Finite Group Databasebeta

← #10699705

Subgroups — #10699705

Order 768.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#70468C211yesmaximal, characteristic, complemented
2#70457C211yesmaximal, characteristic
3C2×#4485C211yesmaximal, characteristic
4C2×#4480C211yesmaximal, characteristic, complemented
5#69580C211yesmaximal, characteristic
6#69567C211yesmaximal, characteristic, complemented
7C3×#1296C211yesmaximal, characteristic, complemented
8#10587—31nomaximal, Sylow
9C2×#874C2211yescharacteristic
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C3×#384C2211yescharacteristic
13C2×#821—21no—
14C2×#821—21no—
15C2×Dic3⋊C8—21no—
16C2×#809—21no—
17C2×#809—21no—
18C2×#807—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22#4550—21no—
23#4529—21no—
24#4485—22no—
25#4480—22no—
26#1982—31no—
27#1957—31no—
28#1947—31no—
29#1939—31no—
30C2×#334—31no—
31C2×#331—31no—
32#1296D311yescharacteristic, complemented
33C22×D12—21no—
34C22×D12—21no—
35C22×D12—21no—
36C22×C3×C8—21no—
37C22×C3×C8—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3D411yescharacteristic
43C2×C4⋊Dic3D411yescharacteristic
44C2×C4×Dic3—21no—
45C22×C3⋊C8—21no—
46C22×C3⋊C8D411yescharacteristic
47C22×C3⋊C8D411yescharacteristic
48#874—21no—
49#874—21no—
50#874—21no—
51#874Q812yes—
52C3×C4⋊C8—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8—21no—
55C3×C4⋊C8—21no—
56C3⋊#95—21no—
57#821—42no—
58#821—42no—
59Dic3⋊C8—42no—
60#809—42no—
61#809—42no—
62#807—42no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)D412yes—
65C4×C3⋊C8—21no—
66C4×C3⋊C8—21no—
67C2×#127—31no—
68C2×#107—61no—
69C2×C4⋊C8—31no—
70C2×C4⋊C8—61no—
71C2×C4⋊C8C2×D311yescharacteristic
72C2×C4⋊C8C2×D311yescharacteristic
73C2×#102—61no—
74C2×#102—61no—
75C2×#102—61no—
76C2×#102—61no—
77C2×C4×C8—31no—
78#384C2×D311yescharacteristic
79C4×C4⋊C4—31no—
80#340—61no—
81#336—61no—
82#334—62no—
83#331—62no—
84C23×D3—41no—
85C22×C3×C4—41no—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4#4111yescharacteristic
89C22×Dic3—21no—
90C22×Dic3—21no—
91C2×D12—22no—
92C2×D12—22no—
93C2×D12—22no—
94C2×D12—41no—
95C2×D12—41no—
96C2×D12—41no—
97C2×D12—41no—
98C2×D12—41no—
99C2×D12—41no—
100C2×D12—41no—
101C2×D12—41no—
102C2×D12—41no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12—81no—
106C2×D12—81no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C2×C3×C8—41no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—81no—
118C3×C42C2×D411yescharacteristic
119C3×C42C2×Q811yescharacteristic
120C3×C42#4111yescharacteristic
121C3×C42#4111yescharacteristic
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3SD1612yes—
126C4×Dic3—41no—
127C4×Dic3—41no—
128C2×C3⋊C8—21no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8—21no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8D812yes—
137C22×D4—61no—
138C22×D4—61no—
139C22×D4—61no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—31no—
143C22×C8—31no—
144C22×C8—61no—
145#127—32no—
146#127—61no—
147#127—61no—
148#127—61no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C42—61no—
155C2×C42C22×D311yescharacteristic
156#107—122no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—32no—
162C4⋊C8—61no—
163C4⋊C8—122no—
164#102—122no—
165#102—122no—
166#102—122no—
167#102—122no—
168C4×C8—61no—
169C4×C8—61no—
170#95—61no—
171C23×C3#12111yescharacteristic
172C22×D3—42no—
173C22×D3—81no—
174C22×D3—81no—
175C22×D3—81no—
176C22×D3—81no—
177C22×D3—81no—
178C22×D3—81no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—81no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4C22⋊Q811yescharacteristic
187C2×C3×C4C22⋊Q811yescharacteristic
188C2×C3×C4C22⋊Q811yescharacteristic
189C2×C3×C4#12611yescharacteristic
190C2×C3×C4C2×D811yescharacteristic
191C2×C3×C4C2×SD1611yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×C3×C4#13511yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C2×Dic3—22no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204D12—42no—
205D12—42no—
206D12—42no—
207D12—42no—
208D12—82no—
209D12—82no—
210D12—81no—
211D12—81no—
212D12—81no—
213D12—81no—
214C3×C8—81no—
215C3×C8—81no—
216C3⋊C8—22no—
217C3⋊C8—42no—
218C3⋊C8—41no—
219C24—121no—
220C2×D4—62no—
221C2×D4—62no—
222C2×D4—62no—
223C2×D4—121no—
224C2×D4—121no—
225C2×D4—121no—
226C2×D4—121no—
227C2×D4—121no—
228C2×D4—121no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4—121no—
233C2×D4—121no—
234C2×D4—241no—
235C2×D4—241no—
236C22×C4—41no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4#23511yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242C2×C8—32no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—62no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—121no—
256C2×C8—121no—
257C4⋊C4—32no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—81no—
266C42—121no—
267C42—121no—
268C42#23511yescharacteristic
269C42#23511yescharacteristic
270C42D3×D411yescharacteristic
271C42D3×Q811yescharacteristic
272C22×C3#39611yescharacteristic
273C22×C3#45911yescharacteristic
274C22×C3#46411yescharacteristic
275C22×C3#46511yescharacteristic
276C22×C3#46911yescharacteristic
277C22×C3#47111yescharacteristic
278C22×C3#47211yescharacteristic
279C2×D3—82no—
280C2×D3—82no—
281C2×D3—82no—
282C2×D3—161no—
283C2×D3—161no—
284C2×D3—161no—
285C2×D3—161no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—161no—
289C3×C4#46011yescharacteristic
290C3×C4#46311yescharacteristic
291C3×C4#46611yescharacteristic
292C3×C4#46811yescharacteristic
293C3×C4#47411yescharacteristic
294C3×C4#47611yescharacteristic
295C3×C4#47811yescharacteristic
296C3×C4#47811yescharacteristic
297Dic3—42no—
298Dic3—81no—
299C23—122no—
300C23—241no—
301C23—241no—
302C23—241no—
303C23—241no—
304C23—241no—
305C23—241no—
306C23#88411yescharacteristic
307D4—122no—
308D4—122no—
309D4—122no—
310D4—122no—
311D4—242no—
312D4—242no—
313D4—241no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—62no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4#87611yescharacteristic
329C2×C4#88411yescharacteristic
330C2×C4#88411yescharacteristic
331C2×C4#89611yescharacteristic
332C2×C4#89611yescharacteristic
333C2×C4#89611yescharacteristic
334C2×C4D3×D811yescharacteristic
335C2×C4#91111yescharacteristic
336C2×C4D3×SD1611yescharacteristic
337C2×C4#91411yescharacteristic
338C2×C4#91411yescharacteristic
339C2×C4#91611yescharacteristic
340C2×C4#91811yescharacteristic
341C2×C4#91911yescharacteristic
342C8—62no—
343C8—81no—
344C8—81no—
345C8—121no—
346C8—122no—
347C2×C3#166511yescharacteristic
348C2×C3#167811yescharacteristic
349C2×C3#168711yescharacteristic
350C2×C3#170211yescharacteristic
351C2×C3#206311yescharacteristic
352C2×C3#206511yescharacteristic
353C2×C3#206711yescharacteristic
354D3—162no—
355C22—242no—
356C22—242no—
357C22—242no—
358C22—481no—
359C22—481no—
360C22—481no—
361C22—481no—
362C22#467611yescharacteristic
363C22#474911yescharacteristic
364C22#475411yescharacteristic
365C22#478211yescharacteristic
366C22#478611yescharacteristic
367C22#479811yescharacteristic
368C22#480711yescharacteristic
369C4—21no—
370C4—21no—
371C4—122no—
372C4—161no—
373C4—241no—
374C4#476211yescharacteristic
375C4#477511yescharacteristic
376C4#481211yescharacteristic
377C4#481911yescharacteristic
378C4#487011yescharacteristic
379C4#487111yescharacteristic
380C4#489011yescharacteristic
381C4#489111yescharacteristic
382C3#1058711yescharacteristic, complemented, Sylow
383C2—482no—
384C2#6839911yescharacteristic
385C2#6844011yescharacteristic
386C2#6847811yescharacteristic
387C2#6851011yescharacteristic
388C2#6890011yescharacteristic
389C2#6890111yescharacteristic
390C2#6899611yescharacteristic
subgroupG/H
centerC23#884
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#121
prime radicalC2×#874C22
hypercenter#1296D3
hypocenterC3#10587
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
1. #70468C2semidirect
4. C2×#4480C2semidirect
6. #69567C2semidirect
7. C3×#1296C2semidirect
32. #1296D3semidirect
382. C3#10587semidirect

Maximal subgroups

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

subgroupindex
#704682
#704572
C2×#44852
C2×#44802
#695802
#695672
C3×#12962
#105873