Finite Group Databasebeta

← #13147426

Subgroups — #13147426

Order 1728.

559 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 57 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#41×#272C211yesmaximal, characteristic
2#11722448C211yesmaximal, characteristic, complemented
3C2×#88627C211yesmaximal, characteristic, complemented
4#11720791C212yesmaximal, complemented
5#11720789C212yesmaximal, complemented
6#41×H(3,𝔽3)C411yescharacteristic, complemented
7#88748C2211yescharacteristic
8Q8×#272C411yescharacteristic, complemented
9#88744—21no—
10#88744C2211yescharacteristic
11D4×#272—21no—
12D4×#272C411yescharacteristic, complemented
13#88740—21no—
14#88740C411yescharacteristic, complemented
15C2×C4×#272—21no—
16C2×C4×#272C2211yescharacteristic
17C22⋊#1088—21no—
18C2×#6188—21no—
19C4⋊#1088—21no—
20C4×#1088C2212yescomplemented
21#88627—21no—
22#88627C2212yescomplemented
23C3×D3×#41—62no—
24C22×#272—21no—
25C22×#272D411yescharacteristic
26C2×#1088—22no—
27Q8×H(3,𝔽3)C2×C411yescharacteristic
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2×C411yescharacteristic
32#6235—21no—
33#6235—21no—
34#6235—21no—
35C2×#1084—21no—
36C2×#1084D411yescharacteristic
37C4⋊#272—21no—
38C4⋊#272C2×C411yescharacteristic
39C4×#272—21no—
40C4×#272—21no—
41C4×#272—21no—
42C4×#272C2×C411yescharacteristic
43C4×#272D411yescharacteristic
44C4×#272D411yescharacteristic
45C4×#272C2311yescharacteristic
46#6229—21no—
47#6229C2×C411yescharacteristic
48#6188—22no—
49C3×#421—91nomaximal
50C32×#41—22no—
51C3×#239—62no—
52C3×D3×Q8—62no—
53C3×#237—62no—
54C3×#237—64no—
55C3×D3×D4—62no—
56C3×D3×D4—64no—
57C3×#235—62no—
58C3×#235—64no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—64no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)C22⋊C411yescharacteristic
63C2×#272—21no—
64C2×#272—41no—
65C2×#272—42no—
66C2×#272C22⋊C411yescharacteristic
67C2×#272C2×D411yescharacteristic
68#1088—42no—
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)C22⋊C411yescharacteristic
71C4×H(3,𝔽3)C22×C411yescharacteristic
72#1084—21no—
73#1084C22⋊C411yescharacteristic
74#1084C2×D411yescharacteristic
75C2×C3×#41—91no—
76D3×#41—62no—
77C2×C3×M16—91no—
78C3×#117—91no—
79C3×#104—92no—
80C3×#104—92no—
81C22×C3×D3—62no—
82C22×C3×D3—64no—
83C32×Q8—22no—
84C32×D4—24no—
85C32×D4—22no—
86C2×C32×C4—24no—
87C2×C32×C4—22no—
88C3×#67—64no—
89C3×#67—64no—
90C3×#67—64no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—64no—
93C3×D12—62no—
94C3×D12—64no—
95C3×C4×D3—62no—
96C3×C4×D3—62no—
97C3×C4×D3—62no—
98C3×C4×D3—62no—
99C3×C4×D3—64no—
100C3×C4×D3—64no—
101C3×C4×D3—64no—
102C3×Dic6—62no—
103C3×Dic6—64no—
104#421—91noSylow
105C2×H(3,𝔽3)—21no—
106C2×H(3,𝔽3)—41no—
107C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
108#272—21no—
109#272—21no—
110#272—41no—
111C3×#41—62no—
112C3×#41—91no—
113C3×#41—91no—
114C3×#41—91no—
115C3×#41—181no—
116C3×#41—181no—
117C3×#41#15711yescharacteristic
118C2×C3×Q8—91no—
119C2×C3×D4—91no—
120C2×C3×D4—181no—
121C22×C3×C4—91no—
122C22×C3×C4—181no—
123#239—62no—
124D3×Q8—62no—
125#237—62no—
126#237—64no—
127D3×D4—62no—
128D3×D4—64no—
129#235—62no—
130#235—64no—
131C2×C4×D3—62no—
132C2×C4×D3—64no—
133C3×M16—92no—
134C3×M16—181no—
135C2×C3×C8—181no—
136C3×C4⋊C4—181no—
137C3×C22⋊C4—181no—
138C3×C42—92no—
139C22×C32—24no—
140C22×C32—22no—
141C2×C3×D3—62no—
142C2×C3×D3—62no—
143C2×C3×D3—64no—
144C2×C3×D3—124no—
145C2×C3×D3—128no—
146C32×C4—24no—
147C32×C4—22no—
148C32×C4—22no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—64no—
152C2×#41—91no—
153C2×M16—91no—
154#117—91no—
155#104—92no—
156#104—92no—
157H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
158C23×C3—91no—
159C23×C3—181no—
160C22×D3—62no—
161C22×D3—64no—
162C3×Q8—62no—
163C3×Q8—91no—
164C3×Q8—181no—
165C3×Q8C2×#15711yescharacteristic
166C3×D4—21no—
167C3×D4—64no—
168C3×D4—62no—
169C3×D4—91no—
170C3×D4—181no—
171C3×D4—181no—
172C3×D4—181no—
173C3×D4—181no—
174C3×D4C2×#15711yescharacteristic
175C2×C3×C4—21no—
176C2×C3×C4—64no—
177C2×C3×C4—62no—
178C2×C3×C4—91no—
179C2×C3×C4—91no—
180C2×C3×C4—91no—
181C2×C3×C4—91no—
182C2×C3×C4—91no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4—182no—
188C2×C3×C4C2×#15711yescharacteristic
189#67—64no—
190#67—64no—
191#67—64no—
192C2×Dic3—62no—
193C2×Dic3—64no—
194D12—62no—
195D12—64no—
196C4×D3—62no—
197C4×D3—62no—
198C4×D3—62no—
199C4×D3—62no—
200C4×D3—64no—
201C4×D3—64no—
202C4×D3—64no—
203Dic6—62no—
204Dic6—64no—
205C3×C8—182no—
206C2×C32—22no—
207C2×C32—42no—
208C2×C32—44no—
209C3×D3—64no—
210C3×D3—122no—
211C3×D3—124no—
212#41—91no—
213#41—91no—
214#41—91no—
215#41—181no—
216#41—181no—
217#41#108811yescharacteristic, complemented
218C2×Q8—91no—
219C2×D4—91no—
220C2×D4—181no—
221C22×C4—91no—
222C22×C4—181no—
223M16—92no—
224M16—181no—
225C2×C8—181no—
226C4⋊C4—181no—
227C22⋊C4—181no—
228C42—92no—
229C22×C3—21no—
230C22×C3—64no—
231C22×C3—62no—
232C22×C3—91no—
233C22×C3—91no—
234C22×C3—181no—
235C22×C3—361no—
236C22×C3—362no—
237C22×C3C22⋊#15711yescharacteristic
238C2×D3—62no—
239C2×D3—62no—
240C2×D3—64no—
241C2×D3—124no—
242C2×D3—128no—
243C3×C4—21no—
244C3×C4—64no—
245C3×C4—62no—
246C3×C4—62no—
247C3×C4—91no—
248C3×C4—91no—
249C3×C4—181no—
250C3×C4—362no—
251C3×C4C22⋊#15711yescharacteristic
252C3×C4C22×#15711yescharacteristic
253Dic3—62no—
254Dic3—62no—
255Dic3—64no—
256C32—22no—
257C23—91no—
258C23—181no—
259Q8—91no—
260Q8—181no—
261Q8C2×#108811yescharacteristic
262D4—21no—
263D4—91no—
264D4—181no—
265D4—181no—
266D4—181no—
267D4—181no—
268D4C2×#108811yescharacteristic
269C2×C4—21no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—91no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—181no—
276C2×C4—181no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—182no—
280C2×C4C2×#108811yescharacteristic
281C8—182no—
282C2×C3—21no—
283C2×C3—41no—
284C2×C3—62no—
285C2×C3—122no—
286C2×C3—124no—
287C2×C3—181no—
288C2×C3—181no—
289C2×C3—361no—
290C2×C3C2×C22⋊#15711yescharacteristic
291D3—64no—
292D3—122no—
293D3—124no—
294C22—21no—
295C22—91no—
296C22—91no—
297C22—181no—
298C22—361no—
299C22—362no—
300C22C22⋊#108811yescharacteristic
301C4—21no—
302C4—91no—
303C4—91no—
304C4—181no—
305C4—362no—
306C4C22⋊#108811yescharacteristic
307C4C22×#108811yescharacteristic
308C3—62no—
309C3#1059384211yescharacteristic
310C2—21no—
311C2—41no—
312C2—181no—
313C2—181no—
314C2—361no—
315C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C22⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
Fitting#41×H(3,𝔽3)C4
prime radical#41×#272C2
cosocleC4×#272C23
hypercenterC3×#41#157
hypocenterH(3,𝔽3)#421
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
2. #11722448C2semidirect
3. C2×#88627C2semidirect
4. #11720791C2semidirect
5. #11720789C2semidirect
6. #41×H(3,𝔽3)C4semidirect
8. Q8×#272C4semidirect
12. D4×#272C4semidirect
14. #88740C4semidirect
20. C4×#1088C22semidirect
22. #88627C22semidirect
157. H(3,𝔽3)#421semidirect
217. #41#1088semidirect

Maximal subgroups

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

subgroupindex
#41×#2722
#117224482
C2×#886272
#117207912
#117207892
C3×#4219