Finite Group Databasebeta

← #13166533

Subgroups — #13166533

Order 1728.

464 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 62 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
1D4×C3⋊#157C211yesmaximal, characteristic, complemented
2#11724259C211yesmaximal, characteristic, complemented
3C3⋊D3×#213C211yesmaximal, characteristic
4C3×#63782C211yesmaximal, characteristic, complemented
5#11722715C211yesmaximal, characteristic, complemented
6#11722707C211yesmaximal, characteristic, complemented
7#11722703C211yesmaximal, characteristic, complemented
8D8×#157—31nomaximal
9C22×C3⋊#157—21no—
10C2×D3×#157—21no—
11C3×D4×C3⋊D3C2211yescharacteristic
12C3⋊D3×D12C2211yescharacteristic
13#88994—21no—
14C4⋊(C3⋊#157)C2211yescharacteristic
15C4×C3⋊#157C2211yescharacteristic
16C3×C4⋊#157C2211yescharacteristic
17C3×#3787C2211yescharacteristic
18(C2×D3)⋊#157—21no—
19#88860C411yescharacteristic, complemented
20C32×#213C411yescharacteristic, complemented
21#88791C411yescharacteristic, complemented
22#88789C411yescharacteristic, complemented
23C3⋊D3×C3⋊C8C2211yescharacteristic
24D4×#157—31no—
25D4×#157—31no—
26D8×C3⋊D3—31no—
27D3×#213—62no—
28#63782—31no—
29#63782D311yescharacteristic, complemented
30#63769—31no—
31C8×#157—31no—
32C22×C3×C3⋊D3—21no—
33C2×D3×C3⋊D3—21no—
34C2×C3⋊#157—21no—
35C2×C3⋊#157—42no—
36C2×C3⋊#157D411yescharacteristic
37C2×C3⋊#157D411yescharacteristic
38C2×C3×#157—21no—
39D3×#157—42no—
40C33×D4C2×C411yescharacteristic, complemented
41C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic, complemented
42C3×#638—21no—
43C3×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
44C3×C4×C3⋊D3C2311yescharacteristic
45C32×D12C2×C411yescharacteristic, complemented
46#6291—21no—
47C3×#622—21no—
48C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic, complemented
49C32×C3⋊C8C2×C411yescharacteristic, complemented
50#4756—91nomaximal
51C22×#157—61no—
52C22×#157—61no—
53D4×C3⋊D3—31no—
54D4×C3⋊D3C2×D311yescharacteristic
55C3×D3×D4—62no—
56D3×D12—62no—
57C22⋊#157—61no—
58C22⋊#157—61no—
59C4⋊#157—31no—
60C4⋊#157C2×D311yescharacteristic
61C4×#157—31no—
62#3787C2×D311yescharacteristic
63C32×D8—31no—
64#3753—31no—
65#3753—31no—
66#3753—62no—
67#3753—122no—
68#3745—31no—
69C8×C3⋊D3—31no—
70C3×#213—22no—
71#3714—62no—
72#3713—62no—
73D3×C3⋊C8—62no—
74C22×C33—21no—
75C2×C3⋊(C3⋊D3)—21no—
76C2×C3×C3⋊D3—21no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3C2×D411yescharacteristic
80C2×C32×D3—21no—
81D3×C3⋊D3—41no—
82D3×C3⋊D3—41no—
83C3⋊#157—21no—
84C3⋊#157—41no—
85C3⋊#157D812yescomplemented
86C3×#157—41no—
87C33×C4C22×C411yescharacteristic
88C3×C3⋊Dic3#4111yescharacteristic
89D4×Dic3—91no—
90C2×#213—91no—
91D3×D8—182no—
92#893—91no—
93C3×#102—91no—
94#821—91no—
95C8×Dic3—91no—
96#807—91no—
97C22×C3⋊D3—21no—
98C22×C3⋊D3—61no—
99C22×C3×D3—122no—
100C2×D32—122no—
101C2×#157—21no—
102C2×#157—31no—
103C2×#157—31no—
104C2×#157—61no—
105C2×#157—122no—
106C2×#157—122no—
107C32×D4—22no—
108C32×D4—31no—
109C32×D4—42no—
110C32×D4C4×D311yescharacteristic, complemented
111#638—21no—
112#638—61no—
113C4⋊(C3⋊D3)—31no—
114C4⋊(C3⋊D3)—62no—
115C4⋊(C3⋊D3)—122no—
116C4⋊(C3⋊D3)C4×D311yescharacteristic, complemented
117C4×C3⋊D3C22×D311yescharacteristic
118C3×#67—122no—
119C3×D12—22no—
120C3×D12—62no—
121C3×C4×D3—62no—
122#626—122no—
123#622—21no—
124C32×C8—31no—
125C3⋊(C3⋊C8)—31no—
126C3⋊(C3⋊C8)—62no—
127C3⋊(C3⋊C8)—122no—
128C3×C3⋊C8—22no—
129C4×D8—271noSylow
130C2×C33—41no—
131C2×C33C4×D411yescharacteristic
132C3⋊(C3⋊D3)—41no—
133C3×C3⋊D3—41no—
134C3×C3⋊D3C2×D811yescharacteristic
135C3×C3⋊D3#13511yescharacteristic
136C32×D3—41no—
137C2×C3×D4—91no—
138C22×Dic3—181no—
139D3×D4—62no—
140D3×D4—182no—
141C2×D12—91no—
142C2×C4×D3—181no—
143C3×D8—62no—
144C2×C3×C8—91no—
145C3×C4⋊C4—91no—
146#217—181no—
147#213—91no—
148#213—91no—
149#213—91no—
150#213—182no—
151#213—182no—
152#213—362no—
153#213#15711yescharacteristic, complemented
154(C2×D3)⋊C4—181no—
155C4⋊Dic3—91no—
156C4×Dic3—91no—
157C2×C3⋊C8—91no—
158D24—182no—
159D3×C8—182no—
160C22×C32—21no—
161C22×C32—42no—
162C22×C32—61no—
163C22×C32—82no—
164C2×C3⋊D3—21no—
165C2×C3⋊D3—41no—
166C2×C3⋊D3—41no—
167C2×C3⋊D3—61no—
168C2×C3⋊D3—122no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—121no—
171C2×C3⋊D3—242no—
172C2×C3⋊D3D3×D411yescharacteristic
173C2×C3×D3—42no—
174C2×C3×D3—62no—
175C2×C3×D3—122no—
176C2×C3×D3—242no—
177C2×C3×D3—242no—
178D32—242no—
179D32—242no—
180#157—32no—
181#157—41no—
182#157—61no—
183#157—121no—
184C32×C4—22no—
185C32×C4—42no—
186C32×C4C2×C4×D311yescharacteristic
187C3⋊Dic3#23711yescharacteristic
188C3×Dic3—62no—
189C2×D8—271no—
190C4×D4—271no—
191C4×D4—271no—
192#107—271no—
193#102—91no—
194#102—271no—
195C4×C8—271no—
196C33C4×D811yescharacteristic, complemented, Sylow
197C23×C3—181no—
198C22×D3—122no—
199C22×D3—181no—
200C22×D3—362no—
201C3×D4—22no—
202C3×D4—42no—
203C3×D4—62no—
204C3×D4—91no—
205C3×D4—181no—
206C3×D4C2×#15711yescharacteristic, complemented
207C2×C3×C4—91no—
208C2×C3×C4—181no—
209#67—122no—
210#67—362no—
211C2×Dic3—91no—
212C2×Dic3—91no—
213C2×Dic3—181no—
214C2×Dic3—362no—
215D12—62no—
216D12—91no—
217D12—182no—
218D12—181no—
219D12—362no—
220D12C2×#15711yescharacteristic, complemented
221C4×D3—62no—
222C4×D3—362no—
223C3×C8—62no—
224C3×C8—181no—
225C3⋊C8—91no—
226C3⋊C8—182no—
227C3⋊C8—362no—
228C3⋊C8C2×#15711yescharacteristic, complemented
229C2×C32—22no—
230C2×C32—42no—
231C2×C32—41no—
232C2×C32—82no—
233C2×C32—121no—
234C2×C32—162no—
235C2×C32#67⋊C411yescharacteristic
236C3⋊D3—41no—
237C3⋊D3—121no—
238C3⋊D3—242no—
239C3⋊D3—482no—
240C3⋊D3D3×D811yescharacteristic
241C3⋊D3#91611yescharacteristic
242C3×D3—62no—
243C3×D3—62no—
244C3×D3—82no—
245C3×D3—242no—
246C2×D4—91no—
247C2×D4—271no—
248C22×C4—541no—
249C22×C4—541no—
250D8—31no—
251D8—271no—
252D8—271no—
253D8—271no—
254C2×C8—91no—
255C2×C8—271no—
256C4⋊C4—91no—
257C4⋊C4—271no—
258C22⋊C4—541no—
259C22⋊C4—541no—
260C42—271no—
261C22×C3—21no—
262C22×C3—42no—
263C22×C3—82no—
264C22×C3—91no—
265C22×C3—122no—
266C22×C3—181no—
267C22×C3—361no—
268C22×C3—361no—
269C2×D3—21no—
270C2×D3—62no—
271C2×D3—122no—
272C2×D3—181no—
273C2×D3—242no—
274C2×D3—242no—
275C2×D3—362no—
276C2×D3—361no—
277C2×D3—361no—
278C2×D3—722no—
279C2×D3—722no—
280C2×D3—722no—
281C3×C4—22no—
282C3×C4—42no—
283C3×C4—91no—
284C3×C4—361no—
285C3×C4C22×#15711yescharacteristic
286Dic3—62no—
287Dic3—92no—
288Dic3—181no—
289Dic3—361no—
290C32—22no—
291C32—42no—
292C32#475611yescharacteristic, complemented
293C23—181no—
294C23—541no—
295D4—31no—
296D4—91no—
297D4—181no—
298D4—271no—
299D4—541no—
300D4D3×#15711yescharacteristic, complemented
301C2×C4—91no—
302C2×C4—181no—
303C2×C4—271no—
304C2×C4—271no—
305C2×C4—541no—
306C2×C4—1082no—
307C2×C4—1082no—
308C8—31no—
309C8—181no—
310C8—271no—
311C2×C3—22no—
312C2×C3—41no—
313C2×C3—42no—
314C2×C3—82no—
315C2×C3—91no—
316C2×C3—91no—
317C2×C3—162no—
318C2×C3—242no—
319C2×C3—361no—
320C2×C3D4×#15711yescharacteristic
321D3—41no—
322D3—62no—
323D3—62no—
324D3—242no—
325D3—361no—
326D3—722no—
327D3—1442no—
328C22—21no—
329C22—61no—
330C22—91no—
331C22—181no—
332C22—361no—
333C22—361no—
334C22—541no—
335C22—1081no—
336C22—1081no—
337C4—91no—
338C4—272no—
339C4—361no—
340C4—541no—
341C4—1081no—
342C4C2×D3×#15711yescharacteristic
343C3—22no—
344C3—42no—
345C3D8×#15711yescharacteristic, complemented
346C2—41no—
347C2—91no—
348C2—91no—
349C2—121no—
350C2—361no—
351C2—1081no—
352C2#67⋊#15711yescharacteristic
subgroupG/H
centerC2#67⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×D4C2×C4
socleC2×C33C4×D4
prime radicalC3⋊D3×#213C2
cosocleC3×C4×C3⋊D3C23
hypercenterD4D3×#157
hypocenterC33C4×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊#157C2semidirect
2. #11724259C2semidirect
4. C3×#63782C2semidirect
5. #11722715C2semidirect
6. #11722707C2semidirect
7. #11722703C2semidirect
19. #88860C4semidirect
20. C32×#213C4semidirect
21. #88791C4semidirect
22. #88789C4semidirect
29. #63782D3semidirect
40. C33×D4C2×C4semidirect
41. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
43. C3×C4⋊(C3⋊D3)C2×C4semidirect
45. C32×D12C2×C4semidirect
48. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
49. C32×C3⋊C8C2×C4semidirect
85. C3⋊#157D8semidirect
110. C32×D4C4×D3semidirect
116. C4⋊(C3⋊D3)C4×D3semidirect
153. #213#157semidirect
196. C33C4×D8semidirect
206. C3×D4C2×#157semidirect
220. D12C2×#157semidirect
228. C3⋊C8C2×#157semidirect
292. C32#4756semidirect
300. D4D3×#157semidirect
345. C3D8×#157semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊#1572
#117242592
C3⋊D3×#2132
C3×#637822
#117227152
#117227072
#117227032
D8×#1573
#47569