Finite Group Databasebeta

← #10694505

Subgroups — #10694505

Order 768.

526 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4694C211yesmaximal, characteristic
2C2×C3×#334C211yesmaximal, characteristic, complemented
3#70457C211yesmaximal, characteristic
4#70456C211yesmaximal, characteristic, complemented
5C2×#4482C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic
7#69757C211yesmaximal, characteristic, complemented
8#9746—31nomaximal, Sylow
9C2×#874C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#821—21no—
15C2×#821—21no—
16C2×#821—22no—
17C2×#816—22no—
18C2×#815—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21#4694—22no—
22#4694C414yescomplemented
23C3×#334—22no—
24#4550—21no—
25#4485—22no—
26#4482—22no—
27C2×#486—31no—
28#1982—31no—
29#1981—31no—
30#1770—31no—
31C2×#334D311yescharacteristic, complemented
32C2×#332—31no—
33C2×#331—31no—
34C22×D12—21no—
35C22×D12—21no—
36C22×D12—21no—
37C22×Dic6—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C22×C3×C8D411yescharacteristic
41C22×C3×C8D411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3D412yes—
46C2×C4×Dic3—21no—
47C22×C3⋊C8—21no—
48#874—21no—
49#874—21no—
50#874—21no—
51#874C2×C412yes—
52#869—21no—
53#869—21no—
54#869—21no—
55#869C2×C412yes—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4×C8C2×C412yes—
61C3×C4×C8D412yes—
62C3⋊#95—21no—
63#821—24no—
64#821—24no—
65#821—44no—
66#816—44no—
67#815—42no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)D412yes—
70C2×C4⋊Q8—31no—
71C2×#127—31no—
72#486—34no—
73#486—62no—
74C2×C4⋊C8—31no—
75C2×C4⋊C8C2×D311yescharacteristic
76C2×C4⋊C8C2×D311yescharacteristic
77C2×#103—62no—
78C2×#102—61no—
79C2×#102—61no—
80C2×#102—62no—
81C2×#97—61no—
82C2×C4×C8C2×D311yescharacteristic
83C4×C4⋊C4—31no—
84#336—61no—
85#334—22no—
86#332—62no—
87#331—62no—
88C23×D3—41no—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×Dic3—22no—
93C22×Dic3—41no—
94C2×D12—22no—
95C2×D12—22no—
96C2×D12—22no—
97C2×D12—41no—
98C2×D12—41no—
99C2×D12—42no—
100C2×D12—41no—
101C2×D12—42no—
102C2×D12—41no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12—41no—
106C2×D12—82no—
107C2×Dic6—22no—
108C2×Dic6—42no—
109C2×Dic6—42no—
110C2×Dic6—41no—
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C2×C3×C8—42no—
118C2×C3×C8—42no—
119C2×C3×C8C22⋊C412yes—
120C2×C3×C8C22⋊C412yes—
121C3×C42C22×C411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42#4111yescharacteristic
125C4⋊Dic3—22no—
126C4⋊Dic3—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4×Dic3—41no—
132C4×Dic3—41no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C22×Q8—61no—
137C22×D4—61no—
138C22×D4—61no—
139C22×D4—61no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—61no—
143C22×C8D1211yescharacteristic
144C22×C8D1211yescharacteristic
145C4⋊Q8—32no—
146C4⋊Q8—61no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149#127—32no—
150#127—61no—
151#127—61no—
152#127—61no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C42—61no—
157C2×C42C22×D311yescharacteristic
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—32no—
163C4⋊C8—61no—
164#103—124no—
165#102—64no—
166#102—64no—
167#102—124no—
168#97—122no—
169C4×C8C4×D312yes—
170C4×C8#6712yes—
171#95—61no—
172C23×C3#12111yescharacteristic
173C22×D3—42no—
174C22×D3—81no—
175C22×D3—81no—
176C22×D3—82no—
177C22×D3—82no—
178C2×C3×C4—22no—
179C2×C3×C4C2×C22⋊C411yescharacteristic
180C2×C3×C4C4×D411yescharacteristic
181C2×C3×C4C4×D411yescharacteristic
182C2×C3×C4#12111yescharacteristic
183C2×C3×C4#12411yescharacteristic
184C2×C3×C4#12711yescharacteristic
185C2×C3×C4#13511yescharacteristic
186C2×C3×C4#13511yescharacteristic
187C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
188C2×C3×C4AGL(1,ℤ/8)12yes—
189C2×C3×C4AGL(1,ℤ/8)12yes—
190C2×C3×C4#13711yescharacteristic
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—82no—
196D12—42no—
197D12—42no—
198D12—44no—
199D12—82no—
200D12—82no—
201D12—81no—
202D12—81no—
203D12—81no—
204D12—81no—
205Dic6—44no—
206Dic6—81no—
207Dic6—81no—
208C3×C8—22no—
209C3×C8—22no—
210C3×C8—81no—
211C3×C8—81no—
212C3⋊C8—42no—
213C24—121no—
214C2×Q8—62no—
215C2×Q8—121no—
216C2×Q8—122no—
217C2×Q8—122no—
218C2×D4—62no—
219C2×D4—62no—
220C2×D4—62no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—121no—
224C2×D4—121no—
225C2×D4—121no—
226C2×D4—122no—
227C2×D4—122no—
228C2×D4—121no—
229C2×D4—121no—
230C2×D4—242no—
231C22×C4—62no—
232C22×C4—121no—
233C22×C4C2×D1211yescharacteristic
234C22×C4D3×D411yescharacteristic
235C22×C4#23911yescharacteristic
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—62no—
245C2×C8—121no—
246C2×C8—121no—
247C2×C8(C2×D3)⋊C412yes—
248C2×C8(C2×D3)⋊C412yes—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C42—121no—
256C42—121no—
257C42C2×C4×D311yescharacteristic
258C42D3×D411yescharacteristic
259C42#23911yescharacteristic
260C42C2×#6711yescharacteristic
261C22×C3#39011yescharacteristic
262C22×C3#46311yescharacteristic
263C22×C3#46411yescharacteristic
264C22×C3#46912yes—
265C22×C3#47112yes—
266C2×D3—82no—
267C2×D3—82no—
268C2×D3—82no—
269C2×D3—162no—
270C2×D3—162no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4#41611yescharacteristic
274C3×C4#41711yescharacteristic
275C3×C4#44212yes—
276C3×C4#46311yescharacteristic
277C3×C4#46411yescharacteristic
278C3×C4#49611yescharacteristic
279C3×C4#49611yescharacteristic
280Dic3—82no—
281Dic3—82no—
282C23—122no—
283C23—242no—
284C23—242no—
285C23—241no—
286C23—241no—
287C23#89511yescharacteristic
288Q8—124no—
289Q8—241no—
290Q8—241no—
291D4—124no—
292D4—122no—
293D4—122no—
294D4—242no—
295D4—242no—
296D4—241no—
297D4—241no—
298D4—241no—
299D4—241no—
300C2×C4—22no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—242no—
306C2×C4#89311yescharacteristic
307C2×C4#89311yescharacteristic
308C2×C4#89511yescharacteristic
309C2×C4#90411yescharacteristic
310C2×C4#90411yescharacteristic
311C2×C4#90811yescharacteristic
312C2×C4#90911yescharacteristic
313C2×C4#91412yes—
314C2×C4#91412yes—
315C2×C4C2×(C2×D3)⋊C411yescharacteristic
316C2×C4#94011yescharacteristic
317C2×C4#94711yescharacteristic
318C8—22no—
319C8—22no—
320C8—81no—
321C8—81no—
322C8—122no—
323C2×C3#170112yes—
324C2×C3#170212yes—
325C2×C3#197211yescharacteristic
326C2×C3#197711yescharacteristic
327C2×C3#198311yescharacteristic
328D3—162no—
329C22—242no—
330C22—242no—
331C22—242no—
332C22—482no—
333C22—482no—
334C22#484411yescharacteristic
335C22#484511yescharacteristic
336C22#486512yes—
337C22#489812yes—
338C22#498811yescharacteristic
339C4—21no—
340C4—21no—
341C4—242no—
342C4—242no—
343C4#484411yescharacteristic
344C4#484511yescharacteristic
345C4#486912yes—
346C4#511311yescharacteristic
347C4#513311yescharacteristic
348C4#517611yescharacteristic
349C4#517611yescharacteristic
350C3#974611yescharacteristic, complemented, Sylow
351C2—482no—
352C2#6854412yes—
353C2#6855412yes—
354C2#7045811yescharacteristic
355C2#7057611yescharacteristic
356C2#7070811yescharacteristic
subgroupG/H
centerC23#895
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#334C2
socleC23×C3#121
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenterC2×#334D3
hypocenterC3#9746
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. C2×C3×#334C2semidirect
4. #70456C2semidirect
5. C2×#4482C2semidirect
7. #69757C2semidirect
22. #4694C4semidirect
31. C2×#334D3semidirect
350. C3#9746semidirect

Maximal subgroups

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

subgroupindex
C2×#46942
C2×C3×#3342
#704572
#704562
C2×#44822
C2×#44852
#697572
#97463