Finite Group Databasebeta

← #10705669

Subgroups — #10705669

Order 768.

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

normal subgroups only

This group has 486 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#82260C211yesmaximal, characteristic
2C3×#1464C211yesmaximal, characteristic, complemented
3#71592C211yesmaximal, characteristic, complemented
4#67449C212yesmaximal, complemented
5#67448C212yesmaximal
6#11758—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#926—21no—
9C2×#875C411yescharacteristic, complemented
10C3×#432—21no—
11#5213—21no—
12#4991C411yescharacteristic, complemented
13#4990—21no—
14#4979C411yescharacteristic, complemented
15#4960C2211yescharacteristic
16#4957—21no—
17#4947C2211yescharacteristic
18#4940—21no—
19#4932C411yescharacteristic, complemented
20C3×#324—21no—
21C3×#323C2212yes—
22#4526—21no—
23#4526C2212yes—
24#2610—31no—
25#1740—31no—
26#1464D311yescharacteristic, complemented
27#1351—32no—
28#1343—32no—
29C23×C3×C4D411yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C22×Dic6C2×C411yescharacteristic
33C3×#117—41no—
34C3×#117—41no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C22⋊C4D411yescharacteristic
38C2×C3×C22⋊C4C2311yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×#217—22no—
42C2×#217—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48#926—24no—
49#926—42no—
50C2×C4⋊Dic3—41no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—41no—
54#875—41no—
55#875—41no—
56C3×C4⋊C8—41no—
57C3×C22⋊C8—22no—
58C3×#97—41no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95C2×C411yescharacteristic
63C3⋊#95C2×C411yescharacteristic
64C2×#124—31no—
65C2×#117—61no—
66C2×#117C2×D311yescharacteristic
67#432—21no—
68#395—31no—
69#394—31no—
70#393—31no—
71#392—31no—
72#390—61no—
73#388—61no—
74#388—61no—
75#386—61no—
76#376—31no—
77#342—32no—
78#342—61no—
79#324—21no—
80#323C2×D312yes—
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×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C2×#67—42no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—81no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—42no—
102C2×D12—41no—
103C2×D12—81no—
104C2×Dic6—41no—
105C2×Dic6—81no—
106C2×C3×C8—42no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—41no—
111C3×C42—21no—
112C3×C42—21no—
113C3×C42—21no—
114C3×C42—21no—
115#217—24no—
116#217—42no—
117#217—81no—
118(C2×D3)⋊C4—42no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—42no—
123(C2×D3)⋊C4—81no—
124C4⋊Dic3—44no—
125Dic3⋊C4—81no—
126C4×Dic3—42no—
127C4×Dic3—44no—
128C22×Q8—31no—
129C22×D4—31no—
130C22×D4—61no—
131C23×C4D1211yescharacteristic
132#124—121no—
133#124—121no—
134#117—41no—
135#117—41no—
136#117—64no—
137#117—122no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—121no—
140C2×C4⋊C4C4×D311yescharacteristic
141C2×C4⋊C4#6711yescharacteristic
142C2×C22⋊C4—62no—
143C2×C22⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4D1211yescharacteristic
150C2×C22⋊C4C22×D311yescharacteristic
151C2×C42—61no—
152C2×C42—121no—
153C2×C42C4×D311yescharacteristic
154C2×C42#6711yescharacteristic
155C4⋊C8—41no—
156C22⋊C8—22no—
157#97—41no—
158#95—31no—
159#95—31no—
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—41no—
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#9911yescharacteristic
194C2×C3×C4#9911yescharacteristic
195C2×C3×C4#10412yes—
196#67—82no—
197#67—82no—
198C2×Dic3—22no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—44no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—82no—
209C2×Dic3—82no—
210D12—82no—
211Dic6—82no—
212C3×C8—82no—
213C24—61no—
214C24C2×D1211yescharacteristic
215C2×Q8—121no—
216C2×Q8—241no—
217C2×D4—121no—
218C2×D4—122no—
219C2×D4—121no—
220C2×D4—121no—
221C2×D4—241no—
222C2×D4—241no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—122no—
232C22×C4(C2×D3)⋊C411yescharacteristic
233C22×C4(C2×D3)⋊C411yescharacteristic
234C22×C4(C2×D3)⋊C411yescharacteristic
235C22×C4(C2×D3)⋊C411yescharacteristic
236C22×C4C2×C4×D311yescharacteristic
237C22×C4C2×#6711yescharacteristic
238C2×C8—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—124no—
242C4⋊C4—241no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—64no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—122no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—122no—
252C22⋊C4—241no—
253C22⋊C4—241no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—124no—
259C42—122no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—81no—
269C22×C3C2×#9911yescharacteristic
270C22×C3#41711yescharacteristic
271C22×C3C2×#10411yescharacteristic
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C2×D3—81no—
277C2×D3—81no—
278C2×D3—81no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—22no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286Dic3—42no—
287Dic3—81no—
288Dic3—84no—
289Dic3—81no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—121no—
295C23—121no—
296C23—121no—
297C23—121no—
298C23—121no—
299C23—121no—
300C23—121no—
301C23#90811yescharacteristic
302C23#90911yescharacteristic
303C23C2×(C2×D3)⋊C411yescharacteristic
304Q8—242no—
305D4—242no—
306D4—242no—
307D4—242no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—62no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—124no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—242no—
335C2×C4—242no—
336C2×C4#80611yescharacteristic
337C2×C4#80611yescharacteristic
338C2×C4#82512yes—
339C8—82no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3#150411yescharacteristic
345C2×C3#212211yescharacteristic
346C2×C3#212611yescharacteristic
347D3—161no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22—241no—
358C22—241no—
359C22—241no—
360C22—241no—
361C22—241no—
362C22—241no—
363C22—241no—
364C22C2×#80611yescharacteristic
365C22#513311yescharacteristic
366C22C2×#82511yescharacteristic
367C4—21no—
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—122no—
375C4—241no—
376C4—244no—
377C4—241no—
378C3#1175811yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—481no—
384C2#6997611yescharacteristic
385C2#7006011yescharacteristic
386C2#7014611yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1464C2
socleC22×C3C2×#99
prime radical#4960C22
cosocleC2×C3×C22⋊C4C23
hypercenter#1464D3
hypocenterC3#11758
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. C3×#1464C2semidirect
3. #71592C2semidirect
4. #67449C2semidirect
9. C2×#875C4semidirect
12. #4991C4semidirect
14. #4979C4semidirect
19. #4932C4semidirect
26. #1464D3semidirect
378. C3#11758semidirect

Maximal subgroups

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

subgroupindex
#822602
C3×#14642
#715922
#674492
#674482
#117583