176#125—61no—
177#125—61no—
178#124—61no—
179#124—61no—
180#124—61no—
181#124—61no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4#6712yes—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C42C22×D311yescharacteristic
195#107—64no—
196#106—64no—
197#97—21no—
198#97—21no—
199#97—32no—
200#97—32no—
201#97—61no—
202#97—61no—
203#95—21no—
204#95—21no—
205#95—62no—
206C23×C3#12011yescharacteristic
207C22×D3—41no—
208C22×D3—42no—
209C22×D3—41no—
210C22×D3—42no—
211C22×D3—41no—
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—82no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—82no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237D12—42no—
238D12—81no—
239Dic6—42no—
240Dic6—81no—
241C3×C8—42no—
242C3⋊C8—24no—
243C3⋊C8—24no—
244C24—61no—
245C2×Q8—22no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—122no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—122no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4D3×D411yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4C2×#6711yescharacteristic
263Q16—128no—
264SD16—128no—
265C2×C8—22no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—64no—
269C2×C8—64no—
270C2×C8—62no—
271C2×C8—62no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—242no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—242no—
291C22⋊C4—242no—
292C42C2×D1211yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C22×C3#39211yescharacteristic
297C22×C3#45312yes—
298C22×C3#45511yescharacteristic
299C22×C3#45612yes—
300C22×C3#45611yescharacteristic
301C2×D3—82no—
302C2×D3—82no—
303C2×D3—81no—
304C2×D3—81no—
305C2×D3—81no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—82no—
313C3×C4—82no—
314Dic3—81no—
315Dic3—81no—
316Dic3—81no—
317C23—122no—
318C23—122no—
319C23—121no—
320C23—121no—
321C23—121no—
322C23#93711yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—82no—
326Q8—122no—
327Q8—241no—
328D4—122no—
329D4—241no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4#88211yescharacteristic
347C2×C4#88211yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4#93011yescharacteristic
351C2×C4#94011yescharacteristic
352C8—42no—
353C8—64no—
354C8—64no—
355C2×C3#166012yes—
356C2×C3#166112yes—
357C2×C3#200211yescharacteristic
358C2×C3#200511yescharacteristic
359C2×C3#200711yescharacteristic
360D3—161no—
361C22—241no—
362C22—242no—
363C22—241no—
364C22—241no—
365C22—242no—
366C22#496011yescharacteristic
367C22#506111yescharacteristic
368C22#509511yescharacteristic
369C22#519812yes—
370C22#520512yes—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—82no—
379C4—241no—
380C4—241no—
381C4—241no—
382C3#1008211yescharacteristic, complemented, Sylow
383C2—481no—
384C2#6969411yescharacteristic
385C2#7032411yescharacteristic
386C2#7034611yescharacteristic
387C2#7124512yes—
388C2#7139112yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1993D3
hypocenterC3#10082
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5081C2semidirect
4. C3×#1993C2semidirect
5. C2×#4465C2semidirect
7. #67630C2semidirect
35. #1993D3semidirect
382. C3#10082semidirect

Maximal subgroups

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

subgroupindex
C2×#50812
C2×#50862
#742552
C3×#19932
C2×#44652
C2×#44642
#676302
#100823