Finite Group Databasebeta

← #13180986

Subgroups — #13180986

Order 1728.

920 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 174 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
1D3×Q8×C3⋊D3C211yesmaximal, characteristic
2Q8×C3⋊#157C211yesmaximal, characteristic, complemented
3C3×Q8×#157C211yesmaximal, characteristic, complemented
4#11724247C213yesmaximal
5#11724246C213yesmaximal
6#11724245C213yesmaximal, complemented
7Dic6×#157C213yesmaximal, complemented
8#10596646—31nomaximal
9Q8×C3⋊(C3⋊D3)C411yescharacteristic, complemented
10C3×Q8×C3⋊D3C2211yescharacteristic
11C32×D3×Q8C411yescharacteristic, complemented
12C4×D3×C3⋊D3C2213yes—
13#89019C413yescomplemented
14C3⋊D3×Dic6C2213yes—
15D3×#634C413yescomplemented
16C4⋊(C3⋊#157)C2213yes—
17C4×C3⋊#157C2213yes—
18C3×C4⋊#157C2213yes—
19C3×C4×#157C2213yes—
20Dic3⋊#157C2213yes—
21Dic3⋊#157C2213yes—
22(C2×D3)⋊#157C2211yescharacteristic
23(C2×D3)⋊#157C2213yes—
24Dic3×#157C2216yes—
25C2×Q8×C3⋊D3—31no—
26D32×Q8—62no—
27Q8×#157—31no—
28Q8×#157—33no—
29Q8×#157—33no—
30Q8×#157D311yescharacteristic, complemented
31#64286—33no—
32#64286—33no—
33C2×D3×C3⋊D3C2311yescharacteristic
34C2×C3⋊#157C2313yes—
35C2×C3⋊#157C2311yescharacteristic
36C2×C3×#157C2313yes—
37C2×C3×#157C2311yescharacteristic
38C33×Q8C2×C411yescharacteristic
39C4×C3⋊(C3⋊D3)C2×C413yes—
40#6308C2×C413yes—
41C3×C4×C3⋊D3C2313yes—
42C3×#634C2×C413yes—
43C32×C4×D3C2×C413yes—
44C32×Dic6C2×C413yes—
45#6293C2×C416yes—
46#6289C2×C413yes—
47Dic3×C3⋊D3C2313yes—
48D3×C3⋊Dic3C2×C413yes—
49#5572—91nomaximal
50C2×C32×Q8—31no—
51Q8×C3⋊D3—31no—
52Q8×C3⋊D3—33no—
53Q8×C3⋊D3—33no—
54Q8×C3⋊D3—62no—
55Q8×C3⋊D3—122no—
56Q8×C3⋊D3C2×D311yescharacteristic
57C2×C4×C3⋊D3—33no—
58C2×#634—33no—
59C3×D3×Q8—22no—
60C3×D3×Q8—62no—
61C4×D32—66no—
62#3797—66no—
63D3×Dic6—66no—
64D3×Dic6—66no—
65C22⋊#157—33no—
66C22⋊#157—31no—
67C4⋊#157—33no—
68C4⋊#157—33no—
69C4⋊#157—33no—
70C4⋊#157C2×D313yes—
71C4×#157—36no—
72C4×#157—33no—
73C4×#157C2×D313yes—
74C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
75C2×C3×C3⋊D3C2411yescharacteristic
76C2×C32×D3C22×C411yescharacteristic
77D3×C3⋊D3—21no—
78D3×C3⋊D3—21no—
79C3⋊#157—23no—
80C3⋊#157—21no—
81C3×#157—21no—
82C3×#157—23no—
83C33×C4C22×C413yes—
84C3⋊(C3⋊Dic3)C22×C413yes—
85C3×C3⋊Dic3C22×C413yes—
86C32×Dic3C22×C413yes—
87C2×D3×Q8—91no—
88C2×D3×Q8—182no—
89C3×C4×Q8—91no—
90Q8×Dic3—91no—
91#892—93no—
92#887—93no—
93#872—93no—
94C4×Dic6—93no—
95C22×C3⋊D3—31no—
96C2×D32—62no—
97C2×#157—33no—
98C2×#157—31no—
99C2×#157C22×D313yes—
100C2×#157C22×D311yescharacteristic
101C32×Q8—22no—
102C32×Q8—33no—
103C32×Q8—42no—
104C32×Q8C4×D311yescharacteristic
105C2×C32×C4—33no—
106C2×C3⋊Dic3—33no—
107C4×C3⋊D3—33no—
108C4×C3⋊D3—33no—
109C4×C3⋊D3—33no—
110C4×C3⋊D3—66no—
111C4×C3⋊D3—126no—
112C4×C3⋊D3C22×D313yes—
113#634—33no—
114#634—36no—
115#634—66no—
116#634—126no—
117#634C4×D313yes—
118C3×C4×D3—26no—
119C3×C4×D3—66no—
120C3×Dic6—26no—
121C3×Dic6—66no—
122#627—612no—
123#624—66no—
124D3×Dic3—66no—
125D3×Dic3—66no—
126#519—271noSylow
127C2×C33C23×C411yescharacteristic
128C3⋊(C3⋊D3)—21no—
129C3×C3⋊D32−1+411yescharacteristic
130C3×C3⋊D32−1+411yescharacteristic
131C32×D3—21no—
132C2×C3×Q8—62no—
133C2×C3×Q8—91no—
134D3×Q8—62no—
135D3×Q8—91no—
136D3×Q8—93no—
137D3×Q8—93no—
138D3×Q8—186no—
139D3×Q8—186no—
140D3×Q8—182no—
141D3×Q8—362no—
142D3×Q8#15711yescharacteristic, complemented
143C2×C4×D3—93no—
144C2×C4×D3—186no—
145C2×Dic6—93no—
146C2×Dic6—186no—
147C3×C4⋊C4—93no—
148C3×C42—93no—
149(C2×D3)⋊C4—91no—
150(C2×D3)⋊C4—93no—
151C4⋊Dic3—93no—
152Dic3⋊C4—93no—
153Dic3⋊C4—93no—
154C4×Dic3—96no—
155C4×Dic3—93no—
156C22×C32—31no—
157C2×C3⋊D3—31no—
158C2×C3⋊D3—62no—
159C2×C3⋊D3—61no—
160C2×C3⋊D3—61no—
161C2×C3⋊D3—122no—
162C2×C3⋊D3C23×D311yescharacteristic
163C2×C3×D3—22no—
164C2×C3×D3—62no—
165D32—64no—
166D32—64no—
167#157—21no—
168#157—23no—
169#157—63no—
170#157—61no—
171C32×C4—26no—
172C32×C4—33no—
173C32×C4—46no—
174C32×C4C2×C4×D313yes—
175C3⋊Dic3—33no—
176C3⋊Dic3—66no—
177C3⋊Dic3—126no—
178C3⋊Dic3C2×C4×D313yes—
179C3×Dic3—26no—
180C3×Dic3—66no—
181C22×Q8—271no—
182C4×Q8—91no—
183C4×Q8—273no—
184C4×Q8—273no—
185C4×Q8—271no—
186#117—273no—
187#117—273no—
188C33#51911yescharacteristic, complemented, Sylow
189C22×D3—91no—
190C22×D3—182no—
191C3×Q8—22no—
192C3×Q8—42no—
193C3×Q8—66no—
194C3×Q8—93no—
195C3×Q8C2×#15711yescharacteristic
196C2×C3×C4—66no—
197C2×C3×C4—93no—
198C2×C3×C4—93no—
199C2×C3×C4—91no—
200C2×Dic3—93no—
201C2×Dic3—91no—
202C2×Dic3—93no—
203C2×Dic3—186no—
204C4×D3—66no—
205C4×D3—93no—
206C4×D3—93no—
207C4×D3—93no—
208C4×D3—186no—
209C4×D3—186no—
210C4×D3—186no—
211C4×D3—366no—
212C4×D3C2×#15713yes—
213Dic6—66no—
214Dic6—93no—
215Dic6—96no—
216Dic6—1812no—
217Dic6—186no—
218Dic6—366no—
219Dic6C2×#15713yes—
220C2×C32—22no—
221C2×C32—42no—
222C2×C32—61no—
223C2×C32C22×C4×D311yescharacteristic
224C3⋊D3—61no—
225C3⋊D3—64no—
226C3⋊D3—124no—
227C3⋊D3#100711yescharacteristic
228C3⋊D3#101011yescharacteristic
229C3×D3—24no—
230C3×D3—62no—
231C3×D3—62no—
232C2×Q8—31no—
233C2×Q8—91no—
234C2×Q8—273no—
235C2×Q8—273no—
236C2×Q8—273no—
237C2×Q8—271no—
238C22×C4—273no—
239C4⋊C4—93no—
240C4⋊C4—273no—
241C4⋊C4—273no—
242C4⋊C4—273no—
243C22⋊C4—271no—
244C22⋊C4—273no—
245C42—93no—
246C42—273no—
247C42—276no—
248C22×C3—62no—
249C22×C3—91no—
250C2×D3—62no—
251C2×D3—91no—
252C2×D3—182no—
253C2×D3—181no—
254C2×D3—181no—
255C2×D3—184no—
256C2×D3—184no—
257C2×D3—362no—
258C2×D3C22×#15711yescharacteristic
259C3×C4—26no—
260C3×C4—46no—
261C3×C4—66no—
262C3×C4—93no—
263C3×C4—181no—
264C3×C4—183no—
265C3×C4C22×#15713yes—
266Dic3—66no—
267Dic3—93no—
268Dic3—186no—
269Dic3—183no—
270Dic3—181no—
271Dic3—366no—
272Dic3C22×#15713yes—
273C32—22no—
274C32—42no—
275C32#557211yescharacteristic, complemented
276C23—271no—
277Q8—33no—
278Q8—93no—
279Q8—276no—
280Q8—273no—
281Q8D3×#15711yescharacteristic
282C2×C4—33no—
283C2×C4—93no—
284C2×C4—91no—
285C2×C4—93no—
286C2×C4—273no—
287C2×C4—273no—
288C2×C4—273no—
289C2×C4—273no—
290C2×C4—271no—
291C2×C4—273no—
292C2×C3—22no—
293C2×C3—42no—
294C2×C3—64no—
295C2×C3—91no—
296C2×C3—91no—
297C2×C3C23×#15711yescharacteristic
298D3—21no—
299D3—62no—
300D3—62no—
301D3—184no—
302D3—181no—
303D3—364no—
304C22—31no—
305C22—91no—
306C22—271no—
307C22—541no—
308C22—541no—
309C4—33no—
310C4—93no—
311C4—183no—
312C4—181no—
313C4—273no—
314C4—541no—
315C4—543no—
316C4C2×D3×#15713yes—
317C3—22no—
318C3—42no—
319C3#1059664611yescharacteristic, complemented
320C2—61no—
321C2—91no—
322C2—91no—
323C2—541no—
324C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC33×Q8C2×C4
prime radicalC2×D3×C3⋊D3C23
cosocleC2×C3×C3⋊D3C24
hypercenterQ8D3×#157
hypocenterC33#519
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
2. Q8×C3⋊#157C2semidirect
3. C3×Q8×#157C2semidirect
6. #11724245C2semidirect
7. Dic6×#157C2semidirect
9. Q8×C3⋊(C3⋊D3)C4semidirect
11. C32×D3×Q8C4semidirect
13. #89019C4semidirect
15. D3×#634C4semidirect
30. Q8×#157D3semidirect
142. D3×Q8#157semidirect
188. C33#519semidirect
275. C32#5572semidirect
319. C3#10596646semidirect

Maximal subgroups

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

subgroupindex
D3×Q8×C3⋊D32
Q8×C3⋊#1572
C3×Q8×#1572
#117242472
#117242462
#117242452
Dic6×#1572
#105966463
#55729