Finite Group Databasebeta

← #13049523

Subgroups — #13049523

Order 1296.

924 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 102 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#10620157C211yesmaximal, characteristic, complemented
2#10620152C211yesmaximal, characteristic, complemented
3#10620147C211yesmaximal, characteristic, complemented
4#10620144C211yesmaximal, characteristic, complemented
5C3×#6293C211yesmaximal, characteristic, complemented
6C3×#6289C211yesmaximal, characteristic, complemented
7C32×#624C211yesmaximal, characteristic, complemented
8#89022—31nomaximal
9#89020—31nomaximal
10#88965—34nomaximal
11C2×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×C3⋊D3C2211yescharacteristic
14C3×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C32×C3⋊Dic3C2211yescharacteristic
16C33×Dic3C2211yescharacteristic
17C33×Dic3C2211yescharacteristic
18C4⋊(C3⋊(C3⋊D3))—31no—
19C4×C3⋊(C3⋊D3)—31no—
20C3×C4×C3⋊D3—31no—
21C3×#634—31no—
22C32×C4×D3—31no—
23C32×C4×D3—31no—
24#6295—34no—
25#6293D311yescharacteristic, complemented
26#6292—34no—
27#6292—31no—
28#6292—34no—
29#6292—64no—
30#6291—31no—
31#6291—31no—
32#6291—31no—
33#6291—61no—
34#6290—31no—
35#6289—31no—
36#6289—34no—
37#6289—64no—
38#6289D311yescharacteristic, complemented
39C3×#627—34no—
40C3×#624—34no—
41C3×#624D314yescomplemented
42C2×C34C2311yescharacteristic
43C3⋊(C3⋊(C3⋊D3))—21no—
44C3×C3⋊(C3⋊D3)—21no—
45C32×C3⋊D3—21no—
46#3828—91no—
47#3799—94no—
48#3799—91no—
49#3798—94no—
50C2×C3⋊(C3⋊D3)—31no—
51C2×C3⋊(C3⋊D3)—31no—
52C2×C3⋊(C3⋊D3)—34no—
53C2×C3⋊(C3⋊D3)—64no—
54C2×C3⋊(C3⋊D3)—61no—
55C2×C3⋊(C3⋊D3)—64no—
56C2×C3⋊(C3⋊D3)—124no—
57C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C3×C3⋊D3—31no—
59C2×C3×C3⋊D3—34no—
60C2×C3×C3⋊D3—64no—
61C2×C3×C3⋊D3C2×D314yes—
62C2×C32×D3—31no—
63C2×C32×D3—31no—
64C2×C32×D3—61no—
65C33×C4—31no—
66C33×C4—31no—
67C3⋊(C3⋊Dic3)C2×D311yescharacteristic
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3—31no—
70C3×C3⋊Dic3—34no—
71C3×C3⋊Dic3—64no—
72C3×C3⋊Dic3C2×D311yescharacteristic
73C3×C3⋊Dic3C2×D311yescharacteristic
74C32×Dic3—24no—
75C32×Dic3—24no—
76C32×Dic3—34no—
77C32×Dic3C2×D314yes—
78C32×Dic3C2×D314yes—
79C32×Dic3C2×D311yescharacteristic
80C32×Dic3C2×D311yescharacteristic
81C34#4111yescharacteristic, complemented, Sylow
82C2×C32×C4—91no—
83#638—91no—
84#638—91no—
85C4⋊(C3⋊D3)—91no—
86C4⋊(C3⋊D3)—94no—
87C4⋊(C3⋊D3)—91no—
88C4⋊(C3⋊D3)—184no—
89C4×C3⋊D3—31no—
90C4×C3⋊D3—91no—
91C4×C3⋊D3—94no—
92C4×C3⋊D3—184no—
93#634—31no—
94C3×C4×D3—34no—
95C3×C4×D3—34no—
96C3×C4×D3—94no—
97C3×C4×D3—91no—
98C3×Dic6—94no—
99C3×Dic6—91no—
100#627—34no—
101#626—94no—
102#626—94no—
103#626—94no—
104#626—91no—
105#626—94no—
106#626—91no—
107#626—184no—
108#626—184no—
109#625—94no—
110#624—34no—
111#624—94no—
112#624—91no—
113#624—184no—
114#624C3⋊D311yescharacteristic, complemented
115C2×C33—24no—
116C2×C33—24no—
117C2×C33—21no—
118C2×C33—44no—
119C2×C33C22×D311yescharacteristic
120C2×C33C22×D311yescharacteristic
121C2×C33C22×D314yes—
122C3⋊(C3⋊D3)—21no—
123C3⋊(C3⋊D3)—61no—
124C3⋊(C3⋊D3)—68no—
125C3⋊(C3⋊D3)—64no—
126C3⋊(C3⋊D3)—61no—
127C3⋊(C3⋊D3)—121no—
128C3⋊(C3⋊D3)—124no—
129C3⋊(C3⋊D3)—128no—
130C3×C3⋊D3—24no—
131C3×C3⋊D3—61no—
132C3×C3⋊D3—64no—
133C3×C3⋊D3—124no—
134C32×D3—61no—
135C32×D3—61no—
136C32×D3—121no—
137#239—271no—
138#235—271no—
139#235—274no—
140C22×C32—91no—
141C2×C3⋊D3—31no—
142C2×C3⋊D3—34no—
143C2×C3⋊D3—64no—
144C2×C3⋊D3—91no—
145C2×C3⋊D3—94no—
146C2×C3⋊D3—94no—
147C2×C3⋊D3—91no—
148C2×C3⋊D3—184no—
149C2×C3⋊D3—184no—
150C2×C3⋊D3—184no—
151C2×C3⋊D3—184no—
152C2×C3⋊D3—184no—
153C2×C3⋊D3—364no—
154C2×C3⋊D3—364no—
155C2×C3⋊D3—3612no—
156C2×C3⋊D3C2×C3⋊D311yescharacteristic
157C2×C3×D3—34no—
158C2×C3×D3—34no—
159C2×C3×D3—64no—
160C2×C3×D3—94no—
161C2×C3×D3—91no—
162C2×C3×D3—184no—
163C32×C4—31no—
164C32×C4—31no—
165C32×C4—34no—
166C32×C4—34no—
167C32×C4—64no—
168C32×C4—64no—
169C32×C4—91no—
170C3⋊Dic3—31no—
171C3⋊Dic3—34no—
172C3⋊Dic3—64no—
173C3⋊Dic3D3211yescharacteristic
174C3×Dic3—24no—
175C3×Dic3—24no—
176C3×Dic3—34no—
177C3×Dic3—34no—
178C3×Dic3—64no—
179C3×Dic3—94no—
180C3×Dic3—91no—
181C3×Dic3—184no—
182C3×Dic3D3214yes—
183C3×Dic3D3214yes—
184C3×Dic3C2×C3⋊D311yescharacteristic
185C3×Dic3C2×C3⋊D311yescharacteristic
186C33—24no—
187C33—21no—
188C33—24no—
189C33—44no—
190C33#23511yescharacteristic, complemented
191C33#23514yescomplemented
192C33#23911yescharacteristic, complemented
193C3×Q8—271no—
194C2×C3×C4—94no—
195C2×C3×C4—271no—
196#67—274no—
197#67—271no—
198#67—274no—
199#67—271no—
200D12—271no—
201D12—274no—
202D12—271no—
203D12—271no—
204D12—271no—
205D12—544no—
206D12—541no—
207C4×D3—31no—
208C4×D3—31no—
209C4×D3—94no—
210C4×D3—91no—
211C4×D3—271no—
212C4×D3—274no—
213C4×D3—271no—
214C4×D3—544no—
215Dic6—94no—
216Dic6—91no—
217C2×C32—24no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32—24no—
222C2×C32—44no—
223C2×C32—412no—
224C2×C32—44no—
225C2×C32—181no—
226C2×C32C2×D3214yes—
227C2×C32C2×D3214yes—
228C2×C32C2×D3211yescharacteristic
229C2×C32C22×C3⋊D311yescharacteristic
230C3⋊D3—21no—
231C3⋊D3—61no—
232C3⋊D3—64no—
233C3⋊D3—124no—
234C3⋊D3—188no—
235C3⋊D3—184no—
236C3⋊D3—181no—
237C3⋊D3—188no—
238C3⋊D3—184no—
239C3⋊D3—181no—
240C3⋊D3—364no—
241C3⋊D3—364no—
242C3⋊D3—364no—
243C3⋊D3—368no—
244C3⋊D3—3624no—
245C3⋊D3—368no—
246C3×D3—64no—
247C3×D3—64no—
248C3×D3—124no—
249C3×D3—184no—
250C3×D3—181no—
251C3×D3—364no—
252#41—811noSylow
253C22×C3—94no—
254C22×C3—271no—
255C2×D3—31no—
256C2×D3—31no—
257C2×D3—61no—
258C2×D3—94no—
259C2×D3—91no—
260C2×D3—184no—
261C2×D3—274no—
262C2×D3—271no—
263C2×D3—271no—
264C2×D3—544no—
265C2×D3—544no—
266C2×D3—541no—
267C2×D3—1084no—
268C3×C4—34no—
269C3×C4—34no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—64no—
273C3×C4—64no—
274C3×C4—91no—
275C3×C4—91no—
276C3×C4—181no—
277C3×C4—271no—
278Dic3—34no—
279Dic3—94no—
280Dic3—91no—
281Dic3—184no—
282Dic3D3×C3⋊D311yescharacteristic
283Dic3D3×C3⋊D311yescharacteristic
284C32—24no—
285C32—24no—
286C32—24no—
287C32—24no—
288C32—24no—
289C32—44no—
290C32—412no—
291C32—44no—
292C32#379814yescomplemented
293C32#379914yescomplemented
294C32#379911yescharacteristic, complemented
295C32#382811yescharacteristic, complemented
296Q8—271no—
297D4—811no—
298D4—811no—
299D4—811no—
300C2×C4—91no—
301C2×C4—271no—
302C2×C4—811no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—21no—
306C2×C3—44no—
307C2×C3—184no—
308C2×C3—541no—
309C2×C3D3314yes—
310C2×C3C2×D3×C3⋊D311yescharacteristic
311C2×C3C2×D3×C3⋊D311yescharacteristic
312D3—61no—
313D3—61no—
314D3—121no—
315D3—184no—
316D3—181no—
317D3—364no—
318D3—548no—
319D3—544no—
320D3—541no—
321D3—541no—
322D3—1081no—
323D3—1084no—
324D3—1088no—
325C22—91no—
326C22—271no—
327C22—811no—
328C4—31no—
329C4—31no—
330C4—91no—
331C4—271no—
332C3—21no—
333C3—24no—
334C3—24no—
335C3—44no—
336C3#8896514yescomplemented
337C3#8902011yescharacteristic, complemented
338C3#8902211yescharacteristic, complemented
339C2—181no—
340C2—541no—
341C2—1621no—
342C2D32×C3⋊D311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×C3⋊D3
commutator, Fitting, socle, cosocleC2×C34C23
hypocenterC34#41
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #10620157C2semidirect
2. #10620152C2semidirect
3. #10620147C2semidirect
4. #10620144C2semidirect
5. C3×#6293C2semidirect
6. C3×#6289C2semidirect
7. C32×#624C2semidirect
25. #6293D3semidirect
38. #6289D3semidirect
41. C3×#624D3semidirect
81. C34#41semidirect
114. #624C3⋊D3semidirect
190. C33#235semidirect
191. C33#235semidirect
192. C33#239semidirect
292. C32#3798semidirect
293. C32#3799semidirect
294. C32#3799semidirect
295. C32#3828semidirect
336. C3#88965semidirect
337. C3#89020semidirect
338. C3#89022semidirect

Maximal subgroups

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

subgroupindex
#106201572
#106201522
#106201472
#106201442
C3×#62932
C3×#62892
C32×#6242
#890223
#890203
#889653