Finite Group Databasebeta

← #13166532

Subgroups — #13166532

Order 1728.

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

normal subgroups only

This group has 460 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
1C3×D4×#157C211yesmaximal, characteristic, complemented
2D12×#157C211yesmaximal, characteristic, complemented
3C3⋊D3×#213C211yesmaximal, characteristic
4#11723107C211yesmaximal, characteristic, complemented
5#11722714C211yesmaximal, characteristic, complemented
6#11722706C211yesmaximal, characteristic, complemented
7#11722704C211yesmaximal, characteristic, complemented
8#10593778—31nomaximal
9C22×C3×#157—21no—
10C2×D3×#157—21no—
11C3×D4×C3⋊D3C2211yescharacteristic
12C3⋊D3×D12C2211yescharacteristic
13C4⋊(C3⋊#157)C2211yescharacteristic
14#88988C2211yescharacteristic
15C3×C22⋊#157—21no—
16C3×C4⋊#157C2211yescharacteristic
17C3×C4×#157C2211yescharacteristic
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×#157D311yescharacteristic, complemented
26D8×C3⋊D3—31no—
27D3×#213—62no—
28#63782—31no—
29#63782—31no—
30#63768—31no—
31#63767—31no—
32C22×C3×C3⋊D3—21no—
33C2×D3×C3⋊D3—21no—
34C2×C3⋊#157—21no—
35C2×C3×#157—21no—
36C2×C3×#157—42no—
37C2×C3×#157D411yescharacteristic
38C2×C3×#157D411yescharacteristic
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#5014—91nomaximal
51C22×#157—21no—
52C22×#157—61no—
53D4×C3⋊D3—31no—
54D4×C3⋊D3C2×D311yescharacteristic
55C3×D3×D4—62no—
56D3×D12—62no—
57C22⋊#157—21no—
58C22⋊#157—61no—
59C4⋊#157—31no—
60C4⋊#157C2×D311yescharacteristic
61C4×#157C2×D311yescharacteristic
62#3787—31no—
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—41no—
84C3×#157—21no—
85C3×#157—21no—
86C3×#157—41no—
87C33×C4C22×C411yescharacteristic
88C3×C3⋊Dic3#4111yescharacteristic
89C3×C4×D4—91no—
90C2×#213—91no—
91D3×D8—182no—
92C4×D12—91no—
93#832—91no—
94#809—91no—
95#808—91no—
96#803—91no—
97C22×C3⋊D3—21no—
98C22×C3⋊D3—61no—
99C22×C3×D3—122no—
100C2×D32—122no—
101C2×#157—21no—
102C2×#157—42no—
103C2×#157—61no—
104C2×#157—122no—
105C2×#157#6711yescharacteristic
106C2×#157#6711yescharacteristic
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—61no—
124C32×C8—31no—
125C3⋊(C3⋊C8)—31no—
126C3⋊(C3⋊C8)—62no—
127C3⋊(C3⋊C8)—122no—
128C3×C3⋊C8—22no—
129#442—271noSylow
130C2×C33—41no—
131C2×C33C4×D411yescharacteristic
132C3⋊(C3⋊D3)—41no—
133C3×C3⋊D3—41no—
134C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
135C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
136C32×D3—41no—
137C2×C3×D4—91no—
138C22×C3×C4—181no—
139D3×D4—62no—
140D3×D4—182no—
141C2×D12—91no—
142C2×C4×D3—181no—
143C3×D8—62no—
144C3×C4⋊C4—91no—
145C3×C22⋊C4—181no—
146C3×C42—91no—
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—
156C2×C3⋊C8—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⋊D3C2×#6711yescharacteristic
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—21no—
181#157—21no—
182#157—41no—
183#157—121no—
184C32×C4—22no—
185C32×C4—42no—
186C32×C4C2×C4×D311yescharacteristic
187C3⋊Dic3#23511yescharacteristic
188C3×Dic3—62no—
189C2×D8—271no—
190C4×D4—91no—
191C4×D4—271no—
192#106—271no—
193#102—271no—
194#102—271no—
195#97—271no—
196C33#44211yescharacteristic, 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—91no—
209C2×C3×C4—91no—
210C2×C3×C4—181no—
211C2×C3×C4—362no—
212#67—122no—
213#67—362no—
214C2×Dic3—181no—
215D12—62no—
216D12—91no—
217D12—182no—
218D12—181no—
219D12—362no—
220D12C2×#15711yescharacteristic, complemented
221C4×D3—62no—
222C4×D3—362no—
223C3×C8—62no—
224C3⋊C8—91no—
225C3⋊C8—182no—
226C3⋊C8—181no—
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×C32C4×#6711yescharacteristic
236C3⋊D3—41no—
237C3⋊D3—121no—
238C3⋊D3—242no—
239C3⋊D3—482no—
240C3⋊D3#93211yescharacteristic
241C3⋊D3#94911yescharacteristic
242C3×D3—62no—
243C3×D3—62no—
244C3×D3—82no—
245C3×D3—242no—
246C2×D4—91no—
247C2×D4—271no—
248C22×C4—181no—
249C22×C4—541no—
250D8—31no—
251D8—271no—
252D8—271no—
253D8—271no—
254C2×C8—271no—
255C2×C8—271no—
256C4⋊C4—91no—
257C4⋊C4—271no—
258C22⋊C4—181no—
259C22⋊C4—541no—
260C42—91no—
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—181no—
285C3×C4—181no—
286C3×C4—361no—
287C3×C4C22×#15711yescharacteristic
288Dic3—62no—
289Dic3—361no—
290C32—22no—
291C32—42no—
292C32#501411yescharacteristic, complemented
293C23—181no—
294C23—541no—
295D4—31no—
296D4—91no—
297D4—181no—
298D4—271no—
299D4—541no—
300D4D3×#15711yescharacteristic, complemented
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4—362no—
306C2×C4—541no—
307C2×C4—1082no—
308C8—31no—
309C8—271no—
310C8—541no—
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—181no—
339C4—181no—
340C4—361no—
341C4—1081no—
342C4C2×D3×#15711yescharacteristic
343C3—22no—
344C3—42no—
345C3#1059377811yescharacteristic, 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
hypocenterC33#442
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. C3×D4×#157C2semidirect
2. D12×#157C2semidirect
4. #11723107C2semidirect
5. #11722714C2semidirect
6. #11722706C2semidirect
7. #11722704C2semidirect
19. #88860C4semidirect
20. C32×#213C4semidirect
21. #88791C4semidirect
22. #88789C4semidirect
25. D4×#157D3semidirect
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
110. C32×D4C4×D3semidirect
116. C4⋊(C3⋊D3)C4×D3semidirect
153. #213#157semidirect
196. C33#442semidirect
206. C3×D4C2×#157semidirect
220. D12C2×#157semidirect
228. C3⋊C8C2×#157semidirect
292. C32#5014semidirect
300. D4D3×#157semidirect
345. C3#10593778semidirect

Maximal subgroups

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

subgroupindex
C3×D4×#1572
D12×#1572
C3⋊D3×#2132
#117231072
#117227142
#117227062
#117227042
#105937783
#50149