Finite Group Databasebeta

← #13160356

Subgroups — #13160356

Order 1728.

880 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 164 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#11723967C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4#11722055C211yesmaximal, characteristic, complemented
5#11722052C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592422—31nomaximal
9#10591127—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12#88864—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)—21no—
16C3×#3757—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#64135—31no—
22#64135—34no—
23#64135—64no—
24#64133D311yescharacteristic
25C3×#924—34no—
26C32×C22⋊C8—31no—
27#63645—31no—
28#63644—31no—
29#63643—31no—
30#63643—31no—
31C3×#830D314yes—
32#63575—34no—
33#63573—34no—
34#63566—34no—
35#63566—34no—
36C23×C33D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C32×C3⋊C8—21no—
48C32×C3⋊C8—21no—
49#5045—91no—
50#4723—94no—
51C22×C32×C4—24no—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic
55C2×C3×Dic6—34no—
56C2×C32×C8—31no—
57C2×C32×C8—31no—
58#3757—21no—
59#3757—68no—
60#3757—64no—
61#3757—61no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)—64no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—21no—
70C4×C3⋊Dic3—68no—
71C4×C3⋊Dic3—64no—
72C4×C3⋊Dic3—61no—
73C3×#217—64no—
74C3×C4⋊Dic3—34no—
75C3×Dic3⋊C4—64no—
76C2×C3×C3⋊C8C2×D314yes—
77C2×C3×C3⋊C8C2×D314yes—
78C22×C33—21no—
79C22×C33—21no—
80C22×C33—21no—
81C22×C33C2×D411yescharacteristic
82C33×C4—21no—
83C33×C4#4111yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—41no—
86C3⋊(C3⋊Dic3)—41no—
87C3×C3⋊Dic3—41no—
88C3×C3⋊Dic3—41no—
89C3×C22⋊Q8—91no—
90#926—94no—
91#926—91no—
92#926—184no—
93#924—34no—
94C3×C22⋊C8—34no—
95#835—91no—
96#835—91no—
97#830C3⋊D311yescharacteristic
98#818—94no—
99#817—94no—
100#816—94no—
101#816—94no—
102#808—91no—
103#807—91no—
104C23×C32—24no—
105C23×C32D1214yes—
106C23×C32#6711yescharacteristic
107C2×C32×C4—28no—
108C2×C32×C4—28no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4D1214yes—
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—68no—
122C2×C3⋊Dic3—68no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—61no—
127#634—21no—
128#634—21no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—64no—
131C3×Dic6—64no—
132C3×Dic6—64no—
133C32×C8—61no—
134C32×C8—61no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8—24no—
137#485—271noSylow
138C2×C33—41no—
139C2×C33#12311yescharacteristic
140C2×C33#13511yescharacteristic
141C2×C33#13711yescharacteristic
142C2×C3×Q8—91no—
143C22×C3×C4—24no—
144C22×C3×C4D3214yes—
145C22×C3×C4C2×C3⋊D311yescharacteristic
146C2×Dic6—34no—
147C2×C3×C8—34no—
148C2×C3×C8—34no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C22⋊C4—181no—
152#217—64no—
153#217—188no—
154#217—184no—
155#217—181no—
156C4⋊Dic3—34no—
157C4⋊Dic3—94no—
158C4⋊Dic3—94no—
159C4⋊Dic3—91no—
160C4⋊Dic3—91no—
161C4⋊Dic3—184no—
162C4⋊Dic3—184no—
163Dic3⋊C4—64no—
164C4×Dic3—188no—
165C4×Dic3—184no—
166C4×Dic3—181no—
167C2×C3⋊C8C2×C3⋊D311yescharacteristic
168C2×C3⋊C8C2×C3⋊D311yescharacteristic
169C22×C32—28no—
170C22×C32—24no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—24no—
175C22×C32—24no—
176C22×C32—24no—
177C22×C32—44no—
178C22×C32—44no—
179C22×C32C2×D1214yes—
180C22×C32C2×#6711yescharacteristic
181C32×C4—28no—
182C32×C4—24no—
183C32×C4—21no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4#23714yes—
187C32×C4#23714yes—
188C32×C4#23711yescharacteristic
189C32×C4#23711yescharacteristic
190C3⋊Dic3—41no—
191C3⋊Dic3—41no—
192C3⋊Dic3—128no—
193C3⋊Dic3—128no—
194C3⋊Dic3—124no—
195C3⋊Dic3—124no—
196C3⋊Dic3—121no—
197C3⋊Dic3—121no—
198C3×Dic3—124no—
199C3×Dic3—124no—
200C22⋊Q8—91no—
201#117—271no—
202#107—271no—
203#106—271no—
204#103—271no—
205#103—271no—
206C22⋊C8—31no—
207C33#48511yescharacteristic, complemented, Sylow
208C23×C3—24no—
209C23×C3#62614yes—
210C23×C3C4⋊(C3⋊D3)11yescharacteristic
211C3×Q8—181no—
212C3×Q8—181no—
213C2×C3×C4—28no—
214C2×C3×C4—28no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223C2×C3×C4#62614yes—
224C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—64no—
228C2×Dic3—64no—
229C2×Dic3—188no—
230C2×Dic3—188no—
231C2×Dic3—184no—
232C2×Dic3—184no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235Dic6—64no—
236Dic6—64no—
237C3×C8—64no—
238C3×C8—64no—
239C3⋊C8—21no—
240C3⋊C8—21no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32—48no—
245C2×C32—44no—
246C2×C32—41no—
247C2×C32#88614yes—
248C2×C32#90414yes—
249C2×C32#90914yes—
250C2×C32#93511yescharacteristic
251C2×C32#94211yescharacteristic
252C2×C32#95011yescharacteristic
253C2×Q8—91no—
254C22×C4D3×C3⋊D311yescharacteristic
255C2×C8—31no—
256C2×C8—31no—
257C4⋊C4—91no—
258C4⋊C4—181no—
259C4⋊C4—271no—
260C4⋊C4—271no—
261C22⋊C4—181no—
262C22⋊C4—541no—
263C42—541no—
264C22×C3—28no—
265C22×C3—24no—
266C22×C3—24no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—24no—
271C22×C3—24no—
272C22×C3—44no—
273C22×C3—44no—
274C22×C3C2×#62614yes—
275C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
276C3×C4—28no—
277C3×C4—21no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—24no—
281C3×C4—361no—
282C3×C4—361no—
283C3×C4#380514yes—
284C3×C4#380514yes—
285C3×C4#383011yescharacteristic
286C3×C4#383011yescharacteristic
287Dic3—124no—
288Dic3—124no—
289Dic3—368no—
290Dic3—368no—
291Dic3—364no—
292Dic3—364no—
293Dic3—361no—
294Dic3—361no—
295C32—24no—
296C32#472314yescomplemented
297C32#504511yescharacteristic, complemented
298C23#629211yescharacteristic
299Q8—181no—
300Q8—181no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—181no—
304C2×C4—181no—
305C2×C4—541no—
306C2×C4—541no—
307C2×C4#629211yescharacteristic
308C2×C4C2×D3×C3⋊D311yescharacteristic
309C8—61no—
310C8—61no—
311C2×C3—24no—
312C2×C3—24no—
313C2×C3—24no—
314C2×C3—48no—
315C2×C3—44no—
316C2×C3—41no—
317C2×C3#6382914yes—
318C2×C3#6383314yes—
319C2×C3#6396214yes—
320C2×C3#6409511yescharacteristic
321C2×C3#6411311yescharacteristic
322C2×C3#6411811yescharacteristic
323C22—21no—
324C22—21no—
325C22—21no—
326C22C2×#629211yescharacteristic
327C4—21no—
328C4—361no—
329C4—361no—
330C4—1081no—
331C4—1081no—
332C4#8902911yescharacteristic
333C4#8902911yescharacteristic
334C3—24no—
335C3#1059112714yescomplemented
336C3#1059242211yescharacteristic, complemented
337C2—41no—
338C2#1172336911yescharacteristic
339C2#1172338711yescharacteristic
340C2#1172360911yescharacteristic
subgroupG/H
centerC22C2×#6292
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722055C2semidirect
5. #11722052C2semidirect
6. #11722040C2semidirect
7. #11722040C2semidirect
207. C33#485semidirect
296. C32#4723semidirect
297. C32#5045semidirect
335. C3#10591127semidirect
336. C3#10592422semidirect

Maximal subgroups

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

subgroupindex
#117239672
C3×#641332
C32×#8302
#117220552
#117220522
#117220402
#105924223
#105911273