Finite Group Databasebeta

← #10697613

Subgroups — #10697613

Order 768.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 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#79271C211yesmaximal, characteristic, complemented
2#79059C211yesmaximal, characteristic
3#73802C211yesmaximal, characteristic
4#70756C211yesmaximal, characteristic
5#70506C211yesmaximal, characteristic, complemented
6#69471C211yesmaximal, characteristic, complemented
7C3×#1303C211yesmaximal, characteristic, complemented
8#10247—31nomaximal, Sylow
9C2×#875—21no—
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5131—21no—
17#5121—21no—
18#4987—22no—
19#4929—21no—
20#4925C2211yescharacteristic
21#4851—22no—
22#4847—22no—
23#4846—21no—
24#4842—21no—
25#4721—22no—
26#4681—21no—
27#4680—21no—
28#4674—22no—
29#4483—22no—
30#4482—22no—
31#4452—22no—
32#3229—31no—
33#3077—31no—
34#2547—31no—
35#1546—31no—
36#1503—31no—
37#1478—31no—
38#1303D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6—21no—
42C2×C3×M16—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—24no—
47C2×(C2×D3)⋊C4—21no—
48#926—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D412yes—
52#924—21no—
53#924—21no—
54#875—42no—
55#875—42no—
56C4×D12—21no—
57C4×D12—21no—
58C4×D12—21no—
59C4×D12—21no—
60#870—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C4⋊C8D412yes—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74C3×#95D412yes—
75C3⋊#95—24no—
76#830—21no—
77#821—22no—
78#821—22no—
79#818—22no—
80#817—22no—
81#817—24no—
82#816—24no—
83C4⋊(C3⋊C8)—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×C4⋊Q8—31no—
87C2×#124—61no—
88C2×C4×D4—31no—
89#498—61no—
90#487—61no—
91#483—62no—
92#478—62no—
93#476—61no—
94#475—62no—
95#474—61no—
96#431—31no—
97#431C2×D311yescharacteristic
98#428—61no—
99#423C2×D311yescharacteristic
100#418—61no—
101#397—62no—
102#391—61no—
103#380—62no—
104C4×C4⋊C4C2×D311yescharacteristic
105#332—62no—
106#328—62no—
107D4⋊C8—62no—
108C23×D3—21no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—22no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12SD1612yes—
121C2×C4×D3—44no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C3×M16—41no—
133C3×M16—41no—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C3×C4⋊C4—42no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—41no—
144(C2×D3)⋊C4—48no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147C4⋊Dic3—22no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—22no—
154C4⋊Dic3—21no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—41no—
160C2×C3⋊C8—41no—
161C22×Q8—61no—
162C22×D4—31no—
163C23×C4—61no—
164C2×M16—21no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171#125—61no—
172#124—122no—
173#124—122no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182#117—61no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—61no—
187C2×C22⋊C4—64no—
188C2×C22⋊C4—61no—
189C2×C42—21no—
190C2×C42C22×D311yescharacteristic
191#107—62no—
192#106—62no—
193#106—64no—
194C4⋊C8—21no—
195C4⋊C8—61no—
196C4⋊C8D1212yes—
197C4⋊C8D1212yes—
198#103—64no—
199#102—62no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C4×C8—21no—
204C4×C8—61no—
205#95—64no—
206#95D1212yes—
207C23×C3#12011yescharacteristic
208C22×D3—22no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—41no—
212C22×D3—41no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×C3×C4C2×SD1611yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235D12—22no—
236D12—22no—
237D12—41no—
238D12—41no—
239C4×D3—82no—
240C4×D3—82no—
241Dic6—42no—
242Dic6—81no—
243Dic6—81no—
244Dic6—81no—
245Dic6—81no—
246Dic6—81no—
247C3×C8—42no—
248C3×C8—42no—
249C3⋊C8—81no—
250C24—61no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×Q8—121no—
257C2×Q8—121no—
258C2×Q8—121no—
259C2×D4—32no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C22×C4—22no—
266C22×C4—61no—
267C22×C4—62no—
268C22×C4—124no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4C2×D1211yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4D3×D411yescharacteristic
274M16—41no—
275M16—41no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—121no—
279C4⋊C4—42no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—62no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—62no—
287C4⋊C4—61no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C22⋊C4—128no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C42—41no—
297C42—41no—
298C42—121no—
299C42C2×D1211yescharacteristic
300C42C2×D1211yescharacteristic
301C42D3×D411yescharacteristic
302C42#23911yescharacteristic
303C22×C3—21no—
304C22×C3—21no—
305C22×C3#39211yescharacteristic
306C22×C3#45011yescharacteristic
307C22×C3#45211yescharacteristic
308C2×D3—44no—
309C2×D3—42no—
310C2×D3—82no—
311C2×D3—81no—
312C2×D3—81no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—82no—
317C3×C4#44811yescharacteristic
318C3×C4#44911yescharacteristic
319C3×C4#45412yes—
320C3×C4#46511yescharacteristic
321C3×C4#46811yescharacteristic
322Dic3—82no—
323Dic3—81no—
324Dic3—81no—
325Dic3—81no—
326C23—62no—
327C23—122no—
328C23—122no—
329C23—121no—
330C23—121no—
331C23#88211yescharacteristic
332Q8—122no—
333Q8—241no—
334Q8—241no—
335Q8—241no—
336Q8—241no—
337Q8—241no—
338D4—62no—
339D4—62no—
340D4—121no—
341D4—121no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—242no—
355C2×C4—242no—
356C2×C4#87411yescharacteristic
357C2×C4#88211yescharacteristic
358C2×C4#88211yescharacteristic
359C2×C4#89511yescharacteristic
360C2×C4#89511yescharacteristic
361C2×C4#89511yescharacteristic
362C2×C4#91111yescharacteristic
363C2×C4D3×SD1611yescharacteristic
364C2×C4#91611yescharacteristic
365C2×C4#91811yescharacteristic
366C8—42no—
367C8—42no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3#200011yescharacteristic
372C2×C3#200311yescharacteristic
373C2×C3#200911yescharacteristic
374D3—82no—
375C22—21no—
376C22—21no—
377C22—124no—
378C22—122no—
379C22—242no—
380C22—241no—
381C22—241no—
382C22#467111yescharacteristic
383C22#477611yescharacteristic
384C22#480811yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—242no—
390C4—241no—
391C4—241no—
392C4—241no—
393C4#478111yescharacteristic
394C4#480611yescharacteristic
395C4#482712yes—
396C4#486411yescharacteristic
397C4#488611yescharacteristic
398C3#1024711yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—242no—
402C2#6876711yescharacteristic
403C2#6888711yescharacteristic
404C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1303D3
hypocenterC3#10247
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
1. #79271C2semidirect
5. #70506C2semidirect
6. #69471C2semidirect
7. C3×#1303C2semidirect
38. #1303D3semidirect
398. C3#10247semidirect

Maximal subgroups

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

subgroupindex
#792712
#790592
#738022
#707562
#705062
#694712
C3×#13032
#102473