Finite Group Databasebeta

← C2×#11720788 #13147403

Subgroups — C2×#11720788 #13147403

Order 1728.

775 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 71 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×D4×#272C211yesmaximal, characteristic
2C2×C4⋊#1088C211yesmaximal, characteristic, complemented
3C2×#88626C211yesmaximal, characteristic, complemented
4#11720788C214yesmaximal, direct product factor
5C23×#272—21no—
6C22×#1088—21no—
7C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
8C2×#6235—21no—
9D4×#272—21no—
10D4×#272—21no—
11D4×#272C412yescomplemented
12D4×#272C2212yes—
13C2×C4⋊#272C411yescharacteristic, complemented
14C2×C4×#272C2211yescharacteristic
15C2×#6188—21no—
16C4⋊#1088—21no—
17C4⋊#1088C2212yescomplemented
18#88626—21no—
19#88626C2212yescomplemented
20C2×C3×D3×D4—62no—
21C23×H(3,𝔽3)—21no—
22C22×#272—21no—
23C22×#272—22no—
24C22×#272—22no—
25C22×#272—42no—
26C22×#272—42no—
27C22×#272D411yescharacteristic
28C2×#1088—22no—
29C2×#1088—42no—
30D4×H(3,𝔽3)—21no—
31D4×H(3,𝔽3)C2×C412yescomplemented
32C2×C4×H(3,𝔽3)C2×C411yescharacteristic
33#6235—22no—
34#6235—22no—
35C2×#1084D411yescharacteristic
36C4⋊#272—21no—
37C4⋊#272C2×C412yescomplemented
38C4×#272C2×C411yescharacteristic
39C4×#272D411yescharacteristic
40C4×#272D411yescharacteristic
41C4×#272C2311yescharacteristic
42#6188—22no—
43C2×C3×#102—91nomaximal
44C23×C3×D3—64no—
45C2×C32×D4—22no—
46C2×C3×#67—64no—
47C3×D3×D4—64no—
48C3×D3×D4—64no—
49C3×D3×D4—64no—
50C3×D3×D4—64no—
51C2×C3×D12—62no—
52C2×C3×C4×D3—62no—
53C22×H(3,𝔽3)—22no—
54C22×H(3,𝔽3)—42no—
55C22×H(3,𝔽3)C22⋊C411yescharacteristic
56C2×#272—22no—
57C2×#272—42no—
58C2×#272—44no—
59C2×#272—42no—
60C2×#272—42no—
61C2×#272C22⋊C411yescharacteristic
62C2×#272D812yes—
63C2×#272SD1612yes—
64C2×#272C2×D411yescharacteristic
65#1088—42no—
66C4×H(3,𝔽3)C22⋊C411yescharacteristic
67C4×H(3,𝔽3)C22×C411yescharacteristic
68#1084C22⋊C411yescharacteristic
69#1084C2×D411yescharacteristic
70C22×C3×D4—91no—
71C2×D3×D4—62no—
72C22×C3×C8—91no—
73C2×C3×C4⋊C4—91no—
74C3×#102—94no—
75C23×C32—24no—
76C22×C3×D3—62no—
77C22×C3×D3—64no—
78C22×C3×D3—68no—
79C22×C3×D3—68no—
80C22×C3×D3—128no—
81C22×C3×D3—128no—
82C32×D4—24no—
83C32×D4—24no—
84C2×C32×C4—22no—
85C3×#67—68no—
86C3×#67—68no—
87C2×C3×Dic3—62no—
88C3×D12—64no—
89C3×D12—64no—
90C3×C4×D3—62no—
91C3×C4×D3—62no—
92C3×C4×D3—62no—
93C3×C4×D3—62no—
94C2×#102—91noSylow
95C2×H(3,𝔽3)—42no—
96C2×H(3,𝔽3)#10212yescomplemented
97C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
98#272—42no—
99#272#10212yescomplemented
100#272C2×D811yescharacteristic
101#272C2×SD1611yescharacteristic
102C24×C3—181no—
103C23×D3—64no—
104C2×C3×D4—62no—
105C2×C3×D4—92no—
106C2×C3×D4—92no—
107C2×C3×D4—91no—
108C2×C3×D4—181no—
109C2×C3×D4—181no—
110C2×C3×D4—181no—
111C2×C3×D4#15711yescharacteristic
112C22×C3×C4—91no—
113C22×C3×C4—181no—
114C2×#67—64no—
115D3×D4—64no—
116D3×D4—64no—
117D3×D4—64no—
118D3×D4—64no—
119C2×D12—62no—
120C2×C4×D3—62no—
121C2×C3×C8—92no—
122C2×C3×C8—181no—
123C2×C3×C8—181no—
124C3×C4⋊C4—92no—
125C3×C4⋊C4—181no—
126C22×C32—28no—
127C22×C32—22no—
128C22×C32—48no—
129C2×C3×D3—62no—
130C2×C3×D3—64no—
131C2×C3×D3—64no—
132C2×C3×D3—62no—
133C2×C3×D3—68no—
134C2×C3×D3—1216no—
135C2×C3×D3—128no—
136C2×C3×D3—128no—
137C2×C3×D3—128no—
138C32×C4—22no—
139C32×C4—22no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C22×D4—91no—
143C22×C8—91no—
144C2×C4⋊C4—91no—
145#102—94no—
146H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
147C23×C3—21no—
148C23×C3—64no—
149C23×C3—91no—
150C23×C3—182no—
151C23×C3—182no—
152C23×C3—181no—
153C23×C3—362no—
154C23×C3—362no—
155C22×D3—62no—
156C22×D3—64no—
157C22×D3—68no—
158C22×D3—68no—
159C22×D3—128no—
160C22×D3—128no—
161C3×D4—21no—
162C3×D4—64no—
163C3×D4—64no—
164C3×D4—92no—
165C3×D4—182no—
166C3×D4—182no—
167C3×D4—181no—
168C3×D4C2×#15712yes—
169C2×C3×C4—62no—
170C2×C3×C4—91no—
171C2×C3×C4—91no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—182no—
176C2×C3×C4—362no—
177C2×C3×C4C2×#15711yescharacteristic
178#67—68no—
179#67—68no—
180C2×Dic3—62no—
181D12—64no—
182D12—64no—
183C4×D3—62no—
184C4×D3—62no—
185C4×D3—62no—
186C4×D3—62no—
187C3×C8—182no—
188C2×C32—22no—
189C2×C32—24no—
190C2×C32—48no—
191C3×D3—64no—
192C3×D3—62no—
193C3×D3—62no—
194C3×D3—128no—
195C24—181no—
196C2×D4—92no—
197C2×D4—92no—
198C2×D4—91no—
199C2×D4—181no—
200C2×D4—181no—
201C2×D4—181no—
202C2×D4#108811yescharacteristic, complemented
203C22×C4—91no—
204C22×C4—181no—
205C2×C8—92no—
206C2×C8—181no—
207C2×C8—181no—
208C4⋊C4—92no—
209C4⋊C4—181no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—68no—
213C22×C3—62no—
214C22×C3—91no—
215C22×C3—91no—
216C22×C3—92no—
217C22×C3—92no—
218C22×C3—128no—
219C22×C3—182no—
220C22×C3—362no—
221C22×C3—362no—
222C22×C3—364no—
223C22×C3—362no—
224C22×C3C22⋊#15711yescharacteristic
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—64no—
228C2×D3—64no—
229C2×D3—68no—
230C2×D3—128no—
231C2×D3—1216no—
232C2×D3—128no—
233C2×D3—128no—
234C3×C4—62no—
235C3×C4—62no—
236C3×C4—91no—
237C3×C4—91no—
238C3×C4—362no—
239C3×C4C22⋊#15711yescharacteristic
240C3×C4C22×#15711yescharacteristic
241Dic3—62no—
242Dic3—62no—
243C32—22no—
244C23—21no—
245C23—91no—
246C23—182no—
247C23—182no—
248C23—181no—
249C23—362no—
250C23—362no—
251D4—21no—
252D4—92no—
253D4—182no—
254D4—182no—
255D4—181no—
256D4C2×#108812yescomplemented
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—182no—
263C2×C4—362no—
264C2×C4C2×#108811yescharacteristic
265C8—182no—
266C2×C3—42no—
267C2×C3—64no—
268C2×C3—62no—
269C2×C3—91no—
270C2×C3—91no—
271C2×C3—92no—
272C2×C3—128no—
273C2×C3—362no—
274C2×C3#6378212yes—
275C2×C3C2×C22⋊#15711yescharacteristic
276D3—62no—
277D3—64no—
278D3—62no—
279D3—128no—
280C22—22no—
281C22—42no—
282C22—92no—
283C22—92no—
284C22—91no—
285C22—91no—
286C22—182no—
287C22—362no—
288C22—364no—
289C22—362no—
290C22—362no—
291C22C22⋊#108811yescharacteristic
292C4—91no—
293C4—91no—
294C4—362no—
295C4C22⋊#108811yescharacteristic
296C4C22×#108811yescharacteristic
297C3—62no—
298C3C2×#6378211yescharacteristic
299C2—42no—
300C2—91no—
301C2—92no—
302C2—91no—
303C2—362no—
304C2#1172078812yesdirect product factor
305C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×D4×#272C2
cosocleC4×#272C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)C2×#102
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#1088C2semidirect
3. C2×#88626C2semidirect
4. #11720788C2direct (N × H)
7. C2×D4×H(3,𝔽3)C4semidirect
11. D4×#272C4semidirect
13. C2×C4⋊#272C4semidirect
17. C4⋊#1088C22semidirect
19. #88626C22semidirect
31. D4×H(3,𝔽3)C2×C4semidirect
37. C4⋊#272C2×C4semidirect
96. C2×H(3,𝔽3)#102semidirect
99. #272#102semidirect
146. H(3,𝔽3)C2×#102semidirect
202. C2×D4#1088semidirect
256. D4C2×#1088semidirect
304. C2#11720788direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×C4⋊#10882
C2×#886262
#117207882
C2×C3×#1029