Finite Group Databasebeta

← #10694490

Subgroups — #10694490

Order 768.

574 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5085C211yesmaximal, characteristic
2C2×C3×#332C211yesmaximal, characteristic, complemented
3#71780C211yesmaximal, characteristic, complemented
4#70466C211yesmaximal, characteristic, complemented
5#70457C211yesmaximal, characteristic
6C2×#4485C211yesmaximal, characteristic
7C2×#4479C211yesmaximal, characteristic, complemented
8#9744—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C22×#215—21no—
12C2×#874C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#821—22no—
16C2×#815—21no—
17#5085—22no—
18#5085C414yescomplemented
19C4×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22C3×#332—22no—
23#4550—21no—
24#4529—22no—
25#4485—22no—
26#4479—22no—
27C2×#492—31no—
28#1988—31no—
29#1982—31no—
30#1770—31no—
31C2×#335—31no—
32C2×#332D311yescharacteristic, complemented
33C2×#331—31no—
34C22×C3×Q8—21no—
35C22×D12—21no—
36C22×D12—21no—
37C22×D12—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8C2×C412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#215—24no—
48C2×#215—24no—
49C2×#215—42no—
50C2×#215—42no—
51C2×#215—42no—
52C2×#215—42no—
53C2×C4⋊Dic3D412yes—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56C22×C3⋊C8D411yescharacteristic
57C22×C3⋊C8D411yescharacteristic
58#874—21no—
59#874—21no—
60#874—21no—
61#874C2×C412yes—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8D412yes—
66C3⋊#95—21no—
67#821—44no—
68#815—42no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4×C3⋊C8C2×C412yes—
72C4×C3⋊C8D412yes—
73C22×SD16—61no—
74C22×SD16—61no—
75C2×C4⋊Q8C2×D311yescharacteristic
76C2×#127—31no—
77#492—34no—
78#492—62no—
79C2×C4⋊C8—31no—
80C2×C4⋊C8C2×D311yescharacteristic
81C2×C4⋊C8C2×D311yescharacteristic
82C2×#102—62no—
83C2×#97—61no—
84C2×C4×C8—31no—
85C4×C4⋊C4—31no—
86#340—62no—
87#336—61no—
88#335—62no—
89#332—22no—
90#331—62no—
91C23×D3—41no—
92C2×C3×Q8—22no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×Dic3—22no—
103C2×D12—22no—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—41no—
107C2×D12—41no—
108C2×D12—42no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—42no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—82no—
116C2×C3×C8—22no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C2×C3×C8—42no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129#215—48no—
130#215—48no—
131C4⋊Dic3—22no—
132C4⋊Dic3—22no—
133C4×Dic3—41no—
134C4×Dic3—41no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—42no—
139C2×C3⋊C8C22⋊C412yes—
140C2×C3⋊C8C22⋊C412yes—
141C2×C3⋊C8D812yes—
142C2×C3⋊C8SD1612yes—
143C22×Q8—21no—
144C22×D4—61no—
145C22×D4—61no—
146C22×D4—61no—
147C2×SD16—64no—
148C2×SD16—64no—
149C2×SD16—122no—
150C2×SD16—122no—
151C2×SD16—122no—
152C2×SD16—122no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—31no—
156C22×C8—31no—
157C22×C8—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C4×D312yes—
162#127—32no—
163#127—61no—
164#127—61no—
165#127—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—32no—
169C2×C42—61no—
170C2×C42C22×D311yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176C4⋊C8#6712yes—
177#102—124no—
178#97—122no—
179C4×C8—32no—
180C4×C8—32no—
181#95—61no—
182C23×C3#12111yescharacteristic
183C22×D3—42no—
184C22×D3—81no—
185C22×D3—81no—
186C22×D3—82no—
187C22×D3—82no—
188C3×Q8—42no—
189C3×Q8—42no—
190C3×Q8—81no—
191C3×Q8—81no—
192C2×C3×C4—22no—
193C2×C3×C4—42no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4C2×C22⋊C411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12411yescharacteristic
201C2×C3×C4#12711yescharacteristic
202C2×C3×C4C2×D811yescharacteristic
203C2×C3×C4C2×SD1611yescharacteristic
204C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
205C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
206C2×C3×C4AGL(1,ℤ/8)12yes—
207C2×C3×C4#13712yes—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211D12—42no—
212D12—42no—
213D12—44no—
214D12—82no—
215D12—82no—
216D12—81no—
217D12—81no—
218D12—81no—
219D12—81no—
220C3×C8—42no—
221C3×C8—81no—
222C3⋊C8—22no—
223C3⋊C8—81no—
224C3⋊C8#10214yescomplemented
225C24—121no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×D4—62no—
233C2×D4—62no—
234C2×D4—62no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4—122no—
240C2×D4—121no—
241C2×D4—122no—
242C2×D4—121no—
243C2×D4—121no—
244C2×D4—242no—
245C22×C4—41no—
246C22×C4—62no—
247C22×C4#23511yescharacteristic
248C22×C4D3×D411yescharacteristic
249C22×C4D3×D411yescharacteristic
250SD16—128no—
251SD16—128no—
252C2×C8—22no—
253C2×C8—32no—
254C2×C8—32no—
255C2×C8—32no—
256C2×C8—32no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—122no—
264C2×C8—121no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C42—121no—
272C42—121no—
273C42C2×C4×D311yescharacteristic
274C42D3×D411yescharacteristic
275C42#23711yescharacteristic
276C42C2×#6711yescharacteristic
277C22×C3#39011yescharacteristic
278C22×C3#45911yescharacteristic
279C22×C3#46011yescharacteristic
280C22×C3#46812yes—
281C22×C3#47112yes—
282C2×D3—82no—
283C2×D3—82no—
284C2×D3—82no—
285C2×D3—162no—
286C2×D3—162no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—82no—
290C3×C4C2×#10211yescharacteristic
291C3×C4#41811yescharacteristic
292C3×C4#44012yes—
293C3×C4#45911yescharacteristic
294C3×C4#46011yescharacteristic
295C3×C4#49611yescharacteristic
296C3×C4#49711yescharacteristic
297Dic3—82no—
298C23—122no—
299C23—242no—
300C23—241no—
301C23—241no—
302C23—242no—
303C23#88411yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—81no—
307Q8—81no—
308D4—122no—
309D4—122no—
310D4—124no—
311D4—242no—
312D4—242no—
313D4—241no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—22no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4D3×C22⋊C411yescharacteristic
325C2×C4#67⋊C411yescharacteristic
326C2×C4#88511yescharacteristic
327C2×C4D3×D811yescharacteristic
328C2×C4#91111yescharacteristic
329C2×C4D3×SD1611yescharacteristic
330C2×C4#91411yescharacteristic
331C2×C4#91412yes—
332C2×C4#91812yes—
333C2×C4C4×#6711yescharacteristic
334C2×C4#93911yescharacteristic
335C2×C4#94011yescharacteristic
336C8—34no—
337C8—42no—
338C8—62no—
339C8—81no—
340C8—241no—
341C2×C3#168512yes—
342C2×C3#168612yes—
343C2×C3#197311yescharacteristic
344C2×C3#197711yescharacteristic
345C2×C3#198211yescharacteristic
346D3—162no—
347C22—242no—
348C22—242no—
349C22—242no—
350C22—482no—
351C22—482no—
352C22#475412yes—
353C22#476211yescharacteristic
354C22#478611yescharacteristic
355C22#481212yes—
356C22#495711yescharacteristic
357C4—21no—
358C4—21no—
359C4—82no—
360C4—242no—
361C4D3×#10211yescharacteristic
362C4#477011yescharacteristic
363C4#481712yes—
364C4#515011yescharacteristic
365C4#516311yescharacteristic
366C4#517611yescharacteristic
367C4#519011yescharacteristic
368C3#974411yescharacteristic, complemented, Sylow
369C2—482no—
370C2#6844812yes—
371C2#6847212yes—
372C2#7010711yescharacteristic
373C2#7016111yescharacteristic
374C2#7026011yescharacteristic
subgroupG/H
centerC23#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#121
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9744
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. C2×C3×#332C2semidirect
3. #71780C2semidirect
4. #70466C2semidirect
7. C2×#4479C2semidirect
18. #5085C4semidirect
32. C2×#332D3semidirect
224. C3⋊C8#102semidirect
368. C3#9744semidirect

Maximal subgroups

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

subgroupindex
C2×#50852
C2×C3×#3322
#717802
#704662
#704572
C2×#44852
C2×#44792
#97443