Finite Group Databasebeta

← #13137412

Subgroups — #13137412

Order 1728.

521 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C4×C4⋊#272C211yesmaximal, characteristic
2C2×#88435C212yesmaximal, complemented
3C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
4C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
5#11720323C212yesmaximal, complemented
6#10590339—31nomaximal
7#10590337—31nomaximal
8C2×C4⋊#272C411yescharacteristic
9C2×C4×#272C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11#88542C412yes—
12C4⋊#1084C411yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16#88435—24no—
17C3×C4×D12—31no—
18C3×C4×D12—31no—
19C3×C4×D12—61no—
20C2×#3711—32no—
21C2×#3711—32no—
22C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
23C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
24C3×C4⋊(C3⋊C8)—31no—
25C3×C4×C3⋊C8—31no—
26(C2×D3)⋊(C3⋊C8)—32no—
27(C2×D3)⋊(C3⋊C8)—32no—
28C22×#272C2×C412yes—
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084C2×C412yes—
33C4⋊#272—22no—
34C4×#272—22no—
35C4×#272—22no—
36(C3⋊(C3⋊C8))⋊C3—22no—
37(C3⋊(C3⋊C8))⋊C3—21no—
38(C3⋊(C3⋊C8))⋊C3D412yescomplemented
39#5012—91no—
40#4838—91no—
41#4688—91no—
42C2×C3×D12—31no—
43C2×C3×D12—31no—
44C2×C3×D12—61no—
45C2×C3×C4×D3—32no—
46C2×C3×C4×D3—32no—
47C2×C3×C4×D3—62no—
48C32×C42—21no—
49C32×C42C2×D311yescharacteristic
50C32×C42C2×D311yescharacteristic
51C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
52C2×C3⋊(C3⋊C8)C2×D312yes—
53C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
54C3×(C2×D3)⋊C4—32no—
55C3×(C2×D3)⋊C4—32no—
56C3×(C2×D3)⋊C4—62no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—61no—
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8—31no—
62C2×C3×C3⋊C8—32no—
63#3711—64no—
64#3711—64no—
65C22×H(3,𝔽3)C22×C411yescharacteristic
66C2×#272—22no—
67C2×#272—22no—
68C2×#272—22no—
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)M1612yescomplemented
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C4×H(3,𝔽3)#4111yescharacteristic
73#1084—22no—
74C3×C4×D4—91no—
75C2×#208—92no—
76C2×ΓL(1,𝔽25)—92no—
77C2×ΓL(1,𝔽25)—92no—
78C4×D12—31no—
79C4×D12—31no—
80C4×D12—61no—
81C3×C4⋊C8—91no—
82C3×C4×C8—91no—
83#830—92no—
84(C2×D3)⋊C8—92no—
85(C2×D3)⋊C8—92no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—91no—
88C4×C3⋊C8—31no—
89C4×C3⋊C8—91no—
90C22×C3×D3—32no—
91C22×C3×D3—32no—
92C22×C3×D3—62no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4C4×D311yescharacteristic
97C2×C32×C4C4×D311yescharacteristic
98C2×C32×C4C4×D311yescharacteristic
99C2×C32×C4C4×D311yescharacteristic
100C2×C32×C4C22×D311yescharacteristic
101C2×C32×C4C22×D311yescharacteristic
102C2×C3×Dic3—32no—
103C2×C3×Dic3—32no—
104C2×C3×Dic3—62no—
105C3×D12—64no—
106C3×D12—62no—
107C3×D12—62no—
108C3×C4×D3—62no—
109C3×C4×D3—62no—
110C3×C4×D3—62no—
111C3×C4×D3—62no—
112C3×C4×D3—122no—
113C3×C4×D3—122no—
114C3⋊(C3⋊C8)—21no—
115C3⋊(C3⋊C8)—22no—
116C3⋊(C3⋊C8)D1212yescomplemented
117C3×C3⋊C8—32no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8—61no—
120#436—271noSylow
121C2×H(3,𝔽3)C4×D411yescharacteristic
122C2×H(3,𝔽3)C2×M1611yescharacteristic
123C2×H(3,𝔽3)#13111yescharacteristic
124#272—42no—
125C2×C3×D4—91no—
126C22×C3×C4—92no—
127C2×D12—31no—
128C2×D12—31no—
129C2×D12—61no—
130C2×C4×D3—32no—
131C2×C4×D3—32no—
132C2×C4×D3—62no—
133C2×C3×C8—91no—
134C2×C3×C8—91no—
135C2×C3×C8—92no—
136C3×C4⋊C4—91no—
137C3×C22⋊C4—92no—
138C3×C42—31no—
139C3×C42—31no—
140C3×C42—61no—
141C3×C42D3211yescharacteristic
142(C2×D3)⋊C4—32no—
143(C2×D3)⋊C4—32no—
144(C2×D3)⋊C4—62no—
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4⋊Dic3—61no—
148#208—184no—
149C2×C3⋊C8—31no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—92no—
155ΓL(1,𝔽25)—184no—
156ΓL(1,𝔽25)—184no—
157C22×C32—21no—
158C22×C32C2×C4×D311yescharacteristic
159C22×C32C2×C4×D311yescharacteristic
160C2×C3×D3—64no—
161C2×C3×D3—62no—
162C2×C3×D3—62no—
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C2×C3×D3—122no—
168C2×C3×D3—122no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—21no—
173C32×C4—21no—
174C32×C4—41no—
175C32×C4ΓL(1,𝔽25)12yescomplemented
176C32×C4ΓL(1,𝔽25)12yescomplemented
177C32×C4C2×D1211yescharacteristic
178C32×C4#23511yescharacteristic
179C32×C4D3×D411yescharacteristic
180C32×C4#23911yescharacteristic
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—122no—
184C2×M16—272no—
185C4×D4—91no—
186C4⋊C8—91no—
187C22⋊C8—272no—
188C4×C8—91no—
189H(3,𝔽3)#43611yescharacteristic, complemented, Sylow
190C23×C3—92no—
191C22×D3—32no—
192C22×D3—32no—
193C22×D3—62no—
194C3×D4—182no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—92no—
205C2×C3×C4—182no—
206C2×C3×C4—182no—
207C2×C3×C4#62411yescharacteristic
208C2×C3×C4#62411yescharacteristic
209C2×C3×C4C2×D3211yescharacteristic
210C2×Dic3—32no—
211C2×Dic3—32no—
212C2×Dic3—62no—
213D12—62no—
214D12—62no—
215D12—64no—
216C4×D3—62no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—122no—
221C4×D3—122no—
222C3×C8—92no—
223C3×C8—182no—
224C3×C8—181no—
225C3⋊C8—32no—
226C3⋊C8—61no—
227C3⋊C8—62no—
228C3⋊C8—92no—
229C3⋊C8—182no—
230C3⋊C8—181no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32C4×D1211yescharacteristic
235C2×C32#89311yescharacteristic
236C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
237C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
238C2×C32#90111yescharacteristic
239C2×C32#90711yescharacteristic
240C3×D3—124no—
241C3×D3—122no—
242C3×D3—122no—
243C2×D4—91no—
244C22×C4—92no—
245M16—544no—
246C2×C8—91no—
247C2×C8—91no—
248C2×C8—92no—
249C4⋊C4—91no—
250C22⋊C4—92no—
251C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C22×C3—31no—
253C22×C3—31no—
254C22×C3—61no—
255C22×C3—182no—
256C22×C3—182no—
257C22×C3—182no—
258C22×C3C2×#62411yescharacteristic
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—62no—
264C2×D3—62no—
265C2×D3—64no—
266C2×D3—122no—
267C2×D3—122no—
268C3×C4—21no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—32no—
274C3×C4—32no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—61no—
279C3×C4—62no—
280C3×C4—121no—
281C3×C4—182no—
282C3×C4#371212yes—
283C3×C4#379911yescharacteristic
284C3×C4D3×D1211yescharacteristic
285Dic3—62no—
286Dic3—62no—
287Dic3—122no—
288C32—21no—
289C32#468811yescharacteristic, complemented
290C32#483811yescharacteristic, complemented
291C23—92no—
292D4—182no—
293C2×C4—92no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4#619911yescharacteristic
297C2×C4#619911yescharacteristic
298C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C8—92no—
300C8—181no—
301C8—182no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—61no—
309C2×C3—61no—
310C2×C3—61no—
311C2×C3—362no—
312C2×C3#6381711yescharacteristic
313C2×C3C2×#371211yescharacteristic
314C2×C3#6389411yescharacteristic
315D3—122no—
316D3—122no—
317D3—124no—
318C22—182no—
319C22—182no—
320C22—182no—
321C22C2×#619911yescharacteristic
322C4—21no—
323C4—182no—
324C4#8843512yescomplemented
325C4#8865211yescharacteristic
326C4#8865411yescharacteristic
327C3—31no—
328C3—31no—
329C3—61no—
330C3#1059033811yescharacteristic
331C2—362no—
332C2#1172106411yescharacteristic
333C2C2×#8843511yescharacteristic
334C2#1172114911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC2×C4⋊#272C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#436
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×#88435C2semidirect
3. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
4. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
5. #11720323C2semidirect
22. C4⋊(C3⋊(C3⋊C8))D3semidirect
23. C4×C3⋊(C3⋊C8)D3semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
70. C4×H(3,𝔽3)M16semidirect
116. C3⋊(C3⋊C8)D12semidirect
175. C32×C4ΓL(1,𝔽25)semidirect
176. C32×C4ΓL(1,𝔽25)semidirect
189. H(3,𝔽3)#436semidirect
289. C32#4688semidirect
290. C32#4838semidirect
324. C4#88435semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
C2×#884352
C4⋊((C3⋊(C3⋊C8))⋊C3)2
C4×(C3⋊(C3⋊C8))⋊C32
#117203232
#105903393
#105903373