Finite Group Databasebeta

← #13147401

Subgroups — #13147401

Order 1728.

787 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×D4×#272C211yesmaximal, characteristic
2C2×C22⋊#1088C212yesmaximal, complemented
3#11720792C212yesmaximal, complemented
4#11720780C212yesmaximal, complemented
5C23×#272C411yescharacteristic, complemented
6C23×#272C2211yescharacteristic
7C22×#1088—22no—
8C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
9C2×#6235C411yescharacteristic, complemented
10C2×#6235C2211yescharacteristic
11D4×#272—22no—
12D4×#272—22no—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272C411yescharacteristic, complemented
15C22⋊#1088—22no—
16C22⋊#1088C2214yescomplemented
17C2×C3×D3×D4—62no—
18C23×H(3,𝔽3)C2×C411yescharacteristic
19C23×H(3,𝔽3)C2×C411yescharacteristic, complemented
20C22×#272—21no—
21C22×#272—21no—
22C22×#272—42no—
23C22×#272—42no—
24C22×#272C2×C411yescharacteristic
25C22×#272C2×C411yescharacteristic, complemented
26C22×#272D412yes—
27C22×#272D412yes—
28C22×#272C2311yescharacteristic
29C2×#1088—24no—
30C2×#1088—44no—
31D4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)C2×C411yescharacteristic, complemented
33#6235—22no—
34#6235—22no—
35C2×#1084C2×C411yescharacteristic, complemented
36C4⋊#272—22no—
37C4×#272—21no—
38C4×#272—21no—
39C2×C3×#99—91nomaximal
40C23×C3×D3—62no—
41C23×C3×D3—62no—
42C2×C32×D4—22no—
43C2×C3×#67—62no—
44C2×C3×#67—62no—
45C3×D3×D4—68no—
46C3×D3×D4—68no—
47C2×C3×D12—62no—
48C2×C3×C4×D3—62no—
49C22×H(3,𝔽3)—21no—
50C22×H(3,𝔽3)—41no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)C22⋊C412yes—
53C22×H(3,𝔽3)C22×C411yescharacteristic
54C2×#272—22no—
55C2×#272—24no—
56C2×#272—21no—
57C2×#272—42no—
58C2×#272—41no—
59C2×#272—42no—
60C2×#272—42no—
61C2×#272—41no—
62C2×#272C22⋊C411yescharacteristic
63C2×#272C22⋊C411yescharacteristic
64C2×#272C2×D412yes—
65#1088—44no—
66C4×H(3,𝔽3)—21no—
67#1084—21no—
68C22×C3×D4—91no—
69C2×D3×D4—62no—
70C2×C3×C22⋊C4—92no—
71C3×#99—92no—
72C3×#99—92no—
73C23×C32—22no—
74C23×C32—22no—
75C22×C3×D3—62no—
76C22×C3×D3—62no—
77C22×C3×D3—64no—
78C22×C3×D3—64no—
79C22×C3×D3—62no—
80C22×C3×D3—64no—
81C22×C3×D3—64no—
82C22×C3×D3—128no—
83C22×C3×D3—128no—
84C32×D4—28no—
85C2×C32×C4—22no—
86C3×#67—68no—
87C3×#67—68no—
88C2×C3×Dic3—62no—
89C3×D12—68no—
90C3×C4×D3—64no—
91C3×C4×D3—64no—
92C2×#99—91noSylow
93C2×H(3,𝔽3)—21no—
94C2×H(3,𝔽3)—22no—
95C2×H(3,𝔽3)—41no—
96C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
97#272—21no—
98#272—22no—
99#272—41no—
100#272#9912yescomplemented
101C24×C3—91no—
102C24×C3—91no—
103C23×D3—62no—
104C23×D3—62no—
105C2×C3×D4—62no—
106C2×C3×D4—91no—
107C2×C3×D4—91no—
108C2×C3×D4—91no—
109C2×C3×D4—182no—
110C2×C3×D4—182no—
111C2×C3×D4#15711yescharacteristic
112C22×C3×C4—91no—
113C22×C3×C4—182no—
114C2×#67—62no—
115C2×#67—62no—
116D3×D4—68no—
117D3×D4—68no—
118C2×D12—62no—
119C2×C4×D3—62no—
120C3×C22⋊C4—94no—
121C3×C22⋊C4—182no—
122C22×C32—24no—
123C22×C32—22no—
124C22×C32—24no—
125C22×C32—44no—
126C22×C32—44no—
127C2×C3×D3—62no—
128C2×C3×D3—62no—
129C2×C3×D3—64no—
130C2×C3×D3—68no—
131C2×C3×D3—64no—
132C2×C3×D3—128no—
133C2×C3×D3—128no—
134C2×C3×D3—128no—
135C2×C3×D3—128no—
136C2×C3×D3—124no—
137C2×C3×D3—124no—
138C32×C4—24no—
139C3×Dic3—64no—
140C22×D4—91no—
141C2×C22⋊C4—92no—
142#99—92no—
143#99—92no—
144H(3,𝔽3)C2×#9911yescharacteristic, complemented, Sylow
145C23×C3—62no—
146C23×C3—62no—
147C23×C3—91no—
148C23×C3—92no—
149C23×C3—91no—
150C23×C3—92no—
151C23×C3—91no—
152C23×C3—181no—
153C23×C3—181no—
154C23×C3—362no—
155C23×C3—362no—
156C23×C3C2×#15711yescharacteristic
157C23×C3C2×#15711yescharacteristic
158C22×D3—62no—
159C22×D3—62no—
160C22×D3—62no—
161C22×D3—64no—
162C22×D3—64no—
163C22×D3—64no—
164C22×D3—64no—
165C22×D3—128no—
166C22×D3—128no—
167C3×D4—22no—
168C3×D4—68no—
169C3×D4—182no—
170C3×D4—182no—
171C3×D4—182no—
172C2×C3×C4—62no—
173C2×C3×C4—91no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4—184no—
177C2×C3×C4—364no—
178C2×C3×C4C2×#15711yescharacteristic
179#67—68no—
180#67—68no—
181C2×Dic3—62no—
182D12—68no—
183C4×D3—64no—
184C4×D3—64no—
185C2×C32—24no—
186C2×C32—22no—
187C2×C32—44no—
188C2×C32—44no—
189C3×D3—64no—
190C3×D3—64no—
191C3×D3—124no—
192C3×D3—124no—
193C24—91no—
194C24—91no—
195C2×D4—91no—
196C2×D4—91no—
197C2×D4—91no—
198C2×D4—182no—
199C2×D4—182no—
200C2×D4#108811yescharacteristic, complemented
201C22×C4—91no—
202C22×C4—182no—
203C22⋊C4—94no—
204C22⋊C4—182no—
205C22×C3—21no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3—64no—
209C22×C3—64no—
210C22×C3—62no—
211C22×C3—91no—
212C22×C3—92no—
213C22×C3—91no—
214C22×C3—124no—
215C22×C3—124no—
216C22×C3—184no—
217C22×C3—182no—
218C22×C3—181no—
219C22×C3—361no—
220C22×C3—362no—
221C22×C3—361no—
222C22×C3—362no—
223C22×C3—362no—
224C22×C3C22⋊#15712yes—
225C22×C3C22×#15711yescharacteristic
226C2×D3—62no—
227C2×D3—62no—
228C2×D3—64no—
229C2×D3—68no—
230C2×D3—64no—
231C2×D3—128no—
232C2×D3—128no—
233C2×D3—128no—
234C2×D3—128no—
235C2×D3—124no—
236C2×D3—124no—
237C3×C4—21no—
238C3×C4—64no—
239C3×C4—181no—
240C3×C4—364no—
241Dic3—64no—
242C32—22no—
243C23—92no—
244C23—91no—
245C23—92no—
246C23—91no—
247C23—91no—
248C23—181no—
249C23—181no—
250C23—362no—
251C23—362no—
252C23C2×#108811yescharacteristic
253C23C2×#108811yescharacteristic, complemented
254D4—22no—
255D4—182no—
256D4—182no—
257D4—182no—
258C2×C4—91no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—184no—
262C2×C4—364no—
263C2×C4C2×#108811yescharacteristic, complemented
264C2×C3—21no—
265C2×C3—22no—
266C2×C3—41no—
267C2×C3—64no—
268C2×C3—62no—
269C2×C3—92no—
270C2×C3—124no—
271C2×C3—124no—
272C2×C3—182no—
273C2×C3—181no—
274C2×C3—361no—
275C2×C3C2×C22⋊#15711yescharacteristic
276D3—64no—
277D3—64no—
278D3—124no—
279D3—124no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—91no—
284C22—92no—
285C22—91no—
286C22—184no—
287C22—182no—
288C22—181no—
289C22—361no—
290C22—362no—
291C22—361no—
292C22—362no—
293C22—362no—
294C22C22⋊#108812yes—
295C22C22×#108811yescharacteristic
296C4—21no—
297C4—181no—
298C4—364no—
299C3—62no—
300C3#1059381711yescharacteristic
301C2—21no—
302C2—22no—
303C2—41no—
304C2—92no—
305C2—182no—
306C2—181no—
307C2—361no—
308C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C22⋊#157
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC22×C3C22×#157
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×D4×#272C2
cosocleC22×#272C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)C2×#99
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#1088C2semidirect
3. #11720792C2semidirect
4. #11720780C2semidirect
5. C23×#272C4semidirect
8. C2×D4×H(3,𝔽3)C4semidirect
9. C2×#6235C4semidirect
14. C2×C4×#272C4semidirect
16. C22⋊#1088C22semidirect
19. C23×H(3,𝔽3)C2×C4semidirect
25. C22×#272C2×C4semidirect
32. C2×C4×H(3,𝔽3)C2×C4semidirect
35. C2×#1084C2×C4semidirect
100. #272#99semidirect
144. H(3,𝔽3)C2×#99semidirect
200. C2×D4#1088semidirect
253. C23C2×#1088semidirect
263. C2×C4C2×#1088semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×C22⋊#10882
#117207922
#117207802
C2×C3×#999