Finite Group Databasebeta

← #13137425

Subgroups — #13137425

Order 1728.

514 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 87 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
2#64078⋊C3C211yesmaximal, characteristic, complemented
3C2×#88437C212yesmaximal
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
5#11720309C212yesmaximal, complemented
6#10590365—31nomaximal
7#10590363—31nomaximal
8C2×C4⋊#272C2211yescharacteristic
9C2×C4×#272—21no—
10C2×#634⋊C3C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12#88542—21no—
13C4⋊#1084C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#88437—24no—
18#64078D311yescharacteristic, complemented
19C3×C4×D12—31no—
20C3×C4×D12—31no—
21C3×C4×D12—61no—
22C3×#869—31no—
23C2×#3716—32no—
24C2×#3715—32no—
25C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
26C3×C4⋊(C3⋊C8)—31no—
27#63567—32no—
28#63565—32no—
29C22×#272—21no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C4⋊#272—21no—
35C4⋊#272D412yes—
36C4×#272—42no—
37C2×(C3⋊Dic3)⋊C3—22no—
38#634⋊C3—22no—
39#634⋊C3—22no—
40(C3⋊(C3⋊C8))⋊C3—22no—
41#5083—91no—
42#5019—91no—
43#4841—91no—
44C2×#634C2×D312yes—
45C2×C3×D12—31no—
46C2×C3×D12—31no—
47C2×C3×D12—61no—
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—61no—
50C2×C3×C4×D3—61no—
51C2×C3×Dic6—32no—
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C32×C42C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)—21no—
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×(C2×D3)⋊C4—62no—
59C3×(C2×D3)⋊C4—61no—
60C3×(C2×D3)⋊C4—61no—
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3—61no—
65C3×C4⋊Dic3—61no—
66C2×C3×C3⋊C8—32no—
67#3716—64no—
68#3715—64no—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C2×#272—22no—
71C2×#272—41no—
72C2×#272—41no—
73C4×H(3,𝔽3)—21no—
74C4×H(3,𝔽3)SD1612yescomplemented
75C4×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)#4111yescharacteristic
77#1084—41no—
78(C3⋊Dic3)⋊C3—42no—
79C3×C4⋊Q8—91no—
80C3×C4×D4—91no—
81C2×#215—92no—
82C2×#214—92no—
83C2×#204—92no—
84C4×D12—31no—
85C4×D12—31no—
86C4×D12—61no—
87#869—31no—
88#869—91no—
89C3×C4⋊C8—91no—
90#835—92no—
91#816—92no—
92#810—92no—
93C4⋊(C3⋊C8)—31no—
94C4⋊(C3⋊C8)—91no—
95C22×C3×D3—62no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4D1211yescharacteristic
102C2×C32×C4D1211yescharacteristic
103C2×C32×C4#6711yescharacteristic
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—22no—
108#634—22no—
109#634—22no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C3×D12—32no—
115C3×D12—32no—
116C3×D12—62no—
117C3×D12—62no—
118C3×D12—61no—
119C3×D12—61no—
120C3×C4×D3—124no—
121C3×C4×D3—122no—
122C3×C4×D3—122no—
123C3×Dic6—62no—
124C3×Dic6—62no—
125C3⋊(C3⋊C8)—22no—
126C3×C3⋊C8—62no—
127#461—271noSylow
128C2×H(3,𝔽3)#12111yescharacteristic
129C2×H(3,𝔽3)C2×SD1611yescharacteristic
130C2×H(3,𝔽3)#13711yescharacteristic
131#272—42no—
132C2×C3×Q8—92no—
133C2×C3×D4—91no—
134C22×C3×C4—181no—
135C2×D12—31no—
136C2×D12—31no—
137C2×D12—61no—
138C2×C4×D3—61no—
139C2×C4×D3—61no—
140C2×C4×D3—62no—
141C2×Dic6—32no—
142C2×Dic6—92no—
143C2×C3×C8—92no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—181no—
146C3×C4⋊C4—181no—
147C3×C22⋊C4—181no—
148C3×C42—31no—
149C3×C42—31no—
150C3×C42—61no—
151C3×C42D3211yescharacteristic
152#215—184no—
153#214—184no—
154(C2×D3)⋊C4—61no—
155(C2×D3)⋊C4—61no—
156(C2×D3)⋊C4—62no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—61no—
160C4⋊Dic3—61no—
161C4⋊Dic3—61no—
162C4⋊Dic3—181no—
163C4⋊Dic3—181no—
164C2×C3⋊C8—32no—
165C2×C3⋊C8—92no—
166#204—184no—
167C22×C32—21no—
168C22×C32C2×D1211yescharacteristic
169C22×C32C2×#6711yescharacteristic
170C2×C3×D3—64no—
171C2×C3×D3—62no—
172C2×C3×D3—62no—
173C2×C3×D3—122no—
174C2×C3×D3—122no—
175C2×C3×D3—121no—
176C2×C3×D3—121no—
177C2×C3×D3—121no—
178C2×C3×D3—121no—
179C32×C4—21no—
180C32×C4—21no—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4#20412yescomplemented
186C32×C4#21512yescomplemented
187C32×C4D3×D411yescharacteristic
188C32×C4D3×D411yescharacteristic
189C32×C4#23711yescharacteristic
190C32×C4#23911yescharacteristic
191C3⋊Dic3—42no—
192C3×Dic3—122no—
193C3×Dic3—122no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196C2×SD16—272no—
197C4⋊Q8—91no—
198C4×D4—91no—
199C4⋊C8—91no—
200#103—272no—
201H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
202C23×C3—181no—
203C22×D3—61no—
204C22×D3—61no—
205C22×D3—62no—
206C3×Q8—182no—
207C3×Q8—182no—
208C3×D4—92no—
209C3×D4—181no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—182no—
220C2×C3×C4—181no—
221C2×C3×C4—362no—
222C2×C3×C4#62611yescharacteristic
223C2×C3×C4#62611yescharacteristic
224C2×C3×C4C2×D3211yescharacteristic
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—182no—
230D12—32no—
231D12—32no—
232D12—62no—
233D12—61no—
234D12—61no—
235D12—62no—
236C4×D3—122no—
237C4×D3—122no—
238C4×D3—124no—
239Dic6—62no—
240Dic6—62no—
241Dic6—182no—
242Dic6—182no—
243C3×C8—182no—
244C3⋊C8—62no—
245C3⋊C8—182no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32#89511yescharacteristic
250C2×C32C2×#20411yescharacteristic
251C2×C32#90911yescharacteristic
252C2×C32#93811yescharacteristic
253C2×C32C2×#21511yescharacteristic
254C2×C32#94211yescharacteristic
255C3×D3—124no—
256C3×D3—122no—
257C3×D3—122no—
258C2×Q8—92no—
259C2×D4—91no—
260C22×C4—181no—
261SD16—544no—
262C2×C8—92no—
263C4⋊C4—91no—
264C4⋊C4—181no—
265C4⋊C4—181no—
266C22⋊C4—181no—
267C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—61no—
271C22×C3—182no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3C2×#62611yescharacteristic
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—64no—
278C2×D3—121no—
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3—122no—
283C2×D3—122no—
284C3×C4—21no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—32no—
290C3×C4—32no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—61no—
294C3×C4—61no—
295C3×C4—62no—
296C3×C4—121no—
297C3×C4—362no—
298C3×C4—361no—
299C3×C4#371712yes—
300C3×C4#379611yescharacteristic
301C3×C4#380211yescharacteristic
302Dic3—121no—
303Dic3—121no—
304Dic3—122no—
305Dic3—122no—
306Dic3—362no—
307C32—21no—
308C32#484111yescharacteristic, complemented
309C32#508311yescharacteristic, complemented
310C23—181no—
311Q8—182no—
312Q8—182no—
313D4—92no—
314D4—181no—
315C2×C4—181no—
316C2×C4—182no—
317C2×C4—362no—
318C2×C4#620011yescharacteristic
319C2×C4#620011yescharacteristic
320C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C8—182no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—61no—
329C2×C3—61no—
330C2×C3—61no—
331C2×C3—362no—
332C2×C3C2×#371711yescharacteristic
333C2×C3#6383311yescharacteristic
334C2×C3#6391611yescharacteristic
335D3—122no—
336D3—122no—
337D3—124no—
338C22—182no—
339C22—361no—
340C22—361no—
341C22C2×#620011yescharacteristic
342C4—21no—
343C4—361no—
344C4—362no—
345C4#8843712yescomplemented
346C4#8864911yescharacteristic
347C4#8865511yescharacteristic
348C3—31no—
349C3—31no—
350C3—61no—
351C3#1059037911yescharacteristic
352C2—362no—
353C2#1172106911yescharacteristic
354C2C2×#8843711yescharacteristic
355C2#1172115911yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#461
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. #11720309C2semidirect
18. #64078D3semidirect
25. C4⋊(C3⋊(C3⋊C8))D3semidirect
74. C4×H(3,𝔽3)SD16semidirect
185. C32×C4#204semidirect
186. C32×C4#215semidirect
201. H(3,𝔽3)#461semidirect
308. C32#4841semidirect
309. C32#5083semidirect
345. C4#88437semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
#64078⋊C32
C2×#884372
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203092
#105903653
#105903633