Finite Group Databasebeta

← #10705646

Subgroups — #10705646

Order 768.

500 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 83 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#82284C211yesmaximal, characteristic
2C3×#1740C211yesmaximal, characteristic, complemented
3#71572C211yesmaximal, characteristic, complemented
4#67492C212yesmaximal, complemented
5#67489C212yesmaximal
6#11754—31nomaximal, Sylow
7C2×C3×#117—21no—
8C2×C3×#117C2211yescharacteristic
9C2×C4×D12C411yescharacteristic, complemented
10#5213—21no—
11#4991C411yescharacteristic, complemented
12#4989—21no—
13#4986C411yescharacteristic, complemented
14#4960C2211yescharacteristic
15#4956—21no—
16#4949C2211yescharacteristic
17#4940—21no—
18#4937C411yescharacteristic, complemented
19#4926—21no—
20C3×#342—21no—
21C3×#342C2212yes—
22#4528—21no—
23#4525C2212yes—
24#2598—31no—
25#1740D311yescharacteristic, complemented
26#1464—31no—
27#1352—32no—
28#1343—32no—
29C23×C3×C4D411yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C22×C4×D3—21no—
33C3×#117—24no—
34C3×#117—41no—
35C3×#117—41no—
36C3×#117—42no—
37C2×C3×C4⋊C4—41no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4D411yescharacteristic
42C2×C3×C22⋊C4C2311yescharacteristic
43C2×C3×C42—41no—
44C2×C3×C42C2×C411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×#217—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×Dic3⋊C4—21no—
53C4×D12—41no—
54C4×D12—41no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95C2×C411yescharacteristic
60C3⋊#95C2×C411yescharacteristic
61#830—22no—
62C4⋊(C3⋊C8)—41no—
63#803—41no—
64C2×C4×D4—31no—
65C2×#117—21no—
66C2×#117C2×D311yescharacteristic
67#432—61no—
68#400—31no—
69#399—31no—
70#394—31no—
71#392—31no—
72#388—61no—
73#386—61no—
74#386—61no—
75#385—61no—
76#383—31no—
77#342—21no—
78#342C2×D312yes—
79#324—61no—
80#323—32no—
81C24×C3C2×D411yescharacteristic
82C23×D3—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—42no—
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C2×#67—42no—
96C2×#67—41no—
97C2×#67—41no—
98C2×#67—81no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×D12—41no—
103C2×D12—42no—
104C2×C4×D3—42no—
105C2×C4×D3—81no—
106C2×C4×D3—81no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—44no—
110C3×C22⋊C4—24no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—42no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42—21no—
118C3×C42—44no—
119#217—81no—
120(C2×D3)⋊C4—42no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—42no—
124(C2×D3)⋊C4—81no—
125C4⋊Dic3—41no—
126Dic3⋊C4—81no—
127C2×C3⋊C8—42no—
128C22×D4—31no—
129C22×D4—61no—
130C23×C4—61no—
131C23×C4#6711yescharacteristic
132C4×D4—121no—
133C4×D4—121no—
134#117—24no—
135#117—41no—
136#117—41no—
137#117—42no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4C4×D311yescharacteristic
142C2×C4⋊C4D1211yescharacteristic
143C2×C22⋊C4—22no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4#6711yescharacteristic
150C2×C22⋊C4C22×D311yescharacteristic
151C2×C42—41no—
152C2×C42C4×D311yescharacteristic
153C2×C42D1211yescharacteristic
154C4⋊C8—121no—
155C22⋊C8—62no—
156#97—121no—
157#95—31no—
158#95—31no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3C2×C22⋊C411yescharacteristic
168C23×C3AGL(1,ℤ/8)11yescharacteristic
169C23×C3#13711yescharacteristic
170C22×D3—22no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—41no—
176C22×D3—41no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—44no—
194C2×C3×C4—82no—
195C2×C3×C4—82no—
196C2×C3×C4#9911yescharacteristic
197C2×C3×C4#9911yescharacteristic
198C2×C3×C4#10412yes—
199#67—82no—
200#67—82no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210D12—42no—
211D12—42no—
212C4×D3—84no—
213C3⋊C8—82no—
214C24—61no—
215C24C2×#6711yescharacteristic
216C2×D4—121no—
217C2×D4—122no—
218C2×D4—122no—
219C2×D4—121no—
220C2×D4—121no—
221C2×D4—241no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—122no—
232C22×C4—241no—
233C22×C4—241no—
234C22×C4(C2×D3)⋊C411yescharacteristic
235C22×C4(C2×D3)⋊C411yescharacteristic
236C22×C4(C2×D3)⋊C411yescharacteristic
237C22×C4(C2×D3)⋊C411yescharacteristic
238C22×C4C2×C4×D311yescharacteristic
239C22×C4C2×D1211yescharacteristic
240C2×C8—122no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—44no—
244C4⋊C4—121no—
245C4⋊C4—241no—
246C22⋊C4—24no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—122no—
251C22⋊C4—122no—
252C22⋊C4—121no—
253C22⋊C4—121no—
254C22⋊C4—241no—
255C22⋊C4—241no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—44no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—81no—
270C22×C3C2×#9911yescharacteristic
271C22×C3#41711yescharacteristic
272C22×C3C2×#10411yescharacteristic
273C2×D3—42no—
274C2×D3—42no—
275C2×D3—42no—
276C2×D3—81no—
277C2×D3—81no—
278C2×D3—81no—
279C2×D3—81no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—84no—
288Dic3—81no—
289Dic3—81no—
290Dic3—81no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—62no—
296C23—121no—
297C23—121no—
298C23—121no—
299C23—121no—
300C23—121no—
301C23—121no—
302C23C2×(C2×D3)⋊C411yescharacteristic
303C23#93211yescharacteristic
304C23#94211yescharacteristic
305D4—122no—
306D4—122no—
307D4—242no—
308D4—242no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—44no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—244no—
338C2×C4#80512yes—
339C2×C4#80611yescharacteristic
340C2×C4#80611yescharacteristic
341C8—242no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3#150411yescharacteristic
347C2×C3#212011yescharacteristic
348C2×C3#212811yescharacteristic
349D3—82no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—122no—
360C22—122no—
361C22—122no—
362C22—241no—
363C22—241no—
364C22—241no—
365C22—241no—
366C22C2×#80511yescharacteristic
367C22C2×#80611yescharacteristic
368C22#496611yescharacteristic
369C4—22no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—84no—
377C4—241no—
378C4—241no—
379C4—241no—
380C3#1175411yescharacteristic, complemented, Sylow
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—242no—
386C2#6939811yescharacteristic
387C2#7005611yescharacteristic
388C2#7006111yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1740C2
socleC22×C3C2×#99
prime radical#4960C22
cosocleC2×C3×C22⋊C4C23
hypercenter#1740D3
hypocenterC3#11754
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. C3×#1740C2semidirect
3. #71572C2semidirect
4. #67492C2semidirect
9. C2×C4×D12C4semidirect
11. #4991C4semidirect
13. #4986C4semidirect
18. #4937C4semidirect
25. #1740D3semidirect
380. C3#11754semidirect

Maximal subgroups

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

subgroupindex
#822842
C3×#17402
#715722
#674922
#674892
#117543