Finite Group Databasebeta

← #13147912

Subgroups — #13147912

Order 1728.

1263 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1263 conjugacy classes of subgroups — too many to draw. Only its 155 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×#88655C211yesmaximal, characteristic
2C2×#88651C211yesmaximal, characteristic, complemented
3C42×#272C211yesmaximal, characteristic, complemented
4#64084⋊C3C212yesmaximal, complemented
5#11721161C214yesmaximal, complemented
6#11721145C214yesmaximal, complemented
7#11721113C212yesmaximal, complemented
8D3×#875—32nomaximal
9C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×#634⋊C3C2212yescomplemented
14C2×#6199C2212yes—
15C2×#6198C2212yes—
16C2×#6196C2212yescomplemented
17#88655—24no—
18#88651—24no—
19C42×H(3,𝔽3)C2211yescharacteristic, complemented
20C4×#1084C2211yescharacteristic, complemented
21C4×#1084C2212yescomplemented
22#3752⋊C3C2218yescomplemented
23#88450C2218yescomplemented
24C2×D3×D12—32no—
25C2×D3×Dic6—32no—
26#64084D312yescomplemented
27C3×#875—32no—
28C3×C42×D3—32no—
29C3×C42×D3—61no—
30D3×(C2×D3)⋊C4—38no—
31#63864—32no—
32#63860—32no—
33#63852—38no—
34C22×#272C2311yescharacteristic
35C22×(C3⋊D3)⋊C3C2314yes—
36C2×((C3⋊D3)⋊C3)⋊C2—24no—
37C2×((C3⋊D3)⋊C3)⋊C2—28no—
38C2×C4×H(3,𝔽3)C2312yes—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084C2312yes—
41C2×#1084C2311yescharacteristic
42C4×#272—24no—
43C4×#272D414yes—
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C4⋊((C3⋊D3)⋊C3)—24no—
46#634⋊C3—24no—
47#6199—24no—
48#6198—24no—
49#6196—24no—
50D3×#124—92no—
51C2×#875—91no—
52C22×D32—34no—
53C2×C4⋊(C3⋊D3)C2×D312yes—
54C2×#634C2×D312yescomplemented
55C2×C3×D12—32no—
56C2×C3×C4×D3—32no—
57C2×C3×C4×D3—34no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—62no—
60C2×C3×Dic6—32no—
61C2×#627—34no—
62C2×#626—34no—
63C2×D3×Dic3—34no—
64D3×D12—68no—
65D3×Dic6—68no—
66C32×C42—21no—
67C32×C42C2×D312yescomplemented
68#3752C2×D318yescomplemented
69C3×(C2×D3)⋊C4—38no—
70C3×C4×Dic3—34no—
71C3×C4×Dic3—32no—
72C3×C4×Dic3—61no—
73C3×C4×Dic3—62no—
74#3722—38no—
75(C2×D3)⋊Dic3—38no—
76C22×H(3,𝔽3)C2411yescharacteristic
77C2×#272C2×D412yes—
78C2×#272#4114yes—
79C2×(C3⋊D3)⋊C3—24no—
80C2×(C3⋊D3)⋊C3—28no—
81((C3⋊D3)⋊C3)⋊C2—48no—
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)C2×D412yes—
84#1084—22no—
85#1084C2×D412yes—
86(C3⋊Dic3)⋊C3—24no—
87C2×D3×Q8—92no—
88C2×D3×D4—92no—
89C22×D12—91no—
90C22×Dic6—91no—
91C3×#124—92no—
92C2×C3×C42—91no—
93#947—92no—
94#936—92no—
95C2×(C2×D3)⋊C4—94no—
96#885—98no—
97D3×C22⋊C4—98no—
98#875—32no—
99#875—92no—
100#875—94no—
101#875—92no—
102C42×D3—32no—
103C42×D3—61no—
104C22×C3⋊D3C22×D314yes—
105C22×C3×D3—32no—
106C22×C3×D3—34no—
107C22×C3×D3—61no—
108C2×D32—616no—
109C2×D32—68no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D312yes—
114C2×C3⋊Dic3C22×D314yes—
115C4⋊(C3⋊D3)—24no—
116#634—24no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C3×D12—64no—
123C3×C4×D3—38no—
124C3×C4×D3—64no—
125C3×C4×D3—68no—
126C3×C4×D3—68no—
127C3×Dic6—64no—
128#627—68no—
129#626—68no—
130D3×Dic3—68no—
131C2×#124—271noSylow
132C2×H(3,𝔽3)C22×D411yescharacteristic
133C2×H(3,𝔽3)C2×#4112yes—
134#272#12414yescomplemented
135(C3⋊D3)⋊C3—44no—
136C23×D3—92no—
137C23×D3—94no—
138C2×C3×Q8—92no—
139C2×C3×D4—92no—
140C22×C3×C4—91no—
141C22×C3×C4—92no—
142C2×#67—94no—
143C22×Dic3—92no—
144D3×Q8—188no—
145D3×D4—188no—
146C2×D12—32no—
147C2×D12—92no—
148C2×D12—92no—
149C2×D12—184no—
150C2×C4×D3—32no—
151C2×C4×D3—34no—
152C2×C4×D3—61no—
153C2×C4×D3—62no—
154C2×C4×D3—94no—
155C2×Dic6—32no—
156C2×Dic6—92no—
157C2×Dic6—92no—
158C2×Dic6—94no—
159C2×Dic6—184no—
160C3×C22⋊C4—98no—
161C3×C42—32no—
162C3×C42—61no—
163C3×C42—92no—
164C3×C42—91no—
165C3×C42D3211yescharacteristic
166#217—98no—
167(C2×D3)⋊C4—38no—
168(C2×D3)⋊C4—98no—
169(C2×D3)⋊C4—98no—
170(C2×D3)⋊C4—98no—
171C4×Dic3—34no—
172C4×Dic3—32no—
173C4×Dic3—62no—
174C4×Dic3—61no—
175C22×C32—21no—
176C22×C32C23×D312yes—
177C2×C3⋊D3—24no—
178C2×C3⋊D3—28no—
179C2×C3×D3—34no—
180C2×C3×D3—38no—
181C2×C3×D3—62no—
182C2×C3×D3—64no—
183C2×C3×D3—64no—
184C2×C3×D3—68no—
185D32—1216no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—22no—
189C32×C4D3×D414yes—
190C3⋊Dic3—24no—
191C3×Dic3—34no—
192C3×Dic3—62no—
193C3×Dic3—64no—
194C3×Dic3—64no—
195C3×Dic3—64no—
196C22×Q8—271no—
197C22×D4—271no—
198#124—92no—
199#124—274no—
200#124—272no—
201C2×C22⋊C4—274no—
202C2×C42—91no—
203H(3,𝔽3)C2×#12411yescharacteristic, complemented, Sylow
204C23×C3—94no—
205C23×C3—91no—
206C22×D3—32no—
207C22×D3—34no—
208C22×D3—61no—
209C22×D3—94no—
210C22×D3—188no—
211C22×D3—1816no—
212C22×D3—184no—
213C22×D3—188no—
214C3×Q8—184no—
215C3×D4—184no—
216C2×C3×C4—32no—
217C2×C3×C4—34no—
218C2×C3×C4—62no—
219C2×C3×C4—61no—
220C2×C3×C4—91no—
221C2×C3×C4—94no—
222C2×C3×C4—94no—
223C2×C3×C4—92no—
224C2×C3×C4—184no—
225C2×C3×C4C2×D3212yes—
226C2×C3×C4C2×D3211yescharacteristic
227#67—188no—
228C2×Dic3—32no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—61no—
232C2×Dic3—62no—
233C2×Dic3—94no—
234C2×Dic3—184no—
235D12—64no—
236D12—184no—
237D12—184no—
238C4×D3—38no—
239C4×D3—64no—
240C4×D3—68no—
241C4×D3—68no—
242C4×D3—188no—
243Dic6—64no—
244Dic6—184no—
245Dic6—188no—
246Dic6—184no—
247C2×C32—21no—
248C2×C32—22no—
249C2×C32C2×D3×D412yes—
250C2×C32D3×#4114yes—
251C3⋊D3—44no—
252C3×D3—38no—
253C3×D3—64no—
254C3×D3—124no—
255C24—272no—
256C2×Q8—92no—
257C2×Q8—272no—
258C2×Q8—544no—
259C2×D4—92no—
260C2×D4—272no—
261C2×D4—544no—
262C22×C4—91no—
263C22×C4—92no—
264C22×C4—272no—
265C22⋊C4—98no—
266C22⋊C4—278no—
267C42—92no—
268C42—91no—
269C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
270C22×C3—32no—
271C22×C3—61no—
272C22×C3—92no—
273C22×C3—94no—
274C22×C3—184no—
275C22×C3—188no—
276C22×C3C22×D3211yescharacteristic
277C2×D3—34no—
278C2×D3—38no—
279C2×D3—62no—
280C2×D3—64no—
281C2×D3—64no—
282C2×D3—68no—
283C2×D3—184no—
284C2×D3—188no—
285C2×D3—3616no—
286C2×D3—368no—
287C3×C4—22no—
288C3×C4—34no—
289C3×C4—64no—
290C3×C4—64no—
291C3×C4—62no—
292C3×C4—92no—
293C3×C4—184no—
294C3×C4—182no—
295C3×C4#380212yes—
296Dic3—34no—
297Dic3—62no—
298Dic3—64no—
299Dic3—64no—
300Dic3—64no—
301Dic3—184no—
302C32—21no—
303C32D3×#12412yescomplemented
304C23—91no—
305C23—94no—
306C23—544no—
307C23—548no—
308Q8—184no—
309Q8—544no—
310D4—184no—
311D4—544no—
312C2×C4—91no—
313C2×C4—94no—
314C2×C4—92no—
315C2×C4—94no—
316C2×C4—184no—
317C2×C4—544no—
318C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
319C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C2×C3—32no—
321C2×C3—34no—
322C2×C3—61no—
323C2×C3—62no—
324C2×C3—94no—
325C2×C3—364no—
326C2×C3C2×#380211yescharacteristic
327C2×C3#6430612yes—
328D3—38no—
329D3—64no—
330D3—124no—
331D3—364no—
332C22—92no—
333C22—94no—
334C22—184no—
335C22—188no—
336C22—1088no—
337C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
338C4—22no—
339C4—92no—
340C4—184no—
341C4—182no—
342C4#8865512yes—
343C3—32no—
344C3—61no—
345C3#1059400911yescharacteristic
346C2—94no—
347C2—364no—
348C2C2×#8865511yescharacteristic
349C2#1172249112yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88655C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#124
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
2. C2×#88651C2semidirect
3. C42×#272C2semidirect
4. #64084⋊C3C2semidirect
5. #11721161C2semidirect
6. #11721145C2semidirect
7. #11721113C2semidirect
13. C2×#634⋊C3C22semidirect
16. C2×#6196C22semidirect
19. C42×H(3,𝔽3)C22semidirect
20. C4×#1084C22semidirect
21. C4×#1084C22semidirect
22. #3752⋊C3C22semidirect
23. #88450C22semidirect
26. #64084D3semidirect
54. C2×#634C2×D3semidirect
67. C32×C42C2×D3semidirect
68. #3752C2×D3semidirect
134. #272#124semidirect
203. H(3,𝔽3)C2×#124semidirect
269. C42((C3⋊D3)⋊C3)⋊C2semidirect
303. C32D3×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#886512
C42×#2722
#64084⋊C32
#117211612
#117211452
#117211132
D3×#8753