Finite Group Databasebeta

← #13147844

Subgroups — #13147844

Order 1728.

916 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 139 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#11721718C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
3#11721159C212yesmaximal, complemented
4#11721144C211yesmaximal, characteristic, complemented
5#11721140C214yesmaximal, complemented
6#11721139C214yesmaximal, complemented
7#11721095C211yesmaximal, characteristic, complemented
8#10593986—32nomaximal
9C2×C4×#272C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×C4×(C3⋊D3)⋊C3C2214yes—
12C2×#6200C2214yes—
13C2×#6198C2214yes—
14C42×H(3,𝔽3)C2211yescharacteristic
15#88542C2212yes—
16#88540C2212yes—
17C4×#1084C2211yescharacteristic
18#3752⋊C3C2214yes—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20#88450C2214yes—
21#88448C2214yes—
22C4×C4⋊(C3⋊D3)D312yescomplemented
23C3×C4×D12—32no—
24C3×#872—32no—
25C3×#872—61no—
26#63912—34no—
27#63911—32no—
28#63909—32no—
29#63908—32no—
30#63906—34no—
31#63869—32no—
32#63853—34no—
33#63850—34no—
34C22×#272C2311yescharacteristic
35C22×(C3⋊D3)⋊C3C2314yes—
36C2×C4×H(3,𝔽3)C2312yes—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2312yes—
39C2×#1084C2311yescharacteristic
40C4×#272—22no—
41C2×(C3⋊Dic3)⋊C3C2314yes—
42C4⋊((C3⋊D3)⋊C3)—24no—
43C4×(C3⋊D3)⋊C3—44no—
44#6200—44no—
45#6198—44no—
46#5562—92no—
47#5535—91no—
48C2×C4⋊(C3⋊D3)C2×D312yes—
49C2×C4×C3⋊D3C2×D314yes—
50C2×C3×D12—32no—
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—34no—
53C2×C3×C4×D3—61no—
54C2×#626—34no—
55C2×#626—34no—
56C2×#626—34no—
57C2×#625—34no—
58C32×C42—21no—
59C32×C42C2×D312yes—
60#3752C2×D314yes—
61C4⋊(C3⋊Dic3)C2×D312yes—
62C3×(C2×D3)⋊C4—34no—
63C3×(C2×D3)⋊C4—32no—
64C3×(C2×D3)⋊C4—32no—
65C3×(C2×D3)⋊C4—62no—
66C3×C4⋊Dic3—32no—
67C3×Dic3⋊C4—32no—
68C3×Dic3⋊C4—32no—
69C3×Dic3⋊C4—62no—
70C3×C4×Dic3—32no—
71C3×C4×Dic3—61no—
72#3724—34no—
73Dic3⋊Dic3—34no—
74#3722—34no—
75(C2×D3)⋊Dic3—34no—
76C22×H(3,𝔽3)C2411yescharacteristic
77C2×#272—21no—
78C2×#272—22no—
79C2×(C3⋊D3)⋊C3—24no—
80C2×(C3⋊D3)⋊C3—24no—
81C2×(C3⋊D3)⋊C3—24no—
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)#4112yes—
84#1084—21no—
85#1084—22no—
86(C3⋊Dic3)⋊C3—24no—
87C3×C4×D4—92no—
88C3×#117—91no—
89#940—92no—
90#938—92no—
91#930—92no—
92#930—94no—
93#929—94no—
94#895—92no—
95#894—94no—
96#890—92no—
97#884—94no—
98#884—94no—
99#883—94no—
100#874—91no—
101C4×D12—32no—
102C4×D12—92no—
103#872—32no—
104#872—61no—
105#870—91no—
106C22×C3⋊D3C22×D314yes—
107C22×C3×D3—34no—
108C22×C3×D3—32no—
109C22×C3×D3—61no—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4C22×D312yes—
113C2×C32×C4C22×D312yes—
114C2×C32×C4C22×D312yes—
115C2×C3⋊Dic3C22×D314yes—
116C4⋊(C3⋊D3)—24no—
117C4×C3⋊D3—44no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C3×D12—64no—
125C3×C4×D3—64no—
126C3×C4×D3—64no—
127C3×C4×D3—124no—
128#626—124no—
129#626—124no—
130#626—124no—
131#625—124no—
132#540—271noSylow
133C2×H(3,𝔽3)C2×#4111yescharacteristic
134C2×H(3,𝔽3)H(5,𝔽2)12yes—
135#272—41no—
136(C3⋊D3)⋊C3—44no—
137C2×C3×D4—92no—
138C22×C3×C4—94no—
139C22×C3×C4—91no—
140C2×#67—94no—
141C2×#67—94no—
142C2×#67—94no—
143C2×#67—94no—
144C2×D12—32no—
145C2×D12—92no—
146C2×D12—94no—
147C2×D12—94no—
148C2×C4×D3—32no—
149C2×C4×D3—34no—
150C2×C4×D3—61no—
151C2×C4×D3—94no—
152C3×C4⋊C4—92no—
153C3×C4⋊C4—92no—
154C3×C22⋊C4—94no—
155C3×C22⋊C4—92no—
156C3×C42—32no—
157C3×C42—61no—
158C3×C42—91no—
159C3×C42D3211yescharacteristic
160#217—94no—
161(C2×D3)⋊C4—32no—
162(C2×D3)⋊C4—32no—
163(C2×D3)⋊C4—34no—
164(C2×D3)⋊C4—62no—
165(C2×D3)⋊C4—94no—
166(C2×D3)⋊C4—94no—
167(C2×D3)⋊C4—94no—
168C4⋊Dic3—32no—
169C4⋊Dic3—92no—
170C4⋊Dic3—94no—
171Dic3⋊C4—32no—
172Dic3⋊C4—32no—
173Dic3⋊C4—62no—
174Dic3⋊C4—94no—
175Dic3⋊C4—94no—
176C4×Dic3—32no—
177C4×Dic3—61no—
178C22×C32—21no—
179C22×C32C23×D312yes—
180C2×C3⋊D3—24no—
181C2×C3⋊D3—24no—
182C2×C3⋊D3—24no—
183C2×C3×D3—62no—
184C2×C3×D3—64no—
185C2×C3×D3—62no—
186C2×C3×D3—64no—
187C2×C3×D3—62no—
188C2×C3×D3—64no—
189C2×C3×D3—62no—
190C2×C3×D3—122no—
191C32×C4—22no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—42no—
195C32×C4#23514yes—
196C3⋊Dic3—24no—
197C3×Dic3—62no—
198C3×Dic3—62no—
199C3×Dic3—62no—
200C3×Dic3—64no—
201C3×Dic3—62no—
202C3×Dic3—122no—
203#127—271no—
204#125—271no—
205#123—274no—
206#121—274no—
207#121—272no—
208C4×D4—92no—
209#117—91no—
210H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
211C23×C3—94no—
212C23×C3—91no—
213C22×D3—32no—
214C22×D3—34no—
215C22×D3—61no—
216C22×D3—94no—
217C3×D4—184no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—61no—
222C2×C3×C4—62no—
223C2×C3×C4—94no—
224C2×C3×C4—92no—
225C2×C3×C4—91no—
226C2×C3×C4—182no—
227C2×C3×C4—364no—
228C2×C3×C4C2×D3212yes—
229C2×C3×C4C2×D3211yescharacteristic
230#67—364no—
231#67—364no—
232#67—364no—
233#67—364no—
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—32no—
237C2×Dic3—34no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—94no—
241D12—64no—
242D12—184no—
243D12—364no—
244D12—364no—
245C4×D3—64no—
246C4×D3—64no—
247C4×D3—124no—
248C4×D3—364no—
249C2×C32—21no—
250C2×C32—22no—
251C2×C32C2×#23512yes—
252C2×C32#100412yes—
253C2×C32#100912yes—
254C3⋊D3—44no—
255C3×D3—122no—
256C3×D3—124no—
257C3×D3—122no—
258C2×D4—92no—
259C2×D4—274no—
260C2×D4—274no—
261C22×C4—91no—
262C22×C4—94no—
263C4⋊C4—92no—
264C4⋊C4—92no—
265C4⋊C4—274no—
266C22⋊C4—92no—
267C22⋊C4—94no—
268C22⋊C4—274no—
269C42—91no—
270C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
271C22×C3—32no—
272C22×C3—61no—
273C22×C3—184no—
274C22×C3—184no—
275C22×C3—184no—
276C22×C3—181no—
277C22×C3—182no—
278C22×C3C22×D3211yescharacteristic
279C2×D3—62no—
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—64no—
284C2×D3—64no—
285C2×D3—64no—
286C2×D3—122no—
287C2×D3—184no—
288C2×D3—184no—
289C2×D3—184no—
290C3×C4—22no—
291C3×C4—34no—
292C3×C4—62no—
293C3×C4—62no—
294C3×C4—62no—
295C3×C4—122no—
296C3×C4—184no—
297C3×C4—181no—
298C3×C4—182no—
299C3×C4#379812yes—
300Dic3—62no—
301Dic3—62no—
302Dic3—62no—
303Dic3—62no—
304Dic3—64no—
305Dic3—122no—
306Dic3—184no—
307C32—21no—
308C32#556212yescomplemented
309C23—91no—
310C23—94no—
311D4—184no—
312D4—1084no—
313D4—1084no—
314C2×C4—92no—
315C2×C4—94no—
316C2×C4—91no—
317C2×C4—182no—
318C2×C4—364no—
319C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
320C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C2×C3—32no—
322C2×C3—32no—
323C2×C3—32no—
324C2×C3—61no—
325C2×C3—62no—
326C2×C3—364no—
327C2×C3—361no—
328C2×C3C2×#379811yescharacteristic
329C2×C3#6430712yes—
330D3—122no—
331D3—122no—
332D3—124no—
333D3—364no—
334C22—181no—
335C22—184no—
336C22—184no—
337C22—182no—
338C22—184no—
339C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
340C4—22no—
341C4—181no—
342C4—184no—
343C4—182no—
344C4#8865012yes—
345C3—32no—
346C3—61no—
347C3#1059398411yescharacteristic
348C2—361no—
349C2—364no—
350C2C2×#8865011yescharacteristic
351C2#1172249212yes—
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)#540
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. #11721718C2semidirect
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #11721159C2semidirect
4. #11721144C2semidirect
5. #11721140C2semidirect
6. #11721139C2semidirect
7. #11721095C2semidirect
22. C4×C4⋊(C3⋊D3)D3semidirect
210. H(3,𝔽3)#540semidirect
308. C32#5562semidirect

Maximal subgroups

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

subgroupindex
#117217182
C4×C4⋊((C3⋊D3)⋊C3)2
#117211592
#117211442
#117211402
#117211392
#117210952
#105939863