Finite Group Databasebeta

← #13153864

Subgroups — #13153864

Order 1728.

1199 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1199 conjugacy classes of subgroups — too many to draw. Only its 205 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
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721830C211yesmaximal, characteristic
3#11721829C212yesmaximal, complemented
4C4×#6235C212yesmaximal, complemented
5#11721824C211yesmaximal, characteristic, complemented
6#11721766C212yesmaximal, complemented
7#11721764C212yesmaximal, complemented
8#11721762C212yesmaximal, complemented
9#11721757C212yesmaximal, complemented
10C3×#5515—34nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2212yes—
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272C2212yes—
17C2×#6229C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2212yes—
19C4⋊C4×H(3,𝔽3)C2212yes—
20#88552C2212yes—
21#88542C2212yes—
22#88542C2212yes—
23#88542C2212yes—
24#88542C2212yes—
25C4⋊#1084C2211yescharacteristic
26C4⋊#1084C2211yescharacteristic
27#88540C2212yes—
28#88540C2212yes—
29#88540C2212yes—
30#88540C2212yes—
31C4×#1084C2212yes—
32C2×C32×C4⋊C4D314yescomplemented
33C3×#930—34no—
34C3×#929—38no—
35C3×C4×#67—38no—
36C3×#924—34no—
37C3×#898—38no—
38C3×#896—38no—
39C3×#894—38no—
40C3×#889—38no—
41C23×H(3,𝔽3)C2311yescharacteristic
42C22×#272C2312yes—
43C2×C4×H(3,𝔽3)—22no—
44C2×C4×H(3,𝔽3)—41no—
45C2×C4×H(3,𝔽3)—41no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2312yes—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2312yes—
50#6235—24no—
51C2×#1084C2312yes—
52C2×#1084C2312yes—
53C2×#1084C2312yes—
54C4⋊#272—41no—
55C4×#272—42no—
56#6229—41no—
57C3×#539—91no—
58#5515—34no—
59C22×C32×C4C2×D314yes—
60C22×C32×C4C2×D314yes—
61C22×C32×C4C2×D314yes—
62C2×C3×#67—38no—
63C2×C3×D12—34no—
64C2×C3×C4×D3—38no—
65C2×C3×Dic6—34no—
66C32×C4⋊C4C2×D318yes—
67C32×C4⋊C4C2×D318yes—
68C3×#217—38no—
69C3×(C2×D3)⋊C4—38no—
70C3×(C2×D3)⋊C4—38no—
71C3×(C2×D3)⋊C4—38no—
72C3×(C2×D3)⋊C4—38no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—34no—
75C3×Dic3⋊C4—38no—
76C3×Dic3⋊C4—38no—
77C3×Dic3⋊C4—38no—
78C3×Dic3⋊C4—38no—
79C3×C4×Dic3—38no—
80C22×H(3,𝔽3)—21no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)#4112yes—
83C22×H(3,𝔽3)C2411yescharacteristic
84C2×#272—22no—
85C2×#272—22no—
86C2×#272—22no—
87C4×H(3,𝔽3)—22no—
88C4×H(3,𝔽3)—21no—
89C4×H(3,𝔽3)—22no—
90C4×H(3,𝔽3)—21no—
91#1084—22no—
92#1084—22no—
93#1084—22no—
94C3×#126—92no—
95C3×#125—92no—
96C3×#123—92no—
97C3×#123—92no—
98C3×C22⋊Q8—91no—
99C3×C22⋊Q8—92no—
100C3×#121—91no—
101C3×C4×D4—92no—
102C2×C3×C4⋊C4—34no—
103C2×C3×C4⋊C4C3⋊D311yescharacteristic
104#930—34no—
105#929—38no—
106C4×#67—38no—
107#924—34no—
108#898—38no—
109#896—38no—
110#894—38no—
111#889—38no—
112C23×C32C22×D314yes—
113C22×C3×D3—38no—
114C2×C32×C4—28no—
115C2×C32×C4—44no—
116C2×C32×C4—44no—
117C2×C32×C4C22×D318yes—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D318yes—
120C2×C32×C4C22×D314yes—
121C3×#67—616no—
122C2×C3×Dic3—38no—
123C2×C3×Dic3—38no—
124C2×C3×Dic3—38no—
125C3×D12—124no—
126C3×C4×D3—128no—
127C3×Dic6—124no—
128#539—91noSylow
129C2×H(3,𝔽3)—22no—
130C2×H(3,𝔽3)C2×#4111yescharacteristic
131C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
132C2×H(3,𝔽3)2−1+411yescharacteristic
133#272—42no—
134C2×C3×Q8—91no—
135C2×C3×D4—92no—
136C2×C3×D4—91no—
137C22×C3×C4—34no—
138C22×C3×C4—34no—
139C22×C3×C4—34no—
140C22×C3×C4—92no—
141C22×C3×C4C2×C3⋊D311yescharacteristic
142C22×C3×C4C2×C3⋊D311yescharacteristic
143C22×C3×C4C2×C3⋊D311yescharacteristic
144C2×#67—38no—
145C2×D12—34no—
146C2×C4×D3—38no—
147C2×Dic6—34no—
148C3×C4⋊C4—38no—
149C3×C4⋊C4—38no—
150C3×C4⋊C4—92no—
151C3×C4⋊C4—92no—
152C3×C4⋊C4—92no—
153C3×C4⋊C4—92no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4C2×C3⋊D312yes—
157C3×C4⋊C4C2×C3⋊D312yes—
158C3×C22⋊C4—92no—
159C3×C22⋊C4—92no—
160C3×C22⋊C4—92no—
161C3×C22⋊C4—92no—
162C3×C22⋊C4—92no—
163C3×C42—92no—
164#217—38no—
165(C2×D3)⋊C4—38no—
166(C2×D3)⋊C4—38no—
167(C2×D3)⋊C4—38no—
168(C2×D3)⋊C4—38no—
169C4⋊Dic3—34no—
170C4⋊Dic3—34no—
171Dic3⋊C4—38no—
172Dic3⋊C4—38no—
173Dic3⋊C4—38no—
174Dic3⋊C4—38no—
175C4×Dic3—38no—
176C22×C32—24no—
177C22×C32—24no—
178C22×C32#23518yes—
179C22×C32C23×D314yes—
180C2×C3×D3—68no—
181C2×C3×D3—68no—
182C2×C3×D3—68no—
183C32×C4—28no—
184C32×C4—24no—
185C32×C4—28no—
186C32×C4—24no—
187C3×Dic3—68no—
188C3×Dic3—68no—
189C3×Dic3—68no—
190#126—92no—
191#125—92no—
192#123—92no—
193#123—92no—
194C22⋊Q8—91no—
195C22⋊Q8—92no—
196#121—91no—
197C4×D4—92no—
198C2×C4⋊C4#27211yescharacteristic, complemented
199H(3,𝔽3)#53911yescharacteristic, complemented, Sylow
200C23×C3—34no—
201C23×C3—92no—
202C23×C3C22×C3⋊D311yescharacteristic
203C22×D3—38no—
204C3×Q8—361no—
205C3×D4—184no—
206C3×D4—361no—
207C2×C3×C4—22no—
208C2×C3×C4—38no—
209C2×C3×C4—34no—
210C2×C3×C4—38no—
211C2×C3×C4—34no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—68no—
215C2×C3×C4—92no—
216C2×C3×C4—92no—
217C2×C3×C4—92no—
218C2×C3×C4—124no—
219C2×C3×C4—124no—
220C2×C3×C4—362no—
221C2×C3×C4C22×C3⋊D312yes—
222C2×C3×C4C22×C3⋊D312yes—
223C2×C3×C4C22×C3⋊D311yescharacteristic
224C2×C3×C4C22×C3⋊D311yescharacteristic
225#67—616no—
226C2×Dic3—38no—
227C2×Dic3—38no—
228C2×Dic3—38no—
229D12—124no—
230C4×D3—128no—
231Dic6—124no—
232C2×C32—28no—
233C2×C32C2×#23514yes—
234C2×C32#100414yes—
235C2×C32#100714yes—
236C3×D3—128no—
237C2×Q8—91no—
238C2×D4—92no—
239C2×D4—91no—
240C22×C4—92no—
241C22×C4C2×#27211yescharacteristic
242C22×C4C2×#27211yescharacteristic
243C22×C4C2×#27211yescharacteristic
244C4⋊C4—92no—
245C4⋊C4—91no—
246C4⋊C4—92no—
247C4⋊C4—92no—
248C4⋊C4—91no—
249C4⋊C4—92no—
250C4⋊C4C2×#27212yes—
251C4⋊C4C2×#27212yes—
252C22⋊C4—92no—
253C22⋊C4—92no—
254C22⋊C4—92no—
255C22⋊C4—92no—
256C22⋊C4—92no—
257C42—92no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—34no—
261C22×C3—38no—
262C22×C3—64no—
263C22×C3—64no—
264C22×C3—182no—
265C22×C3—182no—
266C22×C3—182no—
267C22×C3#382812yes—
268C22×C3C23×C3⋊D311yescharacteristic
269C2×D3—68no—
270C2×D3—68no—
271C2×D3—68no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—64no—
277C3×C4—68no—
278C3×C4—64no—
279C3×C4—68no—
280C3×C4—182no—
281C3×C4—182no—
282C3×C4—182no—
283Dic3—68no—
284Dic3—68no—
285Dic3—68no—
286C32#551514yescomplemented
287C23—92no—
288C23C22×#27211yescharacteristic
289Q8—361no—
290D4—184no—
291D4—361no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—92no—
298C2×C4—362no—
299C2×C4C22×#27212yes—
300C2×C4C22×#27212yes—
301C2×C4C22×#27211yescharacteristic
302C2×C4C22×#27211yescharacteristic
303C2×C3—22no—
304C2×C3—34no—
305C2×C3—34no—
306C2×C3—34no—
307C2×C3—68no—
308C2×C3—362no—
309C2×C3C2×#382811yescharacteristic
310C2×C3#6436111yescharacteristic
311C2×C3#6436411yescharacteristic
312D3—128no—
313C22—21no—
314C22—21no—
315C22—182no—
316C22—182no—
317C22—182no—
318C22#8874012yes—
319C22C23×#27211yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—182no—
325C4—182no—
326C4—182no—
327C3—34no—
328C3#1059598611yescharacteristic
329C2—22no—
330C2—362no—
331C2C2×#8874011yescharacteristic
332C2#1172266411yescharacteristic
333C2#1172267011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radical#11721830C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#539
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. #11721829C2semidirect
4. C4×#6235C2semidirect
5. #11721824C2semidirect
6. #11721766C2semidirect
7. #11721764C2semidirect
8. #11721762C2semidirect
9. #11721757C2semidirect
32. C2×C32×C4⋊C4D3semidirect
198. C2×C4⋊C4#272semidirect
199. H(3,𝔽3)#539semidirect
286. C32#5515semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
#117218302
#117218292
C4×#62352
#117218242
#117217662
#117217642
#117217622
#117217572
C3×#55153