Finite Group Databasebeta

← #11895520

Subgroups — #11895520

Order 1152.

1008 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C211yesmaximal, characteristic, complemented
2#10596055C211yesmaximal, characteristic
3#10592792C211yesmaximal, characteristic, complemented
4#10592786C211yesmaximal, characteristic
5#10592767C212yesmaximal
6#10592762C212yesmaximal, complemented
7#10592730C211yesmaximal, characteristic, complemented
8C4×#3756C212yesmaximal, complemented
9#10592727C212yesmaximal
10C4×#3755C211yesmaximal, characteristic
11Q8×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
12#83689—34nomaximal
13C32×C4⋊Q8—21no—
14C32×C4⋊Q8C2211yescharacteristic
15C32×#125—22no—
16C32×#125C2212yes—
17C32×C4×Q8—21no—
18C32×C4×Q8C2211yescharacteristic
19C32×C4×Q8C2212yes—
20C32×C4×Q8C2211yescharacteristic
21#64155—21no—
22C2×#3756C2212yes—
23C2×#3755C2211yescharacteristic
24#64107—22no—
25#64101—21no—
26#64084C2212yes—
27C4×C4⋊(C3⋊D3)C2211yescharacteristic
28#64078C2211yescharacteristic
29C4×#634C2212yes—
30#63662C2211yescharacteristic
31#63637C2212yes—
32#63637C2212yes—
33#63637C2212yes—
34#63636C2212yes—
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×#557D314yescomplemented
43#5584—34no—
44#5093—34no—
45#5087—34no—
46#5068—38no—
47#5063—38no—
48#5031—34no—
49C4×#216—38no—
50#5028—38no—
51C4×#215—34no—
52Q8×C3⋊C8—34no—
53C2×C32×Q8—21no—
54C2×C32×Q8D411yescharacteristic
55C2×C32×Q8C2311yescharacteristic
56C2×C4⋊(C3⋊D3)C2311yescharacteristic
57C2×C4×C3⋊D3—21no—
58C2×#634C2312yes—
59C32×C4⋊C4—22no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4—22no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4—22no—
64C32×C4⋊C4D411yescharacteristic
65C32×C4⋊C4D412yes—
66C32×C4⋊C4C2312yes—
67C32×C4⋊C4C2311yescharacteristic
68C32×C42—22no—
69C32×C42—21no—
70C32×C42C2312yes—
71C32×C42C2311yescharacteristic
72#3756—24no—
73#3755—22no—
74#3752—21no—
75#3752—22no—
76#3752—22no—
77C4⋊(C3⋊Dic3)—21no—
78C4⋊(C3⋊Dic3)—21no—
79C4⋊(C3⋊Dic3)C2311yescharacteristic
80#3750—22no—
81C4×C3⋊Dic3—22no—
82C2×C3⋊(C3⋊C8)C2311yescharacteristic
83C2×C3⋊(C3⋊C8)C2312yes—
84C2×C3⋊(C3⋊C8)C2311yescharacteristic
85#3364—91noSylow
86C3×C4⋊Q8—24no—
87C3×C4⋊Q8C2×D314yes—
88C3×#125—28no—
89C3×#125C2×D318yes—
90C3×C4×Q8—24no—
91C3×C4×Q8C2×D314yes—
92C3×C4×Q8C2×D318yes—
93C3×C4×Q8C2×D314yes—
94#946—64no—
95C2×#216—38no—
96C2×#215—34no—
97#898—68no—
98#892—64no—
99#875—38no—
100C4×D12—34no—
101#869—34no—
102C4×Dic6—38no—
103#835—34no—
104#810—38no—
105#810—38no—
106#810—38no—
107#809—34no—
108#809—38no—
109#808—34no—
110#807—38no—
111C4⋊(C3⋊C8)—38no—
112C4⋊(C3⋊C8)—34no—
113C4×C3⋊C8—34no—
114C4×C3⋊C8—38no—
115C22×C3⋊D3—21no—
116C32×Q8—22no—
117C32×Q8—21no—
118C32×Q8—42no—
119C32×Q8#4112yes—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4C2×D412yes—
124C2×C32×C4C2×D412yes—
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2411yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—22no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—21no—
132C4×C3⋊D3—42no—
133#634—22no—
134#634—22no—
135C3⋊(C3⋊C8)—21no—
136C3⋊(C3⋊C8)—22no—
137C3⋊(C3⋊C8)—21no—
138#557C3⋊D311yescharacteristic, complemented
139#556—91no—
140#488—92no—
141#487—91no—
142#479—92no—
143#474—91no—
144#464—92no—
145#462—91no—
146C8×Q8—91no—
147C4×Q16—92no—
148C4×SD16—91no—
149C2×C3×Q8—24no—
150C2×C3×Q8#6714yes—
151C2×C3×Q8C22×D314yes—
152C2×D12—34no—
153C2×C4×D3—64no—
154C2×Dic6—38no—
155C3×C4⋊C4—28no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—28no—
160C3×C4⋊C4#6714yes—
161C3×C4⋊C4#6718yes—
162C3×C4⋊C4C22×D318yes—
163C3×C4⋊C4C22×D314yes—
164C3×C42—28no—
165C3×C42—24no—
166C3×C42C22×D318yes—
167C3×C42C22×D314yes—
168#216—616no—
169#215—68no—
170(C2×D3)⋊C4—64no—
171(C2×D3)⋊C4—68no—
172(C2×D3)⋊C4—68no—
173C4⋊Dic3—34no—
174C4⋊Dic3—64no—
175C4⋊Dic3—64no—
176Dic3⋊C4—68no—
177C4×Dic3—68no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8—38no—
180C2×C3⋊C8—34no—
181C22×C32C22×D411yescharacteristic
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—21no—
189C32×C4—42no—
190C32×C4—41no—
191C32×C4#13512yes—
192C32×C4C2×#4111yescharacteristic
193C32×C42−1+411yescharacteristic
194C3⋊Dic3—42no—
195C3⋊Dic3—41no—
196C2×Q16—92no—
197C2×SD16—91no—
198C4⋊Q8—21no—
199C4⋊Q8—91no—
200C4⋊Q8C2×C3⋊D311yescharacteristic
201#126—182no—
202#125—22no—
203#125C2×C3⋊D312yes—
204#124—92no—
205C22⋊Q8—181no—
206C4×Q8—21no—
207C4×Q8—92no—
208C4×Q8C2×C3⋊D311yescharacteristic
209C4×Q8C2×C3⋊D312yes—
210C4×Q8C2×C3⋊D311yescharacteristic
211C4×D4—91no—
212#117—181no—
213#107—92no—
214#106—91no—
215C4⋊C8—91no—
216C4⋊C8—92no—
217#103—91no—
218#103—92no—
219#103—92no—
220#103—92no—
221#102—91no—
222#102—92no—
223C4×C8—91no—
224C4×C8—92no—
225C22×D3—64no—
226C3×Q8—28no—
227C3×Q8—24no—
228C3×Q8—48no—
229C3×Q8#23918yes—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—28no—
233C2×C3×C4C2×#6718yes—
234C2×C3×C4C2×#6718yes—
235C2×C3×C4C2×#6714yes—
236C2×C3×C4C2×#6714yes—
237C2×C3×C4C23×D314yes—
238C2×Dic3—64no—
239C2×Dic3—68no—
240D12—64no—
241D12—64no—
242C4×D3—128no—
243Dic6—68no—
244Dic6—68no—
245C3⋊C8—64no—
246C3⋊C8—68no—
247C3⋊C8—64no—
248C2×C32#54811yescharacteristic
249C2×C32C2×#13511yescharacteristic
250C2×C32#57811yescharacteristic
251C3⋊D3—81no—
252C2×Q8—21no—
253C2×Q8—92no—
254C2×Q8#63811yescharacteristic
255C2×Q8C22×C3⋊D311yescharacteristic
256C2×D4—91no—
257C22×C4—181no—
258Q16—184no—
259SD16—182no—
260C2×C8—91no—
261C2×C8—92no—
262C2×C8—91no—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—91no—
269C4⋊C4—181no—
270C4⋊C4—182no—
271C4⋊C4—181no—
272C4⋊C4#63811yescharacteristic
273C4⋊C4#63812yes—
274C4⋊C4C22×C3⋊D311yescharacteristic
275C4⋊C4C22×C3⋊D312yes—
276C22⋊C4—182no—
277C22⋊C4—181no—
278C22⋊C4—182no—
279C42—22no—
280C42—21no—
281C42—182no—
282C42C22×C3⋊D311yescharacteristic
283C42C22×C3⋊D312yes—
284C22×C3C22×#6714yes—
285C2×D3—124no—
286C2×D3—124no—
287C2×D3—124no—
288C3×C4—28no—
289C3×C4—28no—
290C3×C4—28no—
291C3×C4—24no—
292C3×C4—48no—
293C3×C4—44no—
294C3×C4#95018yes—
295C3×C4C2×#23914yes—
296C3×C4#100714yes—
297Dic3—128no—
298Dic3—124no—
299C32#336411yescharacteristic, complemented, Sylow
300C23—181no—
301Q8—21no—
302Q8—22no—
303Q8—42no—
304Q8—182no—
305Q8—182no—
306Q8#383212yes—
307D4—181no—
308D4—181no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—181no—
313C2×C4—182no—
314C2×C4—362no—
315C2×C4C2×#63812yes—
316C2×C4C2×#63811yescharacteristic
317C2×C4C2×#63812yes—
318C2×C4C2×#63811yescharacteristic
319C2×C4C23×C3⋊D311yescharacteristic
320C8—181no—
321C8—182no—
322C8—181no—
323C2×C3#581714yes—
324C2×C3C2×#95014yes—
325C2×C3#583814yes—
326D3—244no—
327C22—361no—
328C22—361no—
329C22—361no—
330C22C22×#63811yescharacteristic
331C4—22no—
332C4—22no—
333C4—22no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—362no—
338C4—361no—
339C4#6415912yes—
340C4C2×#383211yescharacteristic
341C4#6436411yescharacteristic
342C3#8368914yescomplemented
343C2—721no—
344C2#1059628811yescharacteristic
345C2C2×#6415911yescharacteristic
346C2#1059630911yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#557C2
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenter#557C3⋊D3
hypocenterC32#3364
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
3. #10592792C2semidirect
6. #10592762C2semidirect
7. #10592730C2semidirect
8. C4×#3756C2semidirect
11. Q8×C3⋊(C3⋊C8)C2semidirect
42. C3×#557D3semidirect
138. #557C3⋊D3semidirect
299. C32#3364semidirect
342. C3#83689semidirect

Maximal subgroups

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

subgroupindex
C32×#5572
#105960552
#105927922
#105927862
#105927672
#105927622
#105927302
C4×#37562
#105927272
C4×#37552
Q8×C3⋊(C3⋊C8)2
#836893