Finite Group Databasebeta

← #10690623

Subgroups — #10690623

Order 768.

504 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5026C211yesmaximal, characteristic
2#79288C211yesmaximal, characteristic
3C3×#1479C211yesmaximal, characteristic, complemented
4#71624C211yesmaximal, characteristic, complemented
5#70757C211yesmaximal, characteristic
6#67741C211yesmaximal, characteristic, complemented
7#67740C211yesmaximal, characteristic, complemented
8#9091—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#215C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×#835C411yescharacteristic
15#5026—22no—
16#5026—22no—
17C2×#809C411yescharacteristic
18C2×#807C411yescharacteristic
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21C2×#803C2211yescharacteristic
22#4838—22no—
23#4726—22no—
24#4688—22no—
25C3×Q8⋊C8—22no—
26#4528—22no—
27#4459—22no—
28#2975—31no—
29C2×#440—31no—
30#1479D311yescharacteristic, complemented
31#1478—31no—
32#1461—31no—
33#1281—31no—
34#1275—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×D12C2×C411yescharacteristic
37C22×C4×D3—21no—
38C3×C4×Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#215—22no—
47C2×#215—22no—
48C2×#215—22no—
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic
51C22×C3⋊C8C2×C411yescharacteristic
52C22×C3⋊C8C2×C411yescharacteristic
53C2×ΓL(1,𝔽25)—24no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12D412yes—
57C4×D12D412yes—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C4×C8—21no—
63C3×C4×C8D412yes—
64#835—22no—
65#830—21no—
66#830D412yes—
67(C2×D3)⋊C8—24no—
68#809—22no—
69#807—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72#803—21no—
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C22×SD16—31no—
77C2×C4×Q8C2×D311yescharacteristic
78C2×C4×D4—31no—
79#440—62no—
80#440—62no—
81#436—62no—
82#436—62no—
83#431—31no—
84#431—31no—
85#431C2×D311yescharacteristic
86#431C2×D311yescharacteristic
87C2×#107—31no—
88C2×#103—31no—
89C2×#102—31no—
90#407—62no—
91C2×#97—31no—
92Q8⋊C8—22no—
93D4⋊C8—62no—
94#324—62no—
95C23×D3—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8M1612yescomplemented
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—21no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12C22⋊C412yes—
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—82no—
117C2×C3×C8—22no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42—41no—
124C3×C42—41no—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129#215—28no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132C4⋊Dic3—21no—
133C4⋊Dic3C22⋊C412yes—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8M1612yescomplemented
142ΓL(1,𝔽25)—84no—
143C22×Q8C4×D311yescharacteristic
144C22×D4—31no—
145C23×C4—61no—
146C2×SD16—62no—
147C2×SD16—62no—
148C2×SD16—62no—
149C2×M16—64no—
150C22×C8—31no—
151C22×C8—31no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×D4—32no—
157C4×D4—32no—
158C4×D4—61no—
159C4×D4—61no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4C4×D311yescharacteristic
163C2×C22⋊C4—61no—
164C2×C42—21no—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171#103—62no—
172#102—62no—
173C22⋊C8—21no—
174C22⋊C8—21no—
175C22⋊C8—32no—
176C22⋊C8—64no—
177C22⋊C8—61no—
178#97—61no—
179#97—61no—
180C4×C8—21no—
181C4×C8—61no—
182C4×C8—61no—
183C4×C8#6712yes—
184C23×C3C4×D411yescharacteristic
185C22×D3—22no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—81no—
189C22×D3—81no—
190C3×Q8—22no—
191C3×Q8—22no—
192C3×Q8—41no—
193C3×Q8—41no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—81no—
201C2×C3×C4C2×C22⋊C411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12311yescharacteristic
207C2×C3×C4C2×M1611yescharacteristic
208C2×C3×C4C2×M1611yescharacteristic
209C2×C3×C4#13111yescharacteristic
210C2×C3×C4#13111yescharacteristic
211C2×Dic3—22no—
212C2×Dic3—81no—
213D12—22no—
214D12—22no—
215D12—41no—
216D12—41no—
217C4×D3—44no—
218C4×D3—82no—
219C3×C8—42no—
220C3×C8—81no—
221C3⋊C8—22no—
222C3⋊C8—42no—
223C3⋊C8—41no—
224C3⋊C8—81no—
225C24—61no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8ΓL(1,𝔽25)12yescomplemented
232C2×D4—32no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—61no—
241C22×C4—62no—
242C22×C4—62no—
243C22×C4—242no—
244C22×C4C2×C4×D311yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4#23711yescharacteristic
247SD16—68no—
248M16—244no—
249C2×C8—22no—
250C2×C8—32no—
251C2×C8—41no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—61no—
257C2×C8—62no—
258C2×C8—121no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—32no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—61no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C42—41no—
268C42—41no—
269C42#23511yescharacteristic
270C42D3×D411yescharacteristic
271C42D3×D411yescharacteristic
272C42C2×#6711yescharacteristic
273C22×C3—21no—
274C22×C3—21no—
275C22×C3#38611yescharacteristic
276C22×C3#43511yescharacteristic
277C22×C3#43611yescharacteristic
278C2×D3—42no—
279C2×D3—42no—
280C2×D3—42no—
281C2×D3—82no—
282C2×D3—82no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—42no—
288C3×C4—81no—
289C3×C4#40711yescharacteristic
290C3×C4#40811yescharacteristic
291C3×C4#43511yescharacteristic
292C3×C4#43611yescharacteristic
293Dic3—42no—
294C23—62no—
295C23—121no—
296C23—121no—
297C23—241no—
298C23—241no—
299C23#67⋊C411yescharacteristic
300Q8—22no—
301Q8—22no—
302Q8—41no—
303Q8—41no—
304D4—62no—
305D4—62no—
306D4—121no—
307D4—121no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—62no—
315C2×C4—81no—
316C2×C4—124no—
317C2×C4—242no—
318C2×C4—241no—
319C2×C4D3×C22⋊C411yescharacteristic
320C2×C4#88311yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
323C2×C4#90111yescharacteristic
324C2×C4D3×M1611yescharacteristic
325C2×C4#90711yescharacteristic
326C2×C4C4×#6711yescharacteristic
327C2×C4#93711yescharacteristic
328C2×C4#93811yescharacteristic
329C8—42no—
330C8—62no—
331C8—81no—
332C8—122no—
333C8—121no—
334C8—241no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3#187011yescharacteristic
338C2×C3#190211yescharacteristic
339C2×C3#190411yescharacteristic
340D3—82no—
341C22—21no—
342C22—21no—
343C22—122no—
344C22—122no—
345C22—122no—
346C22—242no—
347C22—242no—
348C22#472911yescharacteristic
349C22#473011yescharacteristic
350C22#495611yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—81no—
357C4—122no—
358C4#472611yescharacteristic
359C4#472711yescharacteristic
360C4#511111yescharacteristic
361C4#512811yescharacteristic
362C3#909111yescharacteristic, complemented, Sylow
363C2—21no—
364C2—21no—
365C2—242no—
366C2#6995811yescharacteristic
367C2#7034911yescharacteristic
368C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1479D3
hypocenterC3#9091
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
3. C3×#1479C2semidirect
4. #71624C2semidirect
6. #67741C2semidirect
7. #67740C2semidirect
30. #1479D3semidirect
101. C2×C3×Q8M16semidirect
141. C2×C3⋊C8M16semidirect
231. C2×Q8ΓL(1,𝔽25)semidirect
362. C3#9091semidirect

Maximal subgroups

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

subgroupindex
C2×#50262
#792882
C3×#14792
#716242
#707572
#677412
#677402
#90913