Finite Group Databasebeta

← C2×#11721762 #13153855

Subgroups — C2×#11721762 #13153855

Order 1728.

1881 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 1881 conjugacy classes of subgroups — too many to draw. Only its 269 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
1C22×C4⋊#272C211yesmaximal, characteristic
2C22×C4×#272C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#88542C211yesmaximal, characteristic, complemented
5C2×#88542C211yesmaximal, characteristic, complemented
6C2×#88542C211yesmaximal, characteristic, complemented
7C2×#88540C211yesmaximal, characteristic, complemented
8#11721762C218yesmaximal, direct product factor
9C2×C3×#894—34nomaximal
10C23×#272C2211yescharacteristic
11C23×#272C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22×#1084C2211yescharacteristic
17C2×C4⋊#272—24no—
18C2×C4⋊#272C2214yes—
19C2×C4×#272—24no—
20C2×C4×#272C2214yescomplemented
21C4⋊C4×H(3,𝔽3)C2214yescomplemented
22#88542C2214yescomplemented
23#88542C2214yescomplemented
24#88542C2214yescomplemented
25#88540C2214yescomplemented
26C22×C3×D12—34no—
27C22×C3×C4×D3—34no—
28C2×C32×C4⋊C4D314yescomplemented
29C2×C3×(C2×D3)⋊C4—34no—
30C2×C3×(C2×D3)⋊C4—34no—
31C2×C3×(C2×D3)⋊C4—34no—
32C2×C3×Dic3⋊C4—34no—
33C3×#894—332no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C22×#272—22no—
36C22×#272—22no—
37C22×#272—22no—
38C22×#272—22no—
39C22×#272—22no—
40C22×#272D414yes—
41C22×#272C2312yes—
42C22×#272C2312yes—
43C2×C4×H(3,𝔽3)—22no—
44C2×C4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)—22no—
46C2×C4×H(3,𝔽3)C2312yes—
47C2×C4×H(3,𝔽3)C2312yes—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×#1084—22no—
50C2×#1084—22no—
51C2×#1084C2312yes—
52C2×#1084C2312yes—
53C4⋊#272—28no—
54C4×#272—28no—
55C2×C3×#123—91no—
56C2×#894—34no—
57C23×C3×D3—34no—
58C23×C3×D3—34no—
59C22×C32×C4C2×D314yes—
60C22×C32×C4C2×D314yes—
61C22×C32×C4C2×D314yes—
62C22×C3×Dic3—34no—
63C22×C3×Dic3—34no—
64C2×C3×D12—316no—
65C2×C3×D12—616no—
66C2×C3×C4×D3—316no—
67C2×C3×C4×D3—616no—
68C32×C4⋊C4C2×D3116yescomplemented
69C3×(C2×D3)⋊C4—316no—
70C3×(C2×D3)⋊C4—316no—
71C3×(C2×D3)⋊C4—316no—
72C3×Dic3⋊C4—316no—
73C22×H(3,𝔽3)C2×D412yes—
74C22×H(3,𝔽3)#4112yes—
75C22×H(3,𝔽3)#4112yes—
76C22×H(3,𝔽3)C2411yescharacteristic
77C2×#272—28no—
78C2×#272—22no—
79C2×#272—22no—
80C2×#272—22no—
81C2×#272—22no—
82C2×#272—22no—
83C2×#272—44no—
84C2×#272C2×D414yes—
85C4×H(3,𝔽3)—22no—
86C4×H(3,𝔽3)—22no—
87C4×H(3,𝔽3)—22no—
88#1084—22no—
89#1084—22no—
90C22×C3×D4—91no—
91C23×C3×C4—91no—
92C22×D12—34no—
93C22×C4×D3—34no—
94C3×#123—98no—
95C2×C3×C4⋊C4—34no—
96C2×C3×C4⋊C4—91no—
97C2×C3×C4⋊C4C3⋊D311yescharacteristic
98C2×C3×C22⋊C4—91no—
99C2×C3×C22⋊C4—91no—
100C2×C3×C22⋊C4—91no—
101C2×(C2×D3)⋊C4—34no—
102C2×(C2×D3)⋊C4—34no—
103C2×(C2×D3)⋊C4—34no—
104C2×Dic3⋊C4—34no—
105#894—332no—
106C23×C32C22×D314yes—
107C22×C3×D3—316no—
108C22×C3×D3—38no—
109C22×C3×D3—38no—
110C22×C3×D3—68no—
111C22×C3×D3—68no—
112C22×C3×D3—68no—
113C22×C3×D3—68no—
114C22×C3×D3—68no—
115C2×C32×C4—28no—
116C2×C32×C4—28no—
117C2×C32×C4—28no—
118C2×C32×C4C22×D318yes—
119C2×C32×C4C22×D318yes—
120C2×C32×C4C22×D318yes—
121C2×C3×Dic3—38no—
122C2×C3×Dic3—38no—
123C2×C3×Dic3—68no—
124C2×C3×Dic3—68no—
125C3×D12—632no—
126C3×C4×D3—632no—
127C2×#123—91noSylow
128C2×H(3,𝔽3)#12314yescomplemented
129C2×H(3,𝔽3)C22×D411yescharacteristic
130C2×H(3,𝔽3)C2×#4111yescharacteristic
131C2×H(3,𝔽3)C2×#4111yescharacteristic
132#272—24no—
133#272—42no—
134C24×C3—91no—
135C24×C3—91no—
136C23×D3—34no—
137C23×D3—34no—
138C2×C3×D4—94no—
139C2×C3×D4—184no—
140C22×C3×C4—34no—
141C22×C3×C4—34no—
142C22×C3×C4—34no—
143C22×C3×C4—91no—
144C22×C3×C4—91no—
145C22×C3×C4—94no—
146C22×C3×C4—184no—
147C22×C3×C4C2×C3⋊D311yescharacteristic
148C22×C3×C4C2×C3⋊D311yescharacteristic
149C22×C3×C4C2×C3⋊D311yescharacteristic
150C22×Dic3—34no—
151C22×Dic3—34no—
152C2×D12—316no—
153C2×D12—616no—
154C2×C4×D3—316no—
155C2×C4×D3—616no—
156C3×C4⋊C4—316no—
157C3×C4⋊C4—94no—
158C3×C4⋊C4C2×C3⋊D314yes—
159C3×C22⋊C4—94no—
160C3×C22⋊C4—94no—
161C3×C22⋊C4—94no—
162(C2×D3)⋊C4—316no—
163(C2×D3)⋊C4—316no—
164(C2×D3)⋊C4—316no—
165Dic3⋊C4—316no—
166C22×C32#23518yes—
167C22×C32D3×D418yes—
168C22×C32#23918yes—
169C22×C32C23×D314yes—
170C2×C3×D3—316no—
171C2×C3×D3—632no—
172C2×C3×D3—68no—
173C2×C3×D3—68no—
174C2×C3×D3—68no—
175C2×C3×D3—68no—
176C2×C3×D3—68no—
177C2×C3×D3—1216no—
178C32×C4—28no—
179C32×C4—28no—
180C32×C4—28no—
181C3×Dic3—68no—
182C3×Dic3—68no—
183C22×D4—91no—
184C23×C4—91no—
185#123—98no—
186C2×C4⋊C4—91no—
187C2×C4⋊C4#27211yescharacteristic, complemented
188C2×C22⋊C4—91no—
189C2×C22⋊C4—91no—
190C2×C22⋊C4—91no—
191H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
192C23×C3—34no—
193C23×C3—92no—
194C23×C3—94no—
195C23×C3—92no—
196C23×C3—182no—
197C23×C3—182no—
198C23×C3—182no—
199C23×C3—182no—
200C23×C3—182no—
201C23×C3C22×C3⋊D311yescharacteristic
202C22×D3—38no—
203C22×D3—316no—
204C22×D3—38no—
205C22×D3—68no—
206C22×D3—68no—
207C22×D3—68no—
208C22×D3—68no—
209C22×D3—68no—
210C3×D4—188no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—38no—
215C2×C3×C4—38no—
216C2×C3×C4—38no—
217C2×C3×C4—68no—
218C2×C3×C4—68no—
219C2×C3×C4—68no—
220C2×C3×C4—92no—
221C2×C3×C4—92no—
222C2×C3×C4—188no—
223C2×C3×C4—182no—
224C2×C3×C4—182no—
225C2×C3×C4C22×C3⋊D312yes—
226C2×C3×C4C22×C3⋊D312yes—
227C2×C3×C4C22×C3⋊D312yes—
228C2×Dic3—38no—
229C2×Dic3—38no—
230C2×Dic3—68no—
231C2×Dic3—68no—
232D12—632no—
233C4×D3—632no—
234C2×C32#894116yescomplemented
235C2×C32C2×#23514yes—
236C2×C32C2×D3×D414yes—
237C2×C32C2×#23914yes—
238C3×D3—616no—
239C3×D3—128no—
240C24—91no—
241C24—91no—
242C2×D4—94no—
243C2×D4—184no—
244C22×C4—91no—
245C22×C4—91no—
246C22×C4—94no—
247C22×C4—184no—
248C22×C4C2×#27211yescharacteristic
249C22×C4C2×#27211yescharacteristic
250C22×C4C2×#27211yescharacteristic
251C4⋊C4—94no—
252C4⋊C4C2×#27214yescomplemented
253C22⋊C4—94no—
254C22⋊C4—94no—
255C22⋊C4—94no—
256C22×C3—34no—
257C22×C3—38no—
258C22×C3—38no—
259C22×C3—38no—
260C22×C3—94no—
261C22×C3—182no—
262C22×C3—188no—
263C22×C3—182no—
264C22×C3—182no—
265C22×C3—182no—
266C22×C3—182no—
267C22×C3—364no—
268C22×C3#382812yes—
269C22×C3D4×C3⋊D312yes—
270C22×C3#383212yes—
271C22×C3C23×C3⋊D311yescharacteristic
272C2×D3—316no—
273C2×D3—632no—
274C2×D3—68no—
275C2×D3—68no—
276C2×D3—68no—
277C2×D3—68no—
278C2×D3—68no—
279C2×D3—1216no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—68no—
284C3×C4—68no—
285C3×C4—68no—
286C3×C4—182no—
287C3×C4—182no—
288Dic3—68no—
289Dic3—68no—
290C32C2×#89414yescomplemented
291C23—92no—
292C23—94no—
293C23—92no—
294C23—182no—
295C23—182no—
296C23—182no—
297C23—182no—
298C23—182no—
299C23C22×#27211yescharacteristic
300D4—188no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—92no—
305C2×C4—92no—
306C2×C4—188no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4C22×#27212yes—
310C2×C4C22×#27212yes—
311C2×C4C22×#27212yes—
312C2×C3—316no—
313C2×C3—34no—
314C2×C3—34no—
315C2×C3—34no—
316C2×C3—184no—
317C2×C3—362no—
318C2×C3#6410314yes—
319C2×C3C2×#382811yescharacteristic
320C2×C3C2×D4×C3⋊D311yescharacteristic
321C2×C3C2×#383211yescharacteristic
322D3—616no—
323D3—128no—
324C22—94no—
325C22—188no—
326C22—182no—
327C22—182no—
328C22—182no—
329C22—182no—
330C22—182no—
331C22—364no—
332C22#8874012yes—
333C22D4×#27212yes—
334C22#8874812yes—
335C22C23×#27211yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—182no—
340C4—182no—
341C3—34no—
342C3C2×#6410311yescharacteristic
343C2—184no—
344C2—362no—
345C2#1172176214yesdirect product factor
346C2C2×#8874011yescharacteristic
347C2C2×D4×#27211yescharacteristic
348C2C2×#8874811yescharacteristic
subgroupG/H
center, socleC23×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C23×C3⋊D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC22×C4⋊#272C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)C2×#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×#272C2semidirect
3. C2×C4⋊C4×H(3,𝔽3)C2semidirect
4. C2×#88542C2semidirect
5. C2×#88542C2semidirect
6. C2×#88542C2semidirect
7. C2×#88540C2semidirect
8. #11721762C2direct (N × H)
20. C2×C4×#272C22semidirect
21. C4⋊C4×H(3,𝔽3)C22semidirect
22. #88542C22semidirect
23. #88542C22semidirect
24. #88542C22semidirect
25. #88540C22semidirect
28. C2×C32×C4⋊C4D3semidirect
68. C32×C4⋊C4C2×D3semidirect
128. C2×H(3,𝔽3)#123semidirect
187. C2×C4⋊C4#272semidirect
191. H(3,𝔽3)C2×#123semidirect
234. C2×C32#894semidirect
252. C4⋊C4C2×#272semidirect
290. C32C2×#894semidirect
345. C2#11721762direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C22×C4×#2722
C2×C4⋊C4×H(3,𝔽3)2
C2×#885422
C2×#885402
#117217622
C2×C3×#8943