Finite Group Databasebeta

← #11784921

Subgroups — #11784921

Order 1152.

1644 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 1644 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596756C211yesmaximal, characteristic
2C32×#408C211yesmaximal, characteristic, complemented
3#10592939C211yesmaximal, characteristic, complemented
4#10592838C212yesmaximal, complemented
5#10592812C212yesmaximal, complemented
6#70010—34nomaximal
7C2×C32×#41C2211yescharacteristic
8#64367—22no—
9#64366—22no—
10#41×C3⋊D3—24no—
11#64364C412yescomplemented
12#64361—22no—
13#64361C412yescomplemented
14C2×#3828—22no—
15C2×#3828C2212yes—
16C2×C32×M16C2211yescharacteristic, complemented
17C22×C32×C8C2211yescharacteristic, complemented
18#64135C2212yescomplemented
19C32×C22⋊C8—22no—
20#63661—22no—
21#63652—24no—
22#63648—22no—
23#63643—22no—
24#5964—34no—
25C3×#408D314yescomplemented
26#5240—34no—
27#5139—38no—
28#5113—38no—
29C32×#41—22no—
30C32×#41—22no—
31C2×C32×Q8C2×C411yescharacteristic
32C2×C32×D4C2×C411yescharacteristic
33C2×C32×D4D412yes—
34C22×C32×C4D412yes—
35C22×C32×C4C2311yescharacteristic
36C2×#638—24no—
37C2×#638—22no—
38#3832—24no—
39Q8×C3⋊D3—24no—
40#3830—28no—
41#3830—24no—
42D4×C3⋊D3—28no—
43D4×C3⋊D3—24no—
44#3828—24no—
45#3828—24no—
46#3828—24no—
47#3828—22no—
48#3828—22no—
49#3828C2×C414yes—
50#3828D414yes—
51C2×C4⋊(C3⋊D3)—22no—
52C2×C4⋊(C3⋊D3)D412yes—
53C2×C4×C3⋊D3—24no—
54C2×C4×C3⋊D3—22no—
55C2×#634—22no—
56C2×#634D412yes—
57C32×M16—22no—
58C2×C32×C8—22no—
59C2×C32×C8—21no—
60C2×C32×C8—21no—
61#3757—22no—
62C4⋊(C3⋊Dic3)—22no—
63C4×C3⋊Dic3—22no—
64#1794—91noSylow
65C2×C3×#41C2×D314yes—
66#1010—68no—
67#1009—68no—
68D3×#41—616no—
69#1007—38no—
70#1004—38no—
71#1004—68no—
72C2×#235—38no—
73C2×#235—68no—
74C2×C3×M16C2×D314yescomplemented
75C22×C3×C8C2×D314yescomplemented
76#926—38no—
77C3×C22⋊C8—28no—
78#834—68no—
79#825—616no—
80#821—68no—
81#816—68no—
82C23×C32C22⋊C412yes—
83C23×C32C2×D411yescharacteristic
84C22×C3⋊D3—24no—
85C22×C3⋊D3—22no—
86C32×Q8—22no—
87C32×D4—24no—
88C32×D4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4C22⋊C412yes—
92C2×C32×C4C22×C411yescharacteristic
93C2×C32×C4C2×D412yes—
94C2×C32×C4C2×D412yes—
95C2×C32×C4C2×D411yescharacteristic
96#638—28no—
97#638—24no—
98#638—24no—
99C2×C3⋊Dic3—24no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—42no—
102C4⋊(C3⋊D3)—24no—
103C4⋊(C3⋊D3)—22no—
104C4⋊(C3⋊D3)C22⋊C414yes—
105C4×C3⋊D3—28no—
106C4×C3⋊D3—24no—
107C4×C3⋊D3—24no—
108#634—24no—
109#634—22no—
110#634C22⋊C414yes—
111C32×C8—41no—
112C32×C8—41no—
113#585—91no—
114#421—92no—
115#416—92no—
116#410—91no—
117#408C3⋊D311yescharacteristic, complemented
118C3×#41—28no—
119C3×#41—28no—
120C2×C3×Q8C4×D314yes—
121C2×C3×D4C4×D314yes—
122C2×C3×D4D1218yes—
123C22×C3×C4#6718yes—
124C22×C3×C4C22×D314yes—
125C2×#67—616no—
126C2×#67—68no—
127#239—616no—
128D3×Q8—616no—
129#237—616no—
130#237—632no—
131D3×D4—616no—
132D3×D4—632no—
133#235—316no—
134#235—316no—
135#235—616no—
136#235—616no—
137#235—616no—
138#235—68no—
139#235—68no—
140C2×D12—38no—
141C2×D12—68no—
142C2×C4×D3—616no—
143C2×C4×D3—68no—
144C2×Dic6—38no—
145C2×Dic6—68no—
146C3×M16—28no—
147C2×C3×C8—28no—
148C2×C3×C8—24no—
149C2×C3×C8—24no—
150#217—68no—
151C4⋊Dic3—68no—
152C4×Dic3—68no—
153C22×C32—22no—
154C22×C32—41no—
155C22×C32—42no—
156C22×C32C2×C22⋊C411yescharacteristic
157C22×C32#12012yes—
158C2×C3⋊D3—24no—
159C2×C3⋊D3—82no—
160C2×C3⋊D3—84no—
161C32×C4—22no—
162C32×C4C2×C22⋊C412yes—
163C32×C4#12011yescharacteristic
164C32×C4#12011yescharacteristic
165C3⋊Dic3—24no—
166C3⋊Dic3—82no—
1672−1+4—92no—
1682−1+4—182no—
169H(5,𝔽2)—92no—
170H(5,𝔽2)—182no—
171H(5,𝔽2)—182no—
172C2×#41—92no—
173C2×#41—184no—
174C2×#41—182no—
175C2×#41C2×C3⋊D311yescharacteristic
176C2×M16C2×C3⋊D311yescharacteristic, complemented
177C22×C8C2×C3⋊D311yescharacteristic, complemented
178#117—92no—
179#104—184no—
180#103—182no—
181#102—182no—
182#99—182no—
183C22⋊C8—22no—
184C23×C3(C2×D3)⋊C418yes—
185C23×C3C2×D1214yes—
186C22×D3—616no—
187C22×D3—68no—
188C3×Q8—28no—
189C3×D4—216no—
190C3×D4—28no—
191C2×C3×C4—28no—
192C2×C3×C4—28no—
193C2×C3×C4(C2×D3)⋊C418yes—
194C2×C3×C4C2×C4×D314yes—
195C2×C3×C4D3×D418yes—
196C2×C3×C4D3×D418yes—
197C2×C3×C4C2×#6714yes—
198#67—632no—
199#67—616no—
200#67—616no—
201C2×Dic3—616no—
202C2×Dic3—68no—
203C2×Dic3—128no—
204D12—316no—
205D12—616no—
206D12—68no—
207C4×D3—632no—
208C4×D3—616no—
209C4×D3—616no—
210Dic6—316no—
211Dic6—616no—
212Dic6—68no—
213C3×C8—44no—
214C3×C8—44no—
215C2×C32—22no—
216C2×C32—21no—
217C2×C32—42no—
218C2×C32#37911yescharacteristic
219C3⋊D3—44no—
220#41—22no—
221#41—22no—
222#41—94no—
223#41—94no—
224#41—184no—
225#41—184no—
226#41—184no—
227#41—188no—
228#41—184no—
229#41—184no—
230#41—182no—
231#41—182no—
232C2×Q8—92no—
233C2×Q8—182no—
234C2×Q8—184no—
235C2×Q8C4×C3⋊D311yescharacteristic
236C2×D4—92no—
237C2×D4—182no—
238C2×D4—184no—
239C2×D4—188no—
240C2×D4—184no—
241C2×D4—182no—
242C2×D4C4×C3⋊D311yescharacteristic
243C2×D4C4⋊(C3⋊D3)12yes—
244C22×C4—184no—
245C22×C4—182no—
246C22×C4#63812yes—
247C22×C4C22×C3⋊D311yescharacteristic
248M16—22no—
249C2×C8—22no—
250C2×C8—21no—
251C2×C8—21no—
252C4⋊C4—182no—
253C22⋊C4—182no—
254C42—182no—
255C22×C3—28no—
256C22×C3—44no—
257C22×C3—48no—
258C22×C3#88218yes—
259C22×C3C2×(C2×D3)⋊C414yes—
260C2×D3—616no—
261C2×D3—248no—
262C2×D3—2416no—
263C3×C4—28no—
264C3×C4D3×C22⋊C418yes—
265C3×C4#93714yes—
266C3×C4#93714yes—
267Dic3—616no—
268Dic3—248no—
269C32#179411yescharacteristic, complemented, Sylow
270C23—184no—
271C23—182no—
272C23#375212yes—
273C23C2×C4⋊(C3⋊D3)11yescharacteristic
274Q8—22no—
275Q8—94no—
276Q8—184no—
277Q8—182no—
278D4—24no—
279D4—22no—
280D4—94no—
281D4—184no—
282D4—188no—
283D4—184no—
284D4—184no—
285D4—182no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—188no—
289C2×C4—184no—
290C2×C4—182no—
291C2×C4—184no—
292C2×C4—184no—
293C2×C4—362no—
294C2×C4#375212yes—
295C2×C4C2×C4×C3⋊D311yescharacteristic
296C2×C4D4×C3⋊D312yes—
297C2×C4D4×C3⋊D312yes—
298C2×C4C2×#63811yescharacteristic
299C8—41no—
300C8—41no—
301C2×C3—28no—
302C2×C3—24no—
303C2×C3—48no—
304C2×C3#495514yes—
305D3—1216no—
306C22—22no—
307C22—41no—
308C22—42no—
309C22—184no—
310C22—722no—
311C22—724no—
312C22#6409112yes—
313C22C2×#375211yescharacteristic
314C4—22no—
315C4—184no—
316C4—722no—
317C4C22⋊C4×C3⋊D312yes—
318C4#6414611yescharacteristic
319C4#6414611yescharacteristic
320C3#7001014yescomplemented
321C2—22no—
322C2—21no—
323C2—42no—
324C2—364no—
325C2#1059265411yescharacteristic
subgroupG/H
centerC4#64146
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×#408C2
socleC2×C32#379
prime radical#10596756C2
cosocleC22×C32×C4C23
hypercenter#408C3⋊D3
hypocenterC32#1794
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C32×#408C2semidirect
3. #10592939C2semidirect
4. #10592838C2semidirect
5. #10592812C2semidirect
11. #64364C4semidirect
13. #64361C4semidirect
16. C2×C32×M16C22semidirect
17. C22×C32×C8C22semidirect
18. #64135C22semidirect
25. C3×#408D3semidirect
74. C2×C3×M16C2×D3semidirect
75. C22×C3×C8C2×D3semidirect
117. #408C3⋊D3semidirect
176. C2×M16C2×C3⋊D3semidirect
177. C22×C8C2×C3⋊D3semidirect
269. C32#1794semidirect
320. C3#70010semidirect

Maximal subgroups

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

subgroupindex
#105967562
C32×#4082
#105929392
#105928382
#105928122
#700103