Finite Group Databasebeta

← D3×Q8×#157 #13180991

Subgroups — D3×Q8×#157 #13180991

Order 1728.

984 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 194 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
4D3×C4⋊#157C213yesmaximal
5C4×D3×#157C213yesmaximal
6#11724245C213yesmaximal, complemented
7Dic6×#157C213yesmaximal, complemented
8C2×Q8×#157—31nomaximal
9C2×D3×#157C2213yes—
10C2×D3×#157C2211yescharacteristic
11Q8×C3⋊(C3⋊D3)C411yescharacteristic, complemented
12C3×Q8×C3⋊D3C2211yescharacteristic
13C32×D3×Q8C411yescharacteristic, complemented
14C4×D3×C3⋊D3C2213yes—
15#89019C413yescomplemented
16C3⋊D3×Dic6C2213yes—
17D3×#634C413yescomplemented
18C4⋊(C3⋊#157)C2213yes—
19C4×C3⋊#157C2213yes—
20C3×C4⋊#157C2213yes—
21C3×C4×#157C2213yes—
22Dic3⋊#157C2216yes—
23Dic3×#157C2213yes—
24Dic3×#157C2213yes—
25C2×Q8×C3⋊D3—31no—
26D32×Q8—62no—
27Q8×#157—31no—
28Q8×#157—33no—
29Q8×#157—33no—
30Q8×#157D311yescharacteristic, direct product factor
31C2×C4⋊#157—33no—
32C2×C4×#157—33no—
33C2×D3×C3⋊D3C2311yescharacteristic
34C2×C3⋊#157C2313yes—
35C2×C3⋊#157C2311yescharacteristic
36C2×C3×#157C2313yes—
37C2×C3×#157C2311yescharacteristic
38D3×#157—26no—
39D3×#157Q814yesdirect product factor
40C33×Q8C2×C411yescharacteristic, complemented
41C4×C3⋊(C3⋊D3)C2×C413yes—
42#6308C2×C413yescomplemented
43C3×C4×C3⋊D3C2313yes—
44C3×#634C2×C413yescomplemented
45C32×C4×D3C2×C413yes—
46C32×Dic6C2×C413yescomplemented
47#6293C2×C416yescomplemented
48#6289C2×C413yes—
49Dic3×C3⋊D3C2313yes—
50D3×C3⋊Dic3C2×C413yes—
51C4×D3×Q8—91nomaximal
52C22×#157—31no—
53C22×#157—33no—
54C2×C32×Q8—31no—
55Q8×C3⋊D3—31no—
56Q8×C3⋊D3—33no—
57Q8×C3⋊D3—33no—
58Q8×C3⋊D3—62no—
59Q8×C3⋊D3—122no—
60Q8×C3⋊D3C2×D311yescharacteristic
61C2×C4×C3⋊D3—33no—
62C2×#634—33no—
63C3×D3×Q8—22no—
64C3×D3×Q8—62no—
65C4×D32—66no—
66#3797—66no—
67D3×Dic6—66no—
68D3×Dic6—66no—
69C4⋊#157—36no—
70C4⋊#157—33no—
71C4⋊#157C2×D313yes—
72C4×#157—33no—
73C4×#157—33no—
74C4×#157—33no—
75C4×#157C2×D313yes—
76C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
77C2×C3×C3⋊D3C2411yescharacteristic
78C2×C32×D3C22×C411yescharacteristic
79D3×C3⋊D3C2×Q812yes—
80D3×C3⋊D3#4112yes—
81C3⋊#157—23no—
82C3⋊#157C2×Q812yescomplemented
83C3×#157—23no—
84C3×#157C2×Q812yescomplemented
85C33×C4C22×C413yes—
86C3⋊(C3⋊Dic3)C22×C413yes—
87C3×C3⋊Dic3C22×C413yes—
88C32×Dic3C22×C413yes—
89C2×D3×Q8—91no—
90C2×D3×Q8—182no—
91C3×C4×Q8—91no—
92Q8×Dic3—91no—
93D3×C4⋊C4—93no—
94#887—93no—
95C42×D3—93no—
96C4×Dic6—93no—
97C22×C3⋊D3—31no—
98C2×D32—62no—
99C2×#157—31no—
100C2×#157—34no—
101C2×#157—33no—
102C2×#157—66no—
103C2×#157C22×D313yes—
104C2×#157C22×D311yescharacteristic
105C32×Q8—22no—
106C32×Q8—33no—
107C32×Q8—42no—
108C32×Q8C4×D311yescharacteristic, complemented
109C2×C32×C4—33no—
110C2×C3⋊Dic3—33no—
111C4×C3⋊D3—33no—
112C4×C3⋊D3—33no—
113C4×C3⋊D3—33no—
114C4×C3⋊D3—66no—
115C4×C3⋊D3—126no—
116C4×C3⋊D3C22×D313yes—
117#634—33no—
118#634—36no—
119#634—66no—
120#634—126no—
121#634C4×D313yescomplemented
122C3×C4×D3—26no—
123C3×C4×D3—66no—
124C3×Dic6—26no—
125C3×Dic6—66no—
126#627—612no—
127#624—66no—
128D3×Dic3—66no—
129D3×Dic3—66no—
130C2×C4×Q8—271noSylow
131C2×C33C23×C411yescharacteristic
132C3⋊(C3⋊D3)C4×Q812yescomplemented
133C3×C3⋊D3C22×Q811yescharacteristic
134C3×C3⋊D3C2×#4111yescharacteristic
135C32×D3C4×Q812yescomplemented
136C2×C3×Q8—62no—
137C2×C3×Q8—91no—
138D3×Q8—62no—
139D3×Q8—91no—
140D3×Q8—93no—
141D3×Q8—93no—
142D3×Q8—186no—
143D3×Q8—186no—
144D3×Q8—182no—
145D3×Q8—362no—
146D3×Q8#15711yescharacteristic, direct product factor
147C2×C4×D3—91no—
148C2×C4×D3—93no—
149C2×C4×D3—93no—
150C2×C4×D3—186no—
151C2×Dic6—93no—
152C2×Dic6—186no—
153C3×C4⋊C4—93no—
154C3×C42—93no—
155C4⋊Dic3—93no—
156Dic3⋊C4—96no—
157C4×Dic3—93no—
158C4×Dic3—93no—
159C4×Dic3—93no—
160C22×C32—31no—
161C2×C3⋊D3—31no—
162C2×C3⋊D3—32no—
163C2×C3⋊D3—32no—
164C2×C3⋊D3—62no—
165C2×C3⋊D3—122no—
166C2×C3⋊D3C23×D311yescharacteristic
167C2×C3×D3—22no—
168C2×C3×D3—62no—
169D32—64no—
170D32—64no—
171#157—23no—
172#157—32no—
173#157—63no—
174#157D3×Q812yesdirect product factor
175C32×C4—26no—
176C32×C4—33no—
177C32×C4—46no—
178C32×C4C2×C4×D313yes—
179C3⋊Dic3—33no—
180C3⋊Dic3—66no—
181C3⋊Dic3—126no—
182C3⋊Dic3C2×C4×D313yes—
183C3×Dic3—26no—
184C3×Dic3—66no—
185C22×Q8—271no—
186C4×Q8—91no—
187C4×Q8—273no—
188C4×Q8—273no—
189C4×Q8—271no—
190C2×C4⋊C4—273no—
191C2×C42—273no—
192C33C2×C4×Q811yescharacteristic, complemented, Sylow
193C22×D3—91no—
194C22×D3—182no—
195C3×Q8—22no—
196C3×Q8—42no—
197C3×Q8—66no—
198C3×Q8—93no—
199C3×Q8C2×#15711yescharacteristic, complemented
200C2×C3×C4—66no—
201C2×C3×C4—91no—
202C2×C3×C4—93no—
203C2×C3×C4—93no—
204C2×Dic3—91no—
205C2×Dic3—93no—
206C2×Dic3—93no—
207C2×Dic3—186no—
208C4×D3—66no—
209C4×D3—94no—
210C4×D3—93no—
211C4×D3—93no—
212C4×D3—93no—
213C4×D3—186no—
214C4×D3—186no—
215C4×D3—186no—
216C4×D3—186no—
217C4×D3—366no—
218C4×D3C2×#15713yes—
219Dic6—66no—
220Dic6—93no—
221Dic6—96no—
222Dic6—1812no—
223Dic6—186no—
224Dic6—366no—
225Dic6C2×#15713yescomplemented
226C2×C32—22no—
227C2×C32—32no—
228C2×C32—42no—
229C2×C32C22×C4×D311yescharacteristic
230C3⋊D3—32no—
231C3⋊D3—64no—
232C3⋊D3—124no—
233C3⋊D3C2×D3×Q811yescharacteristic
234C3⋊D3D3×#4111yescharacteristic
235C3×D3—24no—
236C3×D3—62no—
237C3×D3—62no—
238C2×Q8—31no—
239C2×Q8—91no—
240C2×Q8—273no—
241C2×Q8—273no—
242C2×Q8—273no—
243C2×Q8—271no—
244C22×C4—273no—
245C22×C4—271no—
246C22×C4—273no—
247C4⋊C4—93no—
248C4⋊C4—276no—
249C4⋊C4—273no—
250C42—93no—
251C42—273no—
252C42—273no—
253C42—273no—
254C22×C3—62no—
255C22×C3—91no—
256C2×D3—62no—
257C2×D3—91no—
258C2×D3—92no—
259C2×D3—92no—
260C2×D3—184no—
261C2×D3—184no—
262C2×D3—182no—
263C2×D3—362no—
264C2×D3C22×#15711yescharacteristic
265C3×C4—26no—
266C3×C4—46no—
267C3×C4—66no—
268C3×C4—92no—
269C3×C4—93no—
270C3×C4—183no—
271C3×C4C22×#15713yes—
272Dic3—66no—
273Dic3—92no—
274Dic3—93no—
275Dic3—186no—
276Dic3—183no—
277Dic3—366no—
278Dic3C22×#15713yes—
279C32—22no—
280C32—42no—
281C32C4×D3×Q811yescharacteristic, complemented
282C23—271no—
283Q8—33no—
284Q8—93no—
285Q8—276no—
286Q8—273no—
287Q8D3×#15711yescharacteristic, direct product factor
288C2×C4—33no—
289C2×C4—91no—
290C2×C4—93no—
291C2×C4—93no—
292C2×C4—273no—
293C2×C4—273no—
294C2×C4—273no—
295C2×C4—274no—
296C2×C4—271no—
297C2×C4—273no—
298C2×C4—273no—
299C2×C4—546no—
300C2×C3—22no—
301C2×C3—42no—
302C2×C3—64no—
303C2×C3—91no—
304C2×C3—91no—
305C2×C3C23×#15711yescharacteristic
306D3—62no—
307D3—62no—
308D3—92no—
309D3—184no—
310D3—364no—
311D3Q8×#15712yesdirect product factor
312C22—31no—
313C22—91no—
314C22—272no—
315C22—272no—
316C22—271no—
317C4—33no—
318C4—92no—
319C4—93no—
320C4—183no—
321C4—272no—
322C4—273no—
323C4—543no—
324C4C2×D3×#15713yes—
325C3—22no—
326C3—42no—
327C3C2×Q8×#15711yescharacteristic, complemented
328C2—32no—
329C2—91no—
330C2—91no—
331C2—272no—
332C2C22×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
hypocenterC33C2×C4×Q8
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
11. Q8×C3⋊(C3⋊D3)C4semidirect
13. C32×D3×Q8C4semidirect
15. #89019C4semidirect
17. D3×#634C4semidirect
30. Q8×#157D3direct (N × H)
39. D3×#157Q8direct (N × H)
40. C33×Q8C2×C4semidirect
42. #6308C2×C4semidirect
44. C3×#634C2×C4semidirect
46. C32×Dic6C2×C4semidirect
47. #6293C2×C4semidirect
82. C3⋊#157C2×Q8semidirect
84. C3×#157C2×Q8semidirect
108. C32×Q8C4×D3semidirect
121. #634C4×D3semidirect
132. C3⋊(C3⋊D3)C4×Q8semidirect
135. C32×D3C4×Q8semidirect
146. D3×Q8#157direct (N × H)
174. #157D3×Q8direct (N × H)
192. C33C2×C4×Q8semidirect
199. C3×Q8C2×#157semidirect
225. Dic6C2×#157semidirect
281. C32C4×D3×Q8semidirect
287. Q8D3×#157direct (N × H)
311. D3Q8×#157direct (N × H)
327. C3C2×Q8×#157semidirect

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
D3×C4⋊#1572
C4×D3×#1572
#117242452
Dic6×#1572
C2×Q8×#1573
C4×D3×Q89