Finite Group Databasebeta

← #11791051

Subgroups — #11791051

Order 1152.

642 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C22×#3774C211yesmaximal, characteristic
2C2×C4×#622C211yesmaximal, characteristic, complemented
3C2×#63743C211yesmaximal, characteristic, complemented
4C2×#63739C212yesmaximal, complemented
5#10591385C212yesmaximal, complemented
6C22×#625C2212yes—
7C22×#622C2211yescharacteristic
8C22×#622C2211yescharacteristic
9C2×#3774—22no—
10C2×#3774—22no—
11C2×#3774C414yescomplemented
12C2×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×(C2×D3)⋊C4—22no—
14C2×#3724C2212yes—
15C2×(C2×D3)⋊Dic3—22no—
16C4×#622—22no—
17#63743—22no—
18#63739—24no—
19#4957—62no—
20C23×C3×D3—22no—
21C22×C32×C4D411yescharacteristic
22C22×C3⋊Dic3D411yescharacteristic
23C22×C3⋊Dic3C2311yescharacteristic
24C22×C3×Dic3—22no—
25C2×#625—22no—
26C2×#625—22no—
27C2×#625—22no—
28C2×#625—22no—
29C2×#625—22no—
30C2×#625C2×C414yes—
31C2×#622—21no—
32C2×#622—21no—
33C2×#622C2×C412yes—
34C2×#622D412yes—
35C2×#622D412yes—
36C2×#622D412yes—
37#3774—28no—
38C4×C3⋊Dic3—21no—
39C4×C3⋊Dic3—21no—
40C3×(C2×D3)⋊C4—44no—
41#3724—22no—
42#3724—22no—
43(C2×D3)⋊Dic3—44no—
44#1970—91nomaximal, Sylow
45C22×#67—62no—
46C2×C3×C22⋊C4—62no—
47C2×#217—62no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—62no—
50C2×Dic3⋊C4—62no—
51C2×C4×Dic3—62no—
52C23×C32C2×D411yescharacteristic
53C22×C3×D3—24no—
54C22×C3×D3—42no—
55C22×C3×D3—42no—
56C22×C3×D3—42no—
57C22×C3×D3—42no—
58C22×C3×D3—42no—
59C22×C3×D3—42no—
60C2×C32×C4—21no—
61C2×C32×C4—21no—
62C2×C32×C4D812yescomplemented
63C2×C3⋊Dic3—21no—
64C2×C3⋊Dic3—21no—
65C2×C3⋊Dic3D812yescomplemented
66C2×C3⋊Dic3C22×C411yescharacteristic
67C2×C3⋊Dic3C2×D411yescharacteristic
68C2×C3⋊Dic3C2×D411yescharacteristic
69C2×C3⋊Dic3C2×D411yescharacteristic
70C2×C3⋊Dic3#4111yescharacteristic
71C2×C3⋊Dic3#4111yescharacteristic
72C2×C3×Dic3—42no—
73C2×C3×Dic3—42no—
74C2×C3×Dic3—42no—
75#625—24no—
76#625—24no—
77#625—42no—
78#625—42no—
79#622—22no—
80#622C22⋊C414yescomplemented
81C22×D8—91no—
82C2×#107—91no—
83C2×#102—92no—
84C2×C4×C8—91no—
85#390—92no—
86C24×C3—62no—
87C23×D3—22no—
88C22×C3×C4—22no—
89C22×C3×C4—62no—
90C2×#67—64no—
91C2×#67—64no—
92C2×#67—122no—
93C2×#67—122no—
94C2×#67—122no—
95C2×#67—122no—
96C22×Dic3—22no—
97C22×Dic3—62no—
98C22×Dic3—62no—
99C3×C22⋊C4—124no—
100#217—124no—
101(C2×D3)⋊C4—44no—
102(C2×D3)⋊C4—124no—
103Dic3⋊C4—122no—
104Dic3⋊C4—122no—
105C4×Dic3—122no—
106C4×Dic3—122no—
107C22×C32C4×D411yescharacteristic
108C22×C32#12111yescharacteristic
109C22×C32#12711yescharacteristic
110C22×C32C2×D811yescharacteristic
111C22×C32C2×D811yescharacteristic
112C22×C32#13511yescharacteristic
113C22×C32#13511yescharacteristic
114C2×C3×D3—44no—
115C2×C3×D3—44no—
116C2×C3×D3—44no—
117C2×C3×D3—82no—
118C2×C3×D3—82no—
119C2×C3×D3—82no—
120C2×C3×D3—82no—
121C32×C4—22no—
122C3⋊Dic3—22no—
123C3⋊Dic3C2×C22⋊C411yescharacteristic
124C3⋊Dic3C4×D411yescharacteristic
125C3⋊Dic3#12111yescharacteristic
126C3⋊Dic3#12411yescharacteristic
127C3×Dic3—82no—
128C22×D4—92no—
129C2×D8—94no—
130C2×D8—182no—
131C2×D8—182no—
132C22×C8—91no—
133C22×C8—91no—
134C2×C4⋊C4—92no—
135C2×C22⋊C4—62no—
136C2×C22⋊C4—182no—
137C2×C42—91no—
138#107—182no—
139#102—184no—
140C4×C8—182no—
141C23×C3—22no—
142C23×C3—64no—
143C23×C3—122no—
144C23×C3—122no—
145C23×C3—122no—
146C23×C3—122no—
147C23×C3—122no—
148C23×C3—122no—
149C22×D3—24no—
150C22×D3—42no—
151C22×D3—42no—
152C22×D3—42no—
153C22×D3—42no—
154C22×D3—42no—
155C22×D3—42no—
156C2×C3×C4—24no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—122no—
160C2×C3×C4—122no—
161C2×C3×C4—122no—
162#67—124no—
163#67—124no—
164#67—124no—
165#67—124no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—62no—
170C2×Dic3—62no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175C2×Dic3—64no—
176C2×Dic3—122no—
177C2×Dic3—122no—
178C2×C32#39011yescharacteristic
179C2×C32C4×D811yescharacteristic
180C2×C32C4×D811yescharacteristic
181C2×C32#46611yescharacteristic
182C2×C32#46611yescharacteristic
183C2×C32#49311yescharacteristic
184C2×C32#49511yescharacteristic
185C3×D3—84no—
186C24—62no—
187C2×D4—94no—
188C2×D4—182no—
189C2×D4—182no—
190C2×D4—182no—
191C2×D4—182no—
192C2×D4—182no—
193C22×C4—62no—
194C22×C4—91no—
195C22×C4—91no—
196C22×C4GO+(4,𝔽2)11yescharacteristic
197D8—188no—
198C2×C8—92no—
199C2×C8—92no—
200C2×C8—92no—
201C2×C8—92no—
202C2×C8—181no—
203C2×C8—181no—
204C4⋊C4—182no—
205C4⋊C4—182no—
206C22⋊C4—124no—
207C22⋊C4—364no—
208C42—181no—
209C42—181no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—124no—
218C22×C3—124no—
219C22×C3—124no—
220C22×C3—242no—
221C22×C3—242no—
222C22×C3—242no—
223C22×C3—242no—
224C2×D3—44no—
225C2×D3—44no—
226C2×D3—44no—
227C2×D3—82no—
228C2×D3—82no—
229C2×D3—82no—
230C2×D3—82no—
231C3×C4—44no—
232C3×C4—242no—
233Dic3—62no—
234Dic3—62no—
235Dic3—62no—
236Dic3—62no—
237Dic3—82no—
238Dic3—124no—
239C32#197011yescharacteristic, complemented, Sylow
240C23—64no—
241C23—122no—
242C23—122no—
243C23—122no—
244C23—122no—
245C23—122no—
246C23—122no—
247C23C2×GO+(4,𝔽2)11yescharacteristic
248D4—184no—
249D4—184no—
250D4—362no—
251D4—362no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—92no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4#377412yescomplemented
267C8—94no—
268C8—182no—
269C2×C3—22no—
270C2×C3—22no—
271C2×C3—22no—
272C2×C3—22no—
273C2×C3—22no—
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—244no—
277D3—84no—
278C22—124no—
279C22—124no—
280C22—124no—
281C22—242no—
282C22—242no—
283C22—242no—
284C22—242no—
285C22#6422311yescharacteristic
286C22#6422511yescharacteristic
287C22C4×GO+(4,𝔽2)11yescharacteristic
288C22#6423111yescharacteristic
289C22C2×#377411yescharacteristic
290C22#6423711yescharacteristic
291C22#6424211yescharacteristic
292C4—22no—
293C4—91no—
294C4—91no—
295C4—91no—
296C4—91no—
297C4—182no—
298C4—242no—
299C3—22no—
300C2—244no—
301C2C4×#377411yescharacteristic
302C2#1059341811yescharacteristic
303C2#1059342011yescharacteristic
304C2#1059342311yescharacteristic
305C2#1059342411yescharacteristic
306C2#1059345411yescharacteristic
307C2#1059352011yescharacteristic
subgroupG/H
center, FrattiniC23C2×GO+(4,𝔽2)
commutatorC2×C3⋊Dic3C22×C4
FittingC22×C32×C4D4
socleC23×C32C2×D4
prime radicalC22×#3774C2
cosocleC22×C3⋊Dic3C23
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32#1970
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#622C2semidirect
3. C2×#63743C2semidirect
4. C2×#63739C2semidirect
5. #10591385C2semidirect
11. C2×#3774C4semidirect
62. C2×C32×C4D8semidirect
65. C2×C3⋊Dic3D8semidirect
80. #622C22⋊C4semidirect
239. C32#1970semidirect
266. C2×C4#3774semidirect

Maximal subgroups

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

subgroupindex
C22×#37742
C2×C4×#6222
C2×#637432
C2×#637392
#105913852
#19709