Finite Group Databasebeta

← #13159579

Subgroups — #13159579

Order 1728.

776 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 162 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#11723452C211yesmaximal, characteristic
2#11722190C211yesmaximal, characteristic
3C3×#63641C211yesmaximal, characteristic, complemented
4C32×#815C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722024C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10592408—31nomaximal
9#10590647—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17#88806C411yescharacteristic
18#88806C411yescharacteristic
19#88804C411yescharacteristic
20#88804C411yescharacteristic
21#64081—31no—
22#63874—34no—
23C32×#97—31no—
24#63647—31no—
25#63647—31no—
26#63647—34no—
27#63647—64no—
28#63641—31no—
29#63641D311yescharacteristic, complemented
30C4×C3⋊(C3⋊C8)—31no—
31C3×#815D314yescomplemented
32C3×Dic3⋊C8—34no—
33Dic3⋊(C3⋊C8)—34no—
34#63557—34no—
35Dic3×C3⋊C8—34no—
36C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
39C4×C3⋊(C3⋊D3)—21no—
40C4×C3⋊(C3⋊D3)—21no—
41C2×C3×C3⋊Dic3C2×C411yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C2×C32×Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45C33×C8—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4840—91no—
50#4709—94no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—34no—
53C2×C4×C3⋊D3—64no—
54C2×C32×C8—24no—
55C2×C32×C8—31no—
56C2×C32×C8C2×D314yes—
57C2×C32×C8C2×D311yescharacteristic
58C32×C42—31no—
59#3752—31no—
60#3752—31no—
61#3750—31no—
62#3750—31no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)—31no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—64no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×C4×Dic3—34no—
69C3×C4×Dic3C2×D314yes—
70C2×C3×C3⋊C8—34no—
71C2×C3×C3⋊C8C2×D314yes—
72#3724—34no—
73#3724—34no—
74#3722—34no—
75#3722—34no—
76C22×C33C22×C411yescharacteristic
77C2×C3⋊(C3⋊D3)—21no—
78C2×C3⋊(C3⋊D3)—21no—
79C2×C3⋊(C3⋊D3)—21no—
80C33×C4#4111yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—21no—
83C3×C3⋊Dic3—21no—
84C3×C3⋊Dic3—21no—
85C32×Dic3—21no—
86C32×Dic3—21no—
87#872—94no—
88#872—91no—
89C3×C4⋊C8—91no—
90C3×#97—34no—
91(C2×D3)⋊C8—94no—
92(C2×D3)⋊C8—94no—
93(C2×D3)⋊C8—91no—
94(C2×D3)⋊C8—91no—
95(C2×D3)⋊C8—184no—
96#815C3⋊D311yescharacteristic, complemented
97Dic3⋊C8—34no—
98Dic3⋊C8—94no—
99C8×Dic3—91no—
100C4⋊(C3⋊C8)—91no—
101C4×C3⋊C8—94no—
102C22×C3⋊D3—31no—
103C22×C3⋊D3—34no—
104C22×C3⋊D3—64no—
105C2×C32×C4—24no—
106C2×C32×C4—31no—
107C2×C32×C4—31no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—34no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3C4×D311yescharacteristic
114C2×C3⋊Dic3C4×D311yescharacteristic
115C4×C3⋊D3—61no—
116C4×C3⋊D3—61no—
117C4×C3⋊D3—64no—
118C4×C3⋊D3—64no—
119C4×C3⋊D3—124no—
120C4×C3⋊D3—124no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3C4×D314yes—
124C2×C3×Dic3C4×D314yes—
125C32×C8—21no—
126C32×C8—24no—
127C32×C8—44no—
128C32×C8—61no—
129C3⋊(C3⋊C8)—21no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133C3×C3⋊C8—24no—
134C3×C3⋊C8—64no—
135#433—271noSylow
136C2×C33#11711yescharacteristic
137C2×C33#13111yescharacteristic
138C2×C33#13111yescharacteristic
139C3⋊(C3⋊D3)—41no—
140C2×C4×D3—94no—
141C2×C4×D3—91no—
142C2×C4×D3—184no—
143C2×C3×C8—24no—
144C2×C3×C8—34no—
145C2×C3×C8—91no—
146C2×C3×C8D3214yes—
147C2×C3×C8C2×C3⋊D311yescharacteristic
148C3×C42—34no—
149C3×C42—91no—
150(C2×D3)⋊C4—94no—
151(C2×D3)⋊C4—94no—
152(C2×D3)⋊C4—91no—
153(C2×D3)⋊C4—91no—
154Dic3⋊C4—94no—
155Dic3⋊C4—94no—
156Dic3⋊C4—91no—
157Dic3⋊C4—91no—
158C4×Dic3—34no—
159C4×Dic3C2×C3⋊D311yescharacteristic
160C2×C3⋊C8—34no—
161C2×C3⋊C8—94no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—184no—
164C2×C3⋊C8C2×C3⋊D311yescharacteristic
165C22×C32—24no—
166C22×C32C2×C4×D314yes—
167C22×C32C2×C4×D311yescharacteristic
168C2×C3⋊D3—68no—
169C2×C3⋊D3—61no—
170C2×C3⋊D3—61no—
171C2×C3⋊D3—64no—
172C2×C3⋊D3—61no—
173C2×C3⋊D3—64no—
174C2×C3⋊D3—64no—
175C2×C3⋊D3—124no—
176C2×C3⋊D3—124no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—61no—
180C32×C4—61no—
181C32×C4#23514yes—
182C32×C4#23514yes—
183C32×C4#23711yescharacteristic
184C32×C4#23911yescharacteristic
185C3⋊Dic3—21no—
186C3⋊Dic3—21no—
187C3⋊Dic3—61no—
188C3⋊Dic3—64no—
189C3⋊Dic3—124no—
190C3×Dic3—24no—
191C3×Dic3—24no—
192C3×Dic3—64no—
193C3×Dic3—64no—
194#117—271no—
195C4⋊C8—91no—
196C4⋊C8—271no—
197C22⋊C8—271no—
198C22⋊C8—271no—
199#97—31no—
200C4×C8—271no—
201C33#43311yescharacteristic, complemented, Sylow
202C22×D3—94no—
203C22×D3—91no—
204C22×D3—184no—
205C2×C3×C4—24no—
206C2×C3×C4—34no—
207C2×C3×C4—34no—
208C2×C3×C4—91no—
209C2×C3×C4—91no—
210C2×C3×C4C2×D3214yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×Dic3—34no—
213C2×Dic3—34no—
214C2×Dic3—94no—
215C2×Dic3—91no—
216C2×Dic3—184no—
217C2×Dic3C4×C3⋊D311yescharacteristic
218C2×Dic3C4×C3⋊D311yescharacteristic
219C4×D3—184no—
220C4×D3—184no—
221C4×D3—181no—
222C4×D3—181no—
223C4×D3—364no—
224C4×D3—364no—
225C3×C8—24no—
226C3×C8—21no—
227C3×C8—44no—
228C3×C8—64no—
229C3×C8—181no—
230C3⋊C8—21no—
231C3⋊C8—64no—
232C3⋊C8—184no—
233C3⋊C8—181no—
234C3⋊C8—364no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32#87214yes—
239C2×C32#89211yescharacteristic
240C2×C32#90111yescharacteristic
241C2×C32#90714yes—
242C2×C32#90714yes—
243C2×C32#90711yescharacteristic
244C3⋊D3—128no—
245C3⋊D3—124no—
246C3⋊D3—121no—
247C22×C4—271no—
248C2×C8—31no—
249C2×C8—91no—
250C2×C8—271no—
251C2×C8D3×C3⋊D311yescharacteristic
252C4⋊C4—271no—
253C4⋊C4—271no—
254C22⋊C4—271no—
255C22⋊C4—271no—
256C42—31no—
257C42—91no—
258C22×C3—24no—
259C22×C3C4×D3214yes—
260C22×C3C2×C4×C3⋊D311yescharacteristic
261C2×D3—188no—
262C2×D3—184no—
263C2×D3—184no—
264C2×D3—181no—
265C2×D3—184no—
266C2×D3—181no—
267C2×D3—181no—
268C2×D3—364no—
269C2×D3—364no—
270C3×C4—24no—
271C3×C4—24no—
272C3×C4—64no—
273C3×C4—64no—
274C3×C4—181no—
275C3×C4—181no—
276C3×C4#379914yes—
277C3×C4#380414yes—
278C3×C4#382811yescharacteristic
279C3×C4#382811yescharacteristic
280Dic3—21no—
281Dic3—21no—
282Dic3—64no—
283Dic3—64no—
284Dic3—181no—
285Dic3—184no—
286Dic3—364no—
287C32—24no—
288C32#470914yescomplemented
289C32#484011yescharacteristic, complemented
290C23—271no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—271no—
296C2×C4—541no—
297C2×C4—541no—
298C2×C4C2×D3×C3⋊D311yescharacteristic
299C8—21no—
300C8—61no—
301C8—181no—
302C8—541no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3#6380414yes—
307C2×C3#6380514yes—
308C2×C3#6383614yes—
309C2×C3#6408111yescharacteristic
310C2×C3#6411611yescharacteristic
311C2×C3#6411611yescharacteristic
312D3—368no—
313D3—361no—
314D3—364no—
315C22—541no—
316C22—541no—
317C22—541no—
318C22C4×D3×C3⋊D311yescharacteristic
319C4—61no—
320C4—61no—
321C4—181no—
322C4—181no—
323C4—541no—
324C4#8902211yescharacteristic
325C4#8903111yescharacteristic
326C3—24no—
327C3#1059064714yescomplemented
328C3#1059240811yescharacteristic, complemented
329C2—1081no—
330C2#1172333311yescharacteristic
331C2#1172333411yescharacteristic
332C2#1172346811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63641C2semidirect
4. C32×#815C2semidirect
5. #11722051C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
29. #63641D3semidirect
31. C3×#815D3semidirect
96. #815C3⋊D3semidirect
201. C33#433semidirect
288. C32#4709semidirect
289. C32#4840semidirect
327. C3#10590647semidirect
328. C3#10592408semidirect

Maximal subgroups

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

subgroupindex
#117234522
#117221902
C3×#636412
C32×#8152
#117220512
#117220242
Dic3×C3⋊(C3⋊C8)2
#105924083
#105906473