Finite Group Databasebeta

← #13147788

Subgroups — #13147788

Order 1728.

1193 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 151 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×#88649C212yesmaximal, complemented
2C42×#272C211yesmaximal, characteristic, complemented
3#64084⋊C3C212yesmaximal, complemented
4#11721144C211yesmaximal, characteristic, complemented
5#11721142C214yesmaximal, complemented
6#11721138C214yesmaximal, complemented
7#11721099C211yesmaximal, characteristic, complemented
8#10593969—32nomaximal
9C2×C4×#272C2212yes—
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C2×#634⋊C3C2212yes—
13C2×#6200C2214yes—
14C2×#6198C2214yes—
15C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
16#88649—28no—
17C42×H(3,𝔽3)C2211yescharacteristic
18C4×#1084C2211yescharacteristic
19C4×#1084C2212yes—
20#3752⋊C3C2218yes—
21#88449C2214yes—
22C2×#3799—32no—
23C2×#3795—32no—
24#64084D312yescomplemented
25C3×#875—32no—
26C3×C42×D3—32no—
27C3×C42×D3—61no—
28#63911—32no—
29#63906—38no—
30#63880—38no—
31#63866—32no—
32C22×#272C2311yescharacteristic
33C22×(C3⋊D3)⋊C3C2314yes—
34C2×C4×H(3,𝔽3)C2312yes—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084C2311yescharacteristic
37C2×#1084C2312yes—
38C4×#272—24no—
39C4×#272D414yescomplemented
40C2×(C3⋊Dic3)⋊C3C2314yes—
41C4⋊((C3⋊D3)⋊C3)—24no—
42#634⋊C3—24no—
43#6200—28no—
44#6198—28no—
45((C3⋊D3)⋊C3)⋊C4—28no—
46#5668—92no—
47#5477—91no—
48C2×C4⋊(C3⋊D3)C2×D312yes—
49C2×#634C2×D312yes—
50C2×C3×D12—32no—
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—34no—
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—62no—
55C2×C3×Dic6—32no—
56C2×#626—34no—
57C2×#626—38no—
58C2×#625—34no—
59C2×#624—34no—
60C2×D3×Dic3—34no—
61#3799—68no—
62#3795—68no—
63C32×C42—21no—
64C32×C42C2×D312yes—
65#3752C2×D318yes—
66C3×(C2×D3)⋊C4—38no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—34no—
69C3×C4×Dic3—61no—
70C3×C4×Dic3—62no—
71Dic3⋊Dic3—38no—
72C22×H(3,𝔽3)C2411yescharacteristic
73C2×#272—22no—
74C2×#272C2×D412yes—
75C2×(C3⋊D3)⋊C3—24no—
76C2×(C3⋊D3)⋊C3—28no—
77C4×H(3,𝔽3)—22no—
78C4×H(3,𝔽3)C2×D412yes—
79#1084C2×D412yes—
80#1084#4114yescomplemented
81(C3⋊Dic3)⋊C3—24no—
82C2×#239—92no—
83C2×#237—92no—
84C2×#235—92no—
85C3×#124—92no—
86C2×C3×C42—91no—
87#944—92no—
88#940—92no—
89#930—94no—
90#884—98no—
91#67⋊C4—98no—
92#875—32no—
93#875—92no—
94#874—91no—
95C4×D12—94no—
96C42×D3—32no—
97C42×D3—61no—
98#869—91no—
99C22×C3⋊D3C22×D314yes—
100C22×C3×D3—32no—
101C22×C3×D3—34no—
102C22×C3×D3—61no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3C22×D314yes—
108C4⋊(C3⋊D3)—24no—
109#634—24no—
110C2×C3×Dic3—34no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—61no—
115C3×D12—64no—
116C3×C4×D3—38no—
117C3×C4×D3—64no—
118C3×C4×D3—68no—
119C3×C4×D3—68no—
120C3×Dic6—64no—
121#626—616no—
122#626—68no—
123#625—68no—
124#624—68no—
125D3×Dic3—68no—
126#532—271noSylow
127C2×H(3,𝔽3)C22×D411yescharacteristic
128C2×H(3,𝔽3)C2×#4112yes—
129#272—22no—
130(C3⋊D3)⋊C3—44no—
131C2×C3×Q8—92no—
132C2×C3×D4—92no—
133C22×C3×C4—91no—
134C22×C3×C4—92no—
135C2×#67—94no—
136C2×#67—98no—
137C2×#67—94no—
138C22×Dic3—94no—
139#239—188no—
140#237—188no—
141#235—188no—
142C2×D12—32no—
143C2×D12—92no—
144C2×D12—94no—
145C2×D12—94no—
146C2×C4×D3—32no—
147C2×C4×D3—34no—
148C2×C4×D3—61no—
149C2×C4×D3—62no—
150C2×C4×D3—94no—
151C2×C4×D3—94no—
152C2×Dic6—32no—
153C2×Dic6—92no—
154C3×C22⋊C4—98no—
155C3×C42—32no—
156C3×C42—61no—
157C3×C42—91no—
158C3×C42—92no—
159C3×C42D3211yescharacteristic
160(C2×D3)⋊C4—38no—
161(C2×D3)⋊C4—98no—
162C4⋊Dic3—94no—
163Dic3⋊C4—98no—
164C4×Dic3—34no—
165C4×Dic3—32no—
166C4×Dic3—62no—
167C4×Dic3—61no—
168C22×C32—21no—
169C22×C32C23×D312yes—
170C2×C3⋊D3—24no—
171C2×C3⋊D3—28no—
172C2×C3×D3—34no—
173C2×C3×D3—62no—
174C2×C3×D3—64no—
175C2×C3×D3—64no—
176C2×C3×D3—64no—
177C2×C3×D3—68no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—22no—
181C32×C4D3×D414yes—
182C3⋊Dic3—24no—
183C3×Dic3—34no—
184C3×Dic3—38no—
185C3×Dic3—62no—
186C3×Dic3—64no—
187C3×Dic3—64no—
188C2×#41—272no—
189C4⋊Q8—271no—
190#127—271no—
191#124—92no—
192#121—274no—
193C4×D4—274no—
194C2×C42—91no—
195H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
196C23×C3—94no—
197C23×C3—91no—
198C22×D3—32no—
199C22×D3—34no—
200C22×D3—61no—
201C22×D3—94no—
202C3×Q8—184no—
203C3×D4—184no—
204C2×C3×C4—32no—
205C2×C3×C4—34no—
206C2×C3×C4—62no—
207C2×C3×C4—61no—
208C2×C3×C4—92no—
209C2×C3×C4—91no—
210C2×C3×C4—94no—
211C2×C3×C4—94no—
212C2×C3×C4—184no—
213C2×C3×C4C2×D3212yes—
214C2×C3×C4C2×D3211yescharacteristic
215#67—1816no—
216#67—188no—
217#67—188no—
218C2×Dic3—34no—
219C2×Dic3—32no—
220C2×Dic3—34no—
221C2×Dic3—62no—
222C2×Dic3—61no—
223C2×Dic3—94no—
224C2×Dic3—188no—
225D12—64no—
226D12—184no—
227D12—188no—
228D12—188no—
229C4×D3—38no—
230C4×D3—64no—
231C4×D3—68no—
232C4×D3—68no—
233C4×D3—188no—
234C4×D3—188no—
235Dic6—64no—
236Dic6—184no—
237C2×C32—21no—
238C2×C32—22no—
239C2×C32C2×D3×D412yes—
240C2×C32D3×#4114yes—
241C3⋊D3—44no—
242C3×D3—64no—
243C3×D3—64no—
244C3×D3—124no—
245#41—548no—
246C2×Q8—92no—
247C2×D4—92no—
248C2×D4—274no—
249C2×D4—274no—
250C22×C4—91no—
251C22×C4—92no—
252C22×C4—274no—
253C4⋊C4—274no—
254C22⋊C4—98no—
255C42—92no—
256C42—91no—
257C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C22×C3—32no—
259C22×C3—61no—
260C22×C3—92no—
261C22×C3—184no—
262C22×C3—188no—
263C22×C3—182no—
264C22×C3C22×D3211yescharacteristic
265C2×D3—34no—
266C2×D3—62no—
267C2×D3—64no—
268C2×D3—64no—
269C2×D3—64no—
270C2×D3—68no—
271C2×D3—184no—
272C2×D3—188no—
273C3×C4—22no—
274C3×C4—34no—
275C3×C4—64no—
276C3×C4—64no—
277C3×C4—62no—
278C3×C4—92no—
279C3×C4—94no—
280C3×C4—184no—
281C3×C4#380212yes—
282Dic3—34no—
283Dic3—38no—
284Dic3—62no—
285Dic3—64no—
286Dic3—64no—
287Dic3—184no—
288C32—21no—
289C32#566812yescomplemented
290C23—91no—
291C23—94no—
292Q8—184no—
293D4—184no—
294D4—548no—
295D4—548no—
296C2×C4—92no—
297C2×C4—91no—
298C2×C4—94no—
299C2×C4—94no—
300C2×C4—184no—
301C2×C4—548no—
302C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
303C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
304C2×C3—32no—
305C2×C3—34no—
306C2×C3—61no—
307C2×C3—62no—
308C2×C3—182no—
309C2×C3—364no—
310C2×C3C2×#380211yescharacteristic
311C2×C3#6430612yes—
312D3—64no—
313D3—64no—
314D3—124no—
315D3—364no—
316C22—92no—
317C22—182no—
318C22—184no—
319C22—188no—
320C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C4—22no—
322C4—92no—
323C4—94no—
324C4—184no—
325C4#8865512yes—
326C3—32no—
327C3—61no—
328C3#1059397111yescharacteristic
329C2—182no—
330C2—364no—
331C2C2×#8865511yescharacteristic
332C2#1172249112yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#532
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88649C2semidirect
2. C42×#272C2semidirect
3. #64084⋊C3C2semidirect
4. #11721144C2semidirect
5. #11721142C2semidirect
6. #11721138C2semidirect
7. #11721099C2semidirect
24. #64084D3semidirect
39. C4×#272D4semidirect
80. #1084#41semidirect
195. H(3,𝔽3)#532semidirect
289. C32#5668semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
C42×#2722
#64084⋊C32
#117211442
#117211422
#117211382
#117210992
#105939693