Finite Group Databasebeta

← #10696509

Subgroups — #10696509

Order 768.

600 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 97 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×C3×#489C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic, complemented
3#83007C211yesmaximal, characteristic
4#71786C211yesmaximal, characteristic, complemented
5C2×#4540C211yesmaximal, characteristic, complemented
6C2×#4464C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10085—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#489—22no—
13C3×#489—22no—
14C2×C3×#103—21no—
15C2×C3×#102—21no—
16C2×C3×#97C2211yescharacteristic
17#5231—21no—
18C2×#835—22no—
19#5223—21no—
20#5089—22no—
21#5089—22no—
22C2×#810—21no—
23C2×#809—21no—
24#4961—22no—
25#4960—21no—
26#4947—21no—
27#4936C2211yescharacteristic
28C2×#803C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4540—22no—
31#4464—22no—
32C2×#489—31no—
33C2×#489D311yescharacteristic, complemented
34#2819—31no—
35#1960—31no—
36C2×#329—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C22×#67—22no—
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C22×C3×C8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8D412yescomplemented
50C3×#124—21no—
51C3×#124—21no—
52C3×#124—21no—
53C3×#124—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—22no—
63C2×C4⋊Dic3D412yes—
64C22×C3⋊C8—21no—
65C22×C3⋊C8—21no—
66#875—21no—
67#875—21no—
68#875—21no—
69#875—21no—
70C3×#103—24no—
71C3×#102—24no—
72C3×#97—21no—
73C3×#97D412yescomplemented
74#835—44no—
75C3⋊#95—22no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#810—24no—
79#809—24no—
80#803—21no—
81#803—21no—
82#803—21no—
83#803D412yescomplemented
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124—31no—
86C2×#124C2×D311yescharacteristic
87#489—22no—
88#489—22no—
89#489—62no—
90#489—62no—
91C2×#103—21no—
92C2×#103—61no—
93C2×#103—62no—
94C2×#102—21no—
95C2×#102—61no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97C2×D311yescharacteristic
99#394—62no—
100#393—61no—
101#393—61no—
102#392—61no—
103#392—61no—
104#382—31no—
105#329—62no—
106#329—62no—
107C24×C3—21no—
108C23×D3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×Q8—42no—
115C2×C3×Q8—41no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C22×C3×C4—21no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C2×#67—82no—
125C2×#67—82no—
126C2×#67—82no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—42no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—42no—
135C2×C3×C8—22no—
136C2×C3×C8—42no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—82no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—82no—
155C4⋊Dic3—42no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—42no—
160C2×C3⋊C8—41no—
161C22×Q8—21no—
162C22×Q8—31no—
163C22×Q8#6711yescharacteristic
164C22×D4—31no—
165C22×D4—62no—
166C22×D4#6711yescharacteristic
167C22×C8—21no—
168C22×C8—61no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8#6712yescomplemented
174#124—21no—
175#124—21no—
176#124—21no—
177#124—21no—
178#124—61no—
179#124—61no—
180#124—61no—
181#124—61no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—32no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C42C22×D311yescharacteristic
192#103—24no—
193#103—64no—
194#103—124no—
195#102—24no—
196#102—64no—
197#97—21no—
198#97—32no—
199#97—61no—
200#97—61no—
201#97—61no—
202#97#6712yescomplemented
203#95—62no—
204#95—61no—
205#95—61no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3#12011yescharacteristic
212C22×D3—41no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—42no—
216C22×D3—42no—
217C3×Q8—42no—
218C3×Q8—44no—
219C3×Q8—81no—
220C3×Q8—82no—
221C3×D4—42no—
222C3×D4—81no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12711yescharacteristic
238#67—88no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244D12—42no—
245D12—81no—
246Dic6—42no—
247Dic6—81no—
248C3×C8—42no—
249C3⋊C8—42no—
250C3⋊C8—42no—
251C24—21no—
252C24—61no—
253C2×Q8—22no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—122no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—42no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—122no—
269C2×D4—242no—
270C2×D4—242no—
271C2×D4—242no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—62no—
275C22×C4—61no—
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4C2×#6711yescharacteristic
279C2×C8—22no—
280C2×C8—42no—
281C2×C8—62no—
282C2×C8—62no—
283C2×C8—121no—
284C2×C8—122no—
285C2×C8—121no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—122no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—242no—
300C22⋊C4—242no—
301C42D3×D411yescharacteristic
302C42#23711yescharacteristic
303C42C2×#6711yescharacteristic
304C42C2×#6711yescharacteristic
305C22×C3—82no—
306C22×C3—82no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3#39211yescharacteristic
311C22×C3#45311yescharacteristic
312C22×C3#45411yescharacteristic
313C22×C3#45512yes—
314C22×C3#45512yes—
315C2×D3—82no—
316C2×D3—82no—
317C2×D3—81no—
318C2×D3—81no—
319C2×D3—81no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—81no—
327C3×C4—82no—
328Dic3—82no—
329Dic3—81no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—41no—
334C23—42no—
335C23—121no—
336C23—121no—
337C23—122no—
338C23—121no—
339C23—122no—
340C23#93711yescharacteristic
341Q8—42no—
342Q8—44no—
343Q8—81no—
344Q8—82no—
345Q8—122no—
346Q8—241no—
347D4—42no—
348D4—81no—
349D4—122no—
350D4—241no—
351D4—248no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—121no—
365C2×C4—122no—
366C2×C4#93711yescharacteristic
367C2×C4#93811yescharacteristic
368C2×C4#93911yescharacteristic
369C2×C4#93911yescharacteristic
370C2×C4#94011yescharacteristic
371C2×C4#95311yescharacteristic
372C8—42no—
373C8—122no—
374C8—122no—
375C2×C3—161no—
376C2×C3#165912yes—
377C2×C3#166212yes—
378C2×C3#200411yescharacteristic
379C2×C3#200411yescharacteristic
380C2×C3#200611yescharacteristic
381D3—161no—
382C22—81no—
383C22—82no—
384C22—82no—
385C22—81no—
386C22—81no—
387C22—241no—
388C22—242no—
389C22—242no—
390C22—241no—
391C22—241no—
392C22#506112yes—
393C22#519811yescharacteristic
394C22#520111yescharacteristic
395C22#520212yes—
396C22#522311yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—81no—
404C4—82no—
405C4—242no—
406C4—241no—
407C3#1008511yescharacteristic, complemented, Sylow
408C2—161no—
409C2—481no—
410C2#7124412yes—
411C2#7124612yes—
412C2#7173111yescharacteristic
413C2#7209811yescharacteristic
414C2#7210611yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#489C2
socleC23×C3#120
prime radical#83007C2
hypercenterC2×#489D3
hypocenterC3#10085
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×C3×#489C2semidirect
2. C2×#5089C2semidirect
4. #71786C2semidirect
5. C2×#4540C2semidirect
6. C2×#4464C2semidirect
7. #67659C2semidirect
33. C2×#489D3semidirect
49. C3×C4⋊Q8D4semidirect
73. C3×#97D4semidirect
83. #803D4semidirect
173. C4⋊Q8#67semidirect
202. #97#67semidirect
407. C3#10085semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4892
C2×#50892
#830072
#717862
C2×#45402
C2×#44642
#676592
#100853