Finite Group Databasebeta

← #11895298

Subgroups — #11895298

Order 1152.

1086 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#559C211yesmaximal, characteristic, complemented
2#10596050C211yesmaximal, characteristic, complemented
3#10592771C211yesmaximal, characteristic
4#10592769C211yesmaximal, characteristic, complemented
5#10592756C212yesmaximal, complemented
6#10592755C212yesmaximal, complemented
7C4×#3755C211yesmaximal, characteristic, complemented
8Q8×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9#10592720C211yesmaximal, characteristic, complemented
10C4×#3753C212yesmaximal, complemented
11#10592708C212yesmaximal, complemented
12#83713—34nomaximal
13C32×#127C2211yescharacteristic
14C32×#124—21no—
15C32×#124C2212yes—
16C32×#123—22no—
17C32×C4×Q8C2211yescharacteristic
18C32×C4×Q8C2211yescharacteristic
19C32×C4×D4—21no—
20C32×C4×D4C2212yes—
21#64155—21no—
22C2×#3755C2211yescharacteristic
23C2×#3753C2212yes—
24#64105—22no—
25C4⋊C4×C3⋊D3—21no—
26C4×C4⋊(C3⋊D3)C2211yescharacteristic
27C4×C4⋊(C3⋊D3)C2212yes—
28#64079C2212yes—
29#64078C2211yescharacteristic
30#63662C2212yes—
31#63662C2211yescharacteristic
32#63659C2212yes—
33#63659C2212yes—
34#63659C2212yes—
35#63636C2211yescharacteristic
36#63635C2211yescharacteristic
37#63634C2212yes—
38C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
39C4⋊(C3⋊(C3⋊C8))C2212yes—
40C4×C3⋊(C3⋊C8)C2211yescharacteristic
41C4×C3⋊(C3⋊C8)C2212yes—
42C3×#559D314yescomplemented
43#5579—34no—
44#5072—34no—
45#5070—34no—
46#5057—38no—
47#5056—38no—
48C4×#215—34no—
49Q8×C3⋊C8—34no—
50#5021—34no—
51C4×#213—38no—
52#5009—38no—
53C2×C32×Q8D411yescharacteristic
54C2×C32×Q8C2311yescharacteristic
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4C2312yes—
58C22×C32×C4—22no—
59C2×C4⋊(C3⋊D3)C2311yescharacteristic
60C2×C4×C3⋊D3—21no—
61C2×C4×C3⋊D3—22no—
62C2×#634—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4D411yescharacteristic
65C32×C4⋊C4D412yes—
66C32×C4⋊C4C2311yescharacteristic
67C32×C22⋊C4—22no—
68C32×C22⋊C4—22no—
69C32×C22⋊C4—22no—
70C32×C42—21no—
71C32×C42C2311yescharacteristic
72C32×C42C2312yes—
73#3755—22no—
74#3753—24no—
75#3752—21no—
76#3752—22no—
77C4⋊(C3⋊Dic3)—21no—
78C4⋊(C3⋊Dic3)C2312yes—
79C4⋊(C3⋊Dic3)C2311yescharacteristic
80#3750—22no—
81#3750—22no—
82C2×C3⋊(C3⋊C8)C2312yes—
83C2×C3⋊(C3⋊C8)C2311yescharacteristic
84C2×C3⋊(C3⋊C8)C2311yescharacteristic
85#3361—91noSylow
86C3×#127C2×D314yes—
87C3×#124—24no—
88C3×#124C2×D318yes—
89C3×#123—28no—
90C3×C4×Q8C2×D314yes—
91C3×C4×Q8C2×D314yes—
92C3×C4×D4—24no—
93C3×C4×D4C2×D318yes—
94#946—64no—
95C2×#215—34no—
96C2×#213—38no—
97#896—68no—
98D3×C4⋊C4—64no—
99C4×D12—38no—
100C4×D12—34no—
101#870—38no—
102#869—34no—
103#835—38no—
104#835—34no—
105#832—38no—
106#832—38no—
107#832—38no—
108#809—34no—
109#808—34no—
110#807—38no—
111C4⋊(C3⋊C8)—38no—
112C4⋊(C3⋊C8)—34no—
113C4×C3⋊C8—38no—
114C4×C3⋊C8—34no—
115C23×C32—22no—
116C22×C3⋊D3—21no—
117C32×Q8—22no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×D4—22no—
121C32×D4—22no—
122C32×D4—42no—
123C32×D4—81no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—82no—
127C2×C32×C4C2×D412yes—
128C2×C32×C4C2×D412yes—
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2411yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—22no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)#4112yes—
136C4×C3⋊D3—42no—
137C4×C3⋊D3—82no—
138#634—81no—
139C3⋊(C3⋊C8)—22no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)—21no—
142#559C3⋊D311yescharacteristic, complemented
143#554—91no—
144#488—92no—
145#486—91no—
146#478—92no—
147#477—91no—
148#463—92no—
149#459—91no—
150C8×Q8—91no—
151C4×SD16—91no—
152C4×D8—92no—
153C2×C3×Q8#6714yes—
154C2×C3×Q8C22×D314yes—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4C22×D318yes—
158C22×C3×C4—28no—
159C2×D12—34no—
160C2×C4×D3—64no—
161C2×C4×D3—68no—
162C2×Dic6—64no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4#6714yes—
165C3×C4⋊C4#6718yes—
166C3×C4⋊C4C22×D314yes—
167C3×C22⋊C4—28no—
168C3×C22⋊C4—28no—
169C3×C22⋊C4—28no—
170C3×C42—24no—
171C3×C42C22×D314yes—
172C3×C42C22×D318yes—
173#215—68no—
174#213—616no—
175(C2×D3)⋊C4—64no—
176(C2×D3)⋊C4—68no—
177C4⋊Dic3—38no—
178C4⋊Dic3—34no—
179C4⋊Dic3—64no—
180Dic3⋊C4—68no—
181Dic3⋊C4—68no—
182C2×C3⋊C8—34no—
183C2×C3⋊C8—38no—
184C2×C3⋊C8—34no—
185C22×C32—42no—
186C22×C32—42no—
187C22×C32—42no—
188C22×C32C22×D411yescharacteristic
189C2×C3⋊D3—22no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C32×C4—22no—
193C32×C4—21no—
194C32×C4—22no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4#13512yes—
198C32×C4C2×#4111yescharacteristic
199C32×C42−1+411yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3—42no—
202C2×SD16—91no—
203C2×D8—92no—
204C4⋊Q8—91no—
205#127C2×C3⋊D311yescharacteristic
206#125—92no—
207#124—21no—
208#124C2×C3⋊D312yes—
209#123—22no—
210C22⋊Q8—181no—
211C22⋊Q8—182no—
212C4×Q8C2×C3⋊D311yescharacteristic
213C4×Q8C2×C3⋊D311yescharacteristic
214C4×D4—21no—
215C4×D4—92no—
216C4×D4—91no—
217C4×D4C2×C3⋊D312yes—
218C2×C4⋊C4—181no—
219#107—92no—
220#106—91no—
221C4⋊C8—91no—
222C4⋊C8—92no—
223#103—91no—
224#103—92no—
225#102—92no—
226#102—92no—
227#102—92no—
228#102—91no—
229C4×C8—92no—
230C4×C8—91no—
231C23×C3—28no—
232C22×D3—64no—
233C3×Q8—28no—
234C3×Q8—24no—
235C3×Q8—24no—
236C3×D4—28no—
237C3×D4—28no—
238C3×D4—48no—
239C3×D4—84no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—88no—
243C2×C3×C4C2×#6718yes—
244C2×C3×C4C2×#6718yes—
245C2×C3×C4C2×#6714yes—
246C2×C3×C4C2×#6714yes—
247C2×C3×C4C23×D314yes—
248C2×Dic3—64no—
249C2×Dic3—68no—
250D12—38no—
251D12—64no—
252C4×D3—128no—
253C4×D3—248no—
254Dic6—244no—
255C3⋊C8—68no—
256C3⋊C8—64no—
257C3⋊C8—64no—
258C2×C32—82no—
259C2×C32#54811yescharacteristic
260C2×C32C2×#13511yescharacteristic
261C2×C32#57611yescharacteristic
262C3⋊D3—42no—
263C2×Q8—181no—
264C2×Q8#63811yescharacteristic
265C2×Q8C22×C3⋊D311yescharacteristic
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—91no—
269C2×D4C22×C3⋊D312yes—
270C22×C4—22no—
271C22×C4—181no—
272C22×C4—182no—
273SD16—182no—
274D8—184no—
275C2×C8—91no—
276C2×C8—91no—
277C2×C8—92no—
278C4⋊C4—21no—
279C4⋊C4—91no—
280C4⋊C4—92no—
281C4⋊C4—182no—
282C4⋊C4—181no—
283C4⋊C4—182no—
284C4⋊C4#63811yescharacteristic
285C4⋊C4#63812yes—
286C4⋊C4C22×C3⋊D311yescharacteristic
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—181no—
291C22⋊C4—182no—
292C42—21no—
293C42C22×C3⋊D312yes—
294C42C22×C3⋊D311yescharacteristic
295C22×C3—48no—
296C22×C3—48no—
297C22×C3—48no—
298C22×C3C22×#6714yes—
299C2×D3—68no—
300C2×D3—124no—
301C2×D3—124no—
302C3×C4—28no—
303C3×C4—24no—
304C3×C4—28no—
305C3×C4—44no—
306C3×C4—44no—
307C3×C4#95018yes—
308C3×C4D3×#4114yes—
309C3×C4#101014yes—
310Dic3—124no—
311Dic3—128no—
312C32#336111yescharacteristic, complemented, Sylow
313C23—22no—
314C23—181no—
315Q8—22no—
316Q8—21no—
317Q8—21no—
318Q8—721no—
319D4—22no—
320D4—22no—
321D4—42no—
322D4—81no—
323D4—92no—
324D4—181no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—82no—
328C2×C4—181no—
329C2×C4—182no—
330C2×C4—362no—
331C2×C4—722no—
332C2×C4C2×#63812yes—
333C2×C4C2×#63812yes—
334C2×C4C2×#63811yescharacteristic
335C2×C4C2×#63811yescharacteristic
336C2×C4C23×C3⋊D311yescharacteristic
337C8—182no—
338C8—181no—
339C8—181no—
340C2×C3—88no—
341C2×C3C2×#95014yes—
342C2×C3#583114yes—
343C2×C3#583514yes—
344D3—128no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—182no—
349C22—361no—
350C22—361no—
351C22C22×#63811yescharacteristic
352C4—22no—
353C4—22no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—361no—
358C4—362no—
359C4#6415912yes—
360C4#41×C3⋊D311yescharacteristic
361C4#6436711yescharacteristic
362C3#8371314yescomplemented
363C2—82no—
364C2—362no—
365C2C2×#6415911yescharacteristic
366C2#1059630211yescharacteristic
367C2#1059630611yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#559C2
socleC22×C32C22×D4
prime radical#10592771C2
hypercenter#559C3⋊D3
hypocenterC32#3361
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. C32×#559C2semidirect
2. #10596050C2semidirect
4. #10592769C2semidirect
5. #10592756C2semidirect
6. #10592755C2semidirect
7. C4×#3755C2semidirect
8. Q8×C3⋊(C3⋊C8)C2semidirect
9. #10592720C2semidirect
10. C4×#3753C2semidirect
11. #10592708C2semidirect
42. C3×#559D3semidirect
142. #559C3⋊D3semidirect
312. C32#3361semidirect
362. C3#83713semidirect

Maximal subgroups

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

subgroupindex
C32×#5592
#105960502
#105927712
#105927692
#105927562
#105927552
C4×#37552
Q8×C3⋊(C3⋊C8)2
#105927202
C4×#37532
#105927082
#837133