Finite Group Databasebeta

← #13147425

Subgroups — #13147425

Order 1728.

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

normal subgroups only

This group has 583 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
1#41×#272C211yesmaximal, characteristic
2C2×C4×#1088C211yesmaximal, characteristic, complemented
3#11722445C211yesmaximal, characteristic, complemented
4#11720791C212yesmaximal, complemented
5#11720789C212yesmaximal, complemented
6C22×#1088—21no—
7#41×H(3,𝔽3)C411yescharacteristic, complemented
8#88748C2211yescharacteristic
9Q8×#272C411yescharacteristic, complemented
10#88744—21no—
11#88744C2211yescharacteristic
12D4×#272—21no—
13D4×#272C411yescharacteristic, complemented
14#88740—21no—
15#88740C411yescharacteristic, complemented
16C2×C4×#272—21no—
17C2×C4×#272C2211yescharacteristic
18#88631—21no—
19C4×#1088—21no—
20C4×#1088C2212yescomplemented
21#88627C2212yescomplemented
22#88626—21no—
23C3×D3×#41—62no—
24C22×#272—21no—
25C22×#272D411yescharacteristic
26C2×#1088—24no—
27C2×#1088—22no—
28Q8×H(3,𝔽3)C2×C411yescharacteristic, complemented
29D4×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)C2×C411yescharacteristic, complemented
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2×C411yescharacteristic
33#6235—21no—
34#6235—21no—
35#6235—21no—
36C2×#1084—21no—
37C2×#1084D411yescharacteristic
38C4⋊#272—21no—
39C4⋊#272C2×C411yescharacteristic, complemented
40C4×#272—21no—
41C4×#272—21no—
42C4×#272—21no—
43C4×#272C2×C411yescharacteristic
44C4×#272D411yescharacteristic
45C4×#272D411yescharacteristic
46C4×#272C2311yescharacteristic
47#6229—21no—
48#6229C2×C411yescharacteristic, complemented
49#6188—22no—
50C2×C3×#104—91nomaximal
51C32×#41—22no—
52C3×#239—62no—
53C3×D3×Q8—62no—
54C3×#237—62no—
55C3×#237—64no—
56C3×D3×D4—62no—
57C3×D3×D4—64no—
58C3×#235—62no—
59C3×#235—64no—
60C2×C3×C4×D3—62no—
61C2×C3×C4×D3—64no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)C22⋊C411yescharacteristic
64C2×#272—22no—
65C2×#272—21no—
66C2×#272—42no—
67C2×#272C22⋊C411yescharacteristic
68C2×#272C2×D411yescharacteristic
69#1088—24no—
70C4×H(3,𝔽3)—21no—
71C4×H(3,𝔽3)C22⋊C411yescharacteristic
72C4×H(3,𝔽3)C22×C411yescharacteristic
73#1084—21no—
74#1084C22⋊C411yescharacteristic
75#1084C2×D411yescharacteristic
76C2×C3×#41—91no—
77D3×#41—62no—
78C2×C3×M16—91no—
79C2×C3×C42—91no—
80C3×#104—92no—
81C3×#104—92no—
82C22×C3×D3—62no—
83C22×C3×D3—64no—
84C32×Q8—22no—
85C32×D4—24no—
86C32×D4—22no—
87C2×C32×C4—24no—
88C2×C32×C4—22no—
89C3×#67—64no—
90C3×#67—64no—
91C3×#67—64no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—64no—
94C3×D12—62no—
95C3×D12—64no—
96C3×C4×D3—62no—
97C3×C4×D3—62no—
98C3×C4×D3—62no—
99C3×C4×D3—62no—
100C3×C4×D3—64no—
101C3×C4×D3—64no—
102C3×C4×D3—64no—
103C3×Dic6—62no—
104C3×Dic6—64no—
105C2×#104—91noSylow
106C2×H(3,𝔽3)—21no—
107C2×H(3,𝔽3)—41no—
108C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
109#272—21no—
110#272—41no—
111#272#10412yescomplemented
112C3×#41—62no—
113C3×#41—91no—
114C3×#41—91no—
115C3×#41—91no—
116C3×#41—181no—
117C3×#41—181no—
118C3×#41#15711yescharacteristic
119C2×C3×Q8—91no—
120C2×C3×D4—91no—
121C2×C3×D4—181no—
122C22×C3×C4—91no—
123C22×C3×C4—181no—
124C22×C3×C4—181no—
125#239—62no—
126D3×Q8—62no—
127#237—62no—
128#237—64no—
129D3×D4—62no—
130D3×D4—64no—
131#235—62no—
132#235—64no—
133C2×C4×D3—62no—
134C2×C4×D3—64no—
135C3×M16—92no—
136C3×M16—181no—
137C2×C3×C8—181no—
138C3×C42—92no—
139C3×C42—181no—
140C22×C32—24no—
141C22×C32—22no—
142C2×C3×D3—62no—
143C2×C3×D3—62no—
144C2×C3×D3—64no—
145C2×C3×D3—124no—
146C2×C3×D3—128no—
147C32×C4—24no—
148C32×C4—22no—
149C32×C4—22no—
150C3×Dic3—62no—
151C3×Dic3—62no—
152C3×Dic3—64no—
153C2×#41—91no—
154C2×M16—91no—
155C2×C42—91no—
156#104—92no—
157#104—92no—
158H(3,𝔽3)C2×#10411yescharacteristic, complemented, Sylow
159C23×C3—91no—
160C23×C3—181no—
161C22×D3—62no—
162C22×D3—64no—
163C3×Q8—62no—
164C3×Q8—91no—
165C3×Q8—181no—
166C3×Q8C2×#15711yescharacteristic
167C3×D4—21no—
168C3×D4—64no—
169C3×D4—62no—
170C3×D4—91no—
171C3×D4—181no—
172C3×D4—181no—
173C3×D4—181no—
174C3×D4—181no—
175C3×D4C2×#15711yescharacteristic
176C2×C3×C4—21no—
177C2×C3×C4—64no—
178C2×C3×C4—62no—
179C2×C3×C4—91no—
180C2×C3×C4—91no—
181C2×C3×C4—91no—
182C2×C3×C4—91no—
183C2×C3×C4—91no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4—182no—
188C2×C3×C4—184no—
189C2×C3×C4—181no—
190C2×C3×C4C2×#15711yescharacteristic
191#67—64no—
192#67—64no—
193#67—64no—
194C2×Dic3—62no—
195C2×Dic3—64no—
196D12—62no—
197D12—64no—
198C4×D3—62no—
199C4×D3—62no—
200C4×D3—62no—
201C4×D3—62no—
202C4×D3—64no—
203C4×D3—64no—
204C4×D3—64no—
205Dic6—62no—
206Dic6—64no—
207C3×C8—182no—
208C2×C32—22no—
209C2×C32—42no—
210C2×C32—44no—
211C3×D3—64no—
212C3×D3—122no—
213C3×D3—124no—
214#41—91no—
215#41—91no—
216#41—91no—
217#41—181no—
218#41—181no—
219#41#108811yescharacteristic, complemented
220C2×Q8—91no—
221C2×D4—91no—
222C2×D4—181no—
223C22×C4—91no—
224C22×C4—181no—
225C22×C4—181no—
226M16—92no—
227M16—181no—
228C2×C8—181no—
229C42—92no—
230C42—181no—
231C22×C3—21no—
232C22×C3—64no—
233C22×C3—62no—
234C22×C3—91no—
235C22×C3—91no—
236C22×C3—182no—
237C22×C3—181no—
238C22×C3—362no—
239C22×C3C22⋊#15711yescharacteristic
240C2×D3—62no—
241C2×D3—62no—
242C2×D3—64no—
243C2×D3—124no—
244C2×D3—128no—
245C3×C4—21no—
246C3×C4—64no—
247C3×C4—62no—
248C3×C4—62no—
249C3×C4—91no—
250C3×C4—91no—
251C3×C4—184no—
252C3×C4—181no—
253C3×C4C22⋊#15711yescharacteristic
254C3×C4C22×#15711yescharacteristic
255Dic3—62no—
256Dic3—62no—
257Dic3—64no—
258C32—22no—
259C23—91no—
260C23—181no—
261Q8—91no—
262Q8—181no—
263Q8C2×#108811yescharacteristic, complemented
264D4—21no—
265D4—91no—
266D4—181no—
267D4—181no—
268D4—181no—
269D4—181no—
270D4C2×#108811yescharacteristic, complemented
271C2×C4—21no—
272C2×C4—91no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—91no—
276C2×C4—91no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—182no—
281C2×C4—184no—
282C2×C4—181no—
283C2×C4C2×#108811yescharacteristic
284C8—182no—
285C2×C3—21no—
286C2×C3—41no—
287C2×C3—62no—
288C2×C3—92no—
289C2×C3—122no—
290C2×C3—124no—
291C2×C3—181no—
292C2×C3—361no—
293C2×C3C2×C22⋊#15711yescharacteristic
294D3—64no—
295D3—122no—
296D3—124no—
297C22—21no—
298C22—91no—
299C22—91no—
300C22—182no—
301C22—181no—
302C22—362no—
303C22C22⋊#108811yescharacteristic
304C4—21no—
305C4—91no—
306C4—91no—
307C4—184no—
308C4—181no—
309C4C22⋊#108811yescharacteristic
310C4C22×#108811yescharacteristic
311C3—62no—
312C3#1059383911yescharacteristic
313C2—21no—
314C2—41no—
315C2—92no—
316C2—181no—
317C2—361no—
318C2C2×C22⋊#108811yescharacteristic
subgroupG/H
centerC3×C4C22⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
Fitting#41×H(3,𝔽3)C4
socleC2×C3C2×C22⋊#157
prime radical#41×#272C2
cosocleC4×#272C23
hypercenterC3×#41#157
hypocenterH(3,𝔽3)C2×#104
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
2. C2×C4×#1088C2semidirect
3. #11722445C2semidirect
4. #11720791C2semidirect
5. #11720789C2semidirect
7. #41×H(3,𝔽3)C4semidirect
9. Q8×#272C4semidirect
13. D4×#272C4semidirect
15. #88740C4semidirect
20. C4×#1088C22semidirect
21. #88627C22semidirect
28. Q8×H(3,𝔽3)C2×C4semidirect
30. D4×H(3,𝔽3)C2×C4semidirect
39. C4⋊#272C2×C4semidirect
48. #6229C2×C4semidirect
111. #272#104semidirect
158. H(3,𝔽3)C2×#104semidirect
219. #41#1088semidirect
263. Q8C2×#1088semidirect
270. D4C2×#1088semidirect

Maximal subgroups

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

subgroupindex
#41×#2722
C2×C4×#10882
#117224452
#117207912
#117207892
C2×C3×#1049