Finite Group Databasebeta

← #13147344

Subgroups — #13147344

Order 1728.

415 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 63 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×C4⋊#1088C211yesmaximal, characteristic
2C2×C4×#1088C211yesmaximal, characteristic
3C4⋊C4×#272C211yesmaximal, characteristic
4#11720787C211yesmaximal, characteristic
5#11720787C211yesmaximal, characteristic
6#11720787C211yesmaximal, characteristic
7#11720787C211yesmaximal, characteristic
8C22×#1088C2211yescharacteristic
9C22×#1088C2211yescharacteristic
10C22×#1088C2211yescharacteristic
11C22×#1088C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C4⋊#1088—22no—
16C4×#1088—22no—
17C4⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
18C4⋊#1084C411yescharacteristic, complemented
19#88540C411yescharacteristic, complemented
20#88540C411yescharacteristic, complemented
21C3×D3×C4⋊C4—62no—
22C22×#272C2311yescharacteristic
23C2×#1088—21no—
24C2×#1088—21no—
25C2×#1088—21no—
26C2×#1088—21no—
27C2×#1088—21no—
28C2×#1088—21no—
29C2×#1088—21no—
30C2×#1088—21no—
31C2×#1088—21no—
32C2×#1088—21no—
33C2×#1088D412yes—
34C2×#1088Q812yes—
35C2×C4×H(3,𝔽3)C2×C411yescharacteristic
36C2×C4×H(3,𝔽3)C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2×C411yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C2×#1084C2×C411yescharacteristic
40C2×#1084C2×C411yescharacteristic
41C4×#272—21no—
42C4×#272—21no—
43C4×#272—21no—
44C4×#272—21no—
45C4×#272—21no—
46C4×#272—21no—
47C3×#387—91nomaximal
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—62no—
50C2×C3×C4×D3—62no—
51C32×C4⋊C4—22no—
52C3×C4⋊Dic3—62no—
53C3×Dic3⋊C4—62no—
54C3×Dic3⋊C4—62no—
55C22×H(3,𝔽3)C22×C411yescharacteristic
56C2×#272C2×D411yescharacteristic
57C2×#272C2×Q811yescharacteristic
58C2×#272#4111yescharacteristic
59C2×#272#4111yescharacteristic
60C2×#272#4111yescharacteristic
61C2×#272#4111yescharacteristic
62#1088—22no—
63#1088—22no—
64#1088—41no—
65#1088—41no—
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—21no—
69#1084—21no—
70#1084—21no—
71#1084—21no—
72C2×C3×C4⋊C4—91no—
73C2×C3×C4⋊C4—91no—
74C2×C3×C42—91no—
75D3×C4⋊C4—62no—
76C3×#95—91no—
77C3×#95—91no—
78C3×#95—91no—
79C3×#95—91no—
80C22×C3×D3—62no—
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C3×C4×D3—64no—
88C3×C4×D3—64no—
89C3×C4×D3—122no—
90C3×C4×D3—122no—
91C3×C4×D3—122no—
92C3×C4×D3—122no—
93#387—91noSylow
94C2×H(3,𝔽3)#11711yescharacteristic
95C2×H(3,𝔽3)C4×D411yescharacteristic
96C2×H(3,𝔽3)C4×Q811yescharacteristic
97#272C22⋊Q811yescharacteristic
98#272#12311yescharacteristic
99#272#12511yescharacteristic
100#272#12611yescharacteristic
101C22×C3×C4—91no—
102C22×C3×C4—91no—
103C22×C3×C4—91no—
104C22×C3×C4—91no—
105C22×C3×C4—91no—
106C22×C3×C4—91no—
107C22×C3×C4—91no—
108C2×C4×D3—62no—
109C2×C4×D3—62no—
110C2×C4×D3—62no—
111C3×C4⋊C4—62no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4—91no—
114C3×C4⋊C4—91no—
115C3×C4⋊C4—182no—
116C3×C4⋊C4#15711yescharacteristic
117C3×C42—182no—
118C4⋊Dic3—62no—
119Dic3⋊C4—62no—
120Dic3⋊C4—62no—
121C22×C32—22no—
122C2×C3×D3—62no—
123C2×C3×D3—62no—
124C2×C3×D3—62no—
125C2×C3×D3—62no—
126C2×C3×D3—62no—
127C2×C3×D3—62no—
128C32×C4—24no—
129C32×C4—42no—
130C32×C4—42no—
131C3×Dic3—64no—
132C3×Dic3—122no—
133C3×Dic3—122no—
134C2×C4⋊C4—91no—
135C2×C4⋊C4—91no—
136C2×C42—91no—
137#95—91no—
138#95—91no—
139#95—91no—
140#95—91no—
141H(3,𝔽3)#38711yescharacteristic, complemented, Sylow
142C23×C3—91no—
143C22×D3—62no—
144C2×C3×C4—62no—
145C2×C3×C4—62no—
146C2×C3×C4—62no—
147C2×C3×C4—92no—
148C2×C3×C4—92no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—181no—
153C2×C3×C4—181no—
154C2×C3×C4—181no—
155C2×C3×C4—181no—
156C2×C3×C4—181no—
157C2×C3×C4—181no—
158C2×C3×C4—181no—
159C2×C3×C4—181no—
160C2×C3×C4—181no—
161C2×C3×C4—181no—
162C2×C3×C4—181no—
163C2×C3×C4—181no—
164C2×C3×C4—181no—
165C2×C3×C4—181no—
166C2×C3×C4—181no—
167C2×C3×C4—181no—
168C2×C3×C4C2×#15711yescharacteristic
169C2×C3×C4C2×#15711yescharacteristic
170C2×C3×C4C2×#15711yescharacteristic
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C4×D3—64no—
175C4×D3—64no—
176C4×D3—122no—
177C4×D3—122no—
178C4×D3—122no—
179C4×D3—122no—
180C2×C32—22no—
181C2×C32—22no—
182C2×C32—22no—
183C3×D3—62no—
184C3×D3—62no—
185C3×D3—62no—
186C3×D3—62no—
187C22×C4—91no—
188C22×C4—91no—
189C22×C4—91no—
190C22×C4—91no—
191C22×C4—91no—
192C22×C4—91no—
193C22×C4—91no—
194C4⋊C4—91no—
195C4⋊C4—91no—
196C4⋊C4—91no—
197C4⋊C4—182no—
198C4⋊C4#108811yescharacteristic, complemented
199C42—182no—
200C22×C3—62no—
201C22×C3—91no—
202C22×C3—91no—
203C22×C3—91no—
204C22×C3—91no—
205C22×C3—91no—
206C22×C3—91no—
207C22×C3C22×#15711yescharacteristic
208C2×D3—62no—
209C2×D3—62no—
210C2×D3—62no—
211C2×D3—62no—
212C2×D3—62no—
213C2×D3—62no—
214C3×C4—21no—
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—64no—
218C3×C4—122no—
219C3×C4—122no—
220C3×C4—182no—
221C3×C4—182no—
222C3×C4—181no—
223C3×C4—181no—
224C3×C4—181no—
225C3×C4—361no—
226C3×C4—361no—
227Dic3—64no—
228Dic3—122no—
229Dic3—122no—
230C32—22no—
231C23—91no—
232C2×C4—92no—
233C2×C4—92no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—91no—
237C2×C4—181no—
238C2×C4—181no—
239C2×C4—181no—
240C2×C4—181no—
241C2×C4—181no—
242C2×C4—181no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4—181no—
246C2×C4—181no—
247C2×C4—181no—
248C2×C4—181no—
249C2×C4—181no—
250C2×C4—181no—
251C2×C4—181no—
252C2×C4—181no—
253C2×C4C2×#108811yescharacteristic
254C2×C4C2×#108811yescharacteristic
255C2×C4C2×#108811yescharacteristic
256C2×C3—62no—
257C2×C3—62no—
258C2×C3—62no—
259C2×C3—91no—
260C2×C3—91no—
261C2×C3—91no—
262C2×C3—91no—
263C2×C3#6428611yescharacteristic
264C2×C3D4×#15711yescharacteristic
265C2×C3Q8×#15711yescharacteristic
266D3—62no—
267D3—62no—
268D3—62no—
269D3—62no—
270C22—91no—
271C22—91no—
272C22—91no—
273C22—91no—
274C22—91no—
275C22—91no—
276C22C22×#108811yescharacteristic
277C4—21no—
278C4—21no—
279C4—21no—
280C4—182no—
281C4—182no—
282C4—181no—
283C4—181no—
284C4—181no—
285C4—361no—
286C4—361no—
287C3—62no—
288C3#1059375811yescharacteristic
289C2—91no—
290C2—91no—
291C2—91no—
292C2—91no—
293C2#1172244811yescharacteristic
294C2D4×#108811yescharacteristic
295C2Q8×#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC4⋊C4×H(3,𝔽3)C4
prime radical, cosocleC22×#272C23
hypercenterC3×C4⋊C4#157
hypocenterH(3,𝔽3)#387
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
17. C4⋊C4×H(3,𝔽3)C4semidirect
18. C4⋊#1084C4semidirect
19. #88540C4semidirect
20. #88540C4semidirect
141. H(3,𝔽3)#387semidirect
198. C4⋊C4#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
C2×C4×#10882
C4⋊C4×#2722
#117207872
C3×#3879