Finite Group Databasebeta

← #13160365

Subgroups — #13160365

Order 1728.

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

normal subgroups only

This group has 872 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#11723965C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4#11722055C211yesmaximal, characteristic, complemented
5#11722052C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592422—31nomaximal
9#10591134—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12#88864—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14#88857—21no—
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#64135D311yescharacteristic
22#64133—31no—
23#64133—34no—
24#64133—64no—
25C3×#926—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#63567—34no—
35#63567—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—
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C32×C3⋊C8—21no—
48C32×C3⋊C8—21no—
49#4996—91no—
50#4723—94no—
51C22×C32×C4—24no—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C2×D311yescharacteristic
54C2×#634—31no—
55C2×#634—34no—
56C2×#634—64no—
57C2×C32×C8—31no—
58C2×C32×C8—31no—
59#3757—21no—
60#3757—68no—
61#3757—64no—
62#3757—61no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—68no—
69#3750—64no—
70#3750—61no—
71C4×C3⋊Dic3—21no—
72C3×#217—64no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—34no—
75C3×C4×Dic3—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×#117—91no—
90#926—34no—
91#924—94no—
92#924—91no—
93#924—184no—
94C3×C22⋊C8—34no—
95#830C3⋊D311yescharacteristic
96#818—94no—
97#817—94no—
98#816—94no—
99#816—94no—
100#810—91no—
101#810—91no—
102#808—91no—
103#807—91no—
104C23×C32—24no—
105C23×C32D1214yes—
106C23×C32#6711yescharacteristic
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—44no—
114C2×C32×C4—44no—
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—61no—
128#634—61no—
129#634—64no—
130#634—64no—
131#634—124no—
132#634—124no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—64no—
135C32×C8—61no—
136C32×C8—61no—
137C3×C3⋊C8—24no—
138C3×C3⋊C8—24no—
139#485—271noSylow
140C2×C33—41no—
141C2×C33#12311yescharacteristic
142C2×C33#13511yescharacteristic
143C2×C33#13711yescharacteristic
144C22×C3×C4—24no—
145C22×C3×C4D3214yes—
146C22×C3×C4C2×C3⋊D311yescharacteristic
147C2×Dic6—94no—
148C2×Dic6—91no—
149C2×Dic6—184no—
150C2×C3×C8—34no—
151C2×C3×C8—34no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—91no—
154C3×C22⋊C4—181no—
155C3×C42—181no—
156#217—64no—
157#217—188no—
158#217—184no—
159#217—181no—
160C4⋊Dic3—34no—
161C4⋊Dic3—34no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165Dic3⋊C4—188no—
166Dic3⋊C4—184no—
167Dic3⋊C4—181no—
168C4×Dic3—64no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171C22×C32—28no—
172C22×C32—24no—
173C22×C32—21no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—24no—
177C22×C32—24no—
178C22×C32—24no—
179C22×C32—44no—
180C22×C32—44no—
181C22×C32C2×D1214yes—
182C22×C32C2×#6711yescharacteristic
183C32×C4—21no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—44no—
188C32×C4#23511yescharacteristic
189C32×C4#23511yescharacteristic
190C32×C4#23714yes—
191C32×C4#23714yes—
192C3⋊Dic3—41no—
193C3⋊Dic3—41no—
194C3⋊Dic3—128no—
195C3⋊Dic3—128no—
196C3⋊Dic3—124no—
197C3⋊Dic3—124no—
198C3⋊Dic3—121no—
199C3⋊Dic3—121no—
200C3×Dic3—124no—
201C3×Dic3—124no—
202C22⋊Q8—271no—
203#117—91no—
204#107—271no—
205#106—271no—
206#103—271no—
207#103—271no—
208C22⋊C8—31no—
209C33#48511yescharacteristic, complemented, Sylow
210C23×C3—24no—
211C23×C3#62614yes—
212C23×C3C4⋊(C3⋊D3)11yescharacteristic
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—44no—
220C2×C3×C4—44no—
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—184no—
236Dic6—184no—
237Dic6—181no—
238Dic6—181no—
239Dic6—364no—
240Dic6—364no—
241C3×C8—64no—
242C3×C8—64no—
243C3⋊C8—21no—
244C3⋊C8—21no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32—48no—
249C2×C32—44no—
250C2×C32—41no—
251C2×C32#88614yes—
252C2×C32#90414yes—
253C2×C32#90914yes—
254C2×C32#92911yescharacteristic
255C2×C32#95011yescharacteristic
256C2×C32#95111yescharacteristic
257C2×Q8—271no—
258C22×C4D3×C3⋊D311yescharacteristic
259C2×C8—31no—
260C2×C8—31no—
261C4⋊C4—91no—
262C4⋊C4—91no—
263C4⋊C4—271no—
264C4⋊C4—541no—
265C22⋊C4—181no—
266C22⋊C4—541no—
267C42—181no—
268C22×C3—28no—
269C22×C3—24no—
270C22×C3—24no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—24no—
275C22×C3—24no—
276C22×C3—44no—
277C22×C3—44no—
278C22×C3C2×#62614yes—
279C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
280C3×C4—21no—
281C3×C4—24no—
282C3×C4—24no—
283C3×C4—24no—
284C3×C4—44no—
285C3×C4—361no—
286C3×C4—361no—
287C3×C4#380414yes—
288C3×C4#380414yes—
289C3×C4#383011yescharacteristic
290C3×C4#383011yescharacteristic
291Dic3—124no—
292Dic3—124no—
293Dic3—368no—
294Dic3—368no—
295Dic3—364no—
296Dic3—361no—
297Dic3—361no—
298Dic3—364no—
299C32—24no—
300C32#472314yescomplemented
301C32#499611yescharacteristic, complemented
302C23#629211yescharacteristic
303Q8—541no—
304Q8—541no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4—541no—
310C2×C4—541no—
311C2×C4#629211yescharacteristic
312C2×C4C2×D3×C3⋊D311yescharacteristic
313C8—61no—
314C8—61no—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3—48no—
319C2×C3—44no—
320C2×C3—41no—
321C2×C3#6382914yes—
322C2×C3#6383114yes—
323C2×C3#6396614yes—
324C2×C3#6409511yescharacteristic
325C2×C3#6411311yescharacteristic
326C2×C3#6411811yescharacteristic
327C22—21no—
328C22—21no—
329C22—21no—
330C22C2×#629211yescharacteristic
331C4—21no—
332C4—361no—
333C4—361no—
334C4—1081no—
335C4—1081no—
336C4#8902811yescharacteristic
337C4#8902811yescharacteristic
338C3—24no—
339C3#1059113414yescomplemented
340C3#1059242211yescharacteristic, complemented
341C2—41no—
342C2#1172336911yescharacteristic
343C2#1172338811yescharacteristic
344C2#1172360811yescharacteristic
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. #11722042C2semidirect
7. #11722042C2semidirect
209. C33#485semidirect
300. C32#4723semidirect
301. C32#4996semidirect
339. C3#10591134semidirect
340. C3#10592422semidirect

Maximal subgroups

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

subgroupindex
#117239652
C3×#641352
C32×#8302
#117220552
#117220522
#117220422
#105924223
#105911343