Finite Group Databasebeta

← #13137426

Subgroups — #13137426

Order 1728.

556 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#11721720C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×#88437C212yesmaximal, complemented
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
5#11720324C212yesmaximal, complemented
6#10590377—31nomaximal
7#10590364—31nomaximal
8C2×C4⋊#272—21no—
9C2×C4⋊#272—21no—
10C2×C4⋊#272C2212yes—
11C2×#634⋊C3C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14#3750⋊C3—21no—
15C4×(C3⋊Dic3)⋊C3—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#88437—24no—
18C4×#634D311yescharacteristic, complemented
19C3×#874—31no—
20C3×#874—31no—
21C3×#874—61no—
22C3×C4×Dic6—31no—
23C2×#3716—32no—
24C2×#3715—32no—
25C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
26C3×C4⋊(C3⋊C8)—31no—
27#63563—32no—
28#63562—32no—
29C22×#272—22no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C4⋊#272—22no—
34C4⋊#272—22no—
35C4⋊#272—42no—
36C4⋊#272—42no—
37C2×(C3⋊Dic3)⋊C3—21no—
38C2×(C3⋊Dic3)⋊C3—21no—
39#634⋊C3—21no—
40#634⋊C3D412yescomplemented
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5075—91no—
43#5027—91no—
44#4848—91no—
45C2×#634C2×D311yescharacteristic
46C2×C3×D12—32no—
47C2×C3×D12—32no—
48C2×C3×D12—62no—
49C2×C3×D12—62no—
50C2×C3×D12—62no—
51C2×C3×D12—61no—
52C2×C3×D12—61no—
53C2×C3×D12—61no—
54C2×C3×D12—61no—
55C2×C3×Dic6—31no—
56C32×C42—21no—
57C32×C42C2×D311yescharacteristic
58C32×C42C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—21no—
61C4×C3⋊Dic3—21no—
62C2×C3⋊(C3⋊C8)C2×D312yes—
63C3×C4⋊Dic3—31no—
64C3×Dic3⋊C4—61no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—32no—
67#3716—64no—
68#3715—64no—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C2×#272—42no—
71C2×#272—42no—
72C2×#272—42no—
73C4×H(3,𝔽3)—21no—
74C4×H(3,𝔽3)SD1612yescomplemented
75C4×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)#4111yescharacteristic
77(C3⋊Dic3)⋊C3—22no—
78(C3⋊Dic3)⋊C3—41no—
79C3×#127—91no—
80C3×C4×Q8—91no—
81C2×#215—92no—
82C2×#214—92no—
83C2×#204—92no—
84#874—31no—
85#874—31no—
86#874—61no—
87C4×Dic6—31no—
88C4×Dic6—91no—
89C3×C4⋊C8—91no—
90#832—92no—
91#821—92no—
92#809—92no—
93C4⋊(C3⋊C8)—31no—
94C4⋊(C3⋊C8)—91no—
95C22×C3×D3—64no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
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—21no—
108C2×C3⋊Dic3—21no—
109#634—21no—
110#634D1212yescomplemented
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C3×D12—62no—
114C3×D12—62no—
115C3×D12—62no—
116C3×D12—62no—
117C3×D12—122no—
118C3×D12—122no—
119C3×D12—124no—
120C3×D12—122no—
121C3×D12—122no—
122C3×D12—124no—
123C3×D12—122no—
124C3×D12—122no—
125C3×Dic6—32no—
126C3×Dic6—61no—
127C3⋊(C3⋊C8)—22no—
128C3×C3⋊C8—62no—
129#460—271noSylow
130C2×H(3,𝔽3)#12111yescharacteristic
131C2×H(3,𝔽3)C2×SD1611yescharacteristic
132C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
133#272—82no—
134C2×C3×Q8—91no—
135C2×C3×D4—92no—
136C2×C3×D4—181no—
137C2×C3×D4—181no—
138C2×D12—32no—
139C2×D12—32no—
140C2×D12—62no—
141C2×D12—61no—
142C2×D12—61no—
143C2×D12—61no—
144C2×D12—61no—
145C2×D12—62no—
146C2×D12—62no—
147C2×Dic6—31no—
148C2×Dic6—91no—
149C2×C3×C8—92no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C42—31no—
153C3×C42—31no—
154C3×C42—61no—
155C3×C42—181no—
156C3×C42D3211yescharacteristic
157#215—184no—
158#214—184no—
159C4⋊Dic3—31no—
160C4⋊Dic3—91no—
161Dic3⋊C4—61no—
162Dic3⋊C4—181no—
163C4×Dic3—61no—
164C4×Dic3—181no—
165C2×C3⋊C8—32no—
166C2×C3⋊C8—92no—
167#204—184no—
168C22×C32—21no—
169C22×C32C2×D1211yescharacteristic
170C22×C32C2×#6711yescharacteristic
171C2×C3×D3—124no—
172C2×C3×D3—124no—
173C2×C3×D3—124no—
174C2×C3×D3—122no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179C2×C3×D3—122no—
180C32×C4—22no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4#20412yescomplemented
187C32×C4#21512yescomplemented
188C32×C4C2×D1211yescharacteristic
189C32×C4#23511yescharacteristic
190C32×C4D3×D411yescharacteristic
191C32×C4#23911yescharacteristic
192C3⋊Dic3—22no—
193C3⋊Dic3—41no—
194C3×Dic3—62no—
195C3×Dic3—121no—
196C2×SD16—272no—
197#127—91no—
198C4×Q8—91no—
199C4⋊C8—91no—
200#102—272no—
201H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
202C23×C3—182no—
203C22×D3—62no—
204C22×D3—62no—
205C22×D3—64no—
206C3×Q8—92no—
207C3×Q8—181no—
208C3×D4—182no—
209C3×D4—182no—
210C3×D4—362no—
211C3×D4—362no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223C2×C3×C4#62611yescharacteristic
224C2×C3×C4#62611yescharacteristic
225C2×C3×C4C2×D3211yescharacteristic
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—181no—
229C2×Dic3—181no—
230D12—62no—
231D12—62no—
232D12—62no—
233D12—62no—
234D12—122no—
235D12—122no—
236D12—122no—
237D12—122no—
238D12—122no—
239D12—122no—
240D12—124no—
241D12—124no—
242Dic6—32no—
243Dic6—61no—
244Dic6—92no—
245Dic6—181no—
246C3×C8—182no—
247C3⋊C8—62no—
248C3⋊C8—182no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32#89511yescharacteristic
253C2×C32C2×#20411yescharacteristic
254C2×C32#90811yescharacteristic
255C2×C32#93011yescharacteristic
256C2×C32C2×#21511yescharacteristic
257C2×C32#94911yescharacteristic
258C3×D3—244no—
259C3×D3—242no—
260C3×D3—242no—
261C2×Q8—91no—
262C2×D4—92no—
263C2×D4—181no—
264C2×D4—181no—
265SD16—544no—
266C2×C8—92no—
267C4⋊C4—91no—
268C4⋊C4—181no—
269C42—181no—
270C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—61no—
274C22×C3—362no—
275C22×C3—362no—
276C22×C3—362no—
277C22×C3C2×#62611yescharacteristic
278C2×D3—122no—
279C2×D3—122no—
280C2×D3—122no—
281C2×D3—122no—
282C2×D3—122no—
283C2×D3—122no—
284C2×D3—124no—
285C2×D3—124no—
286C2×D3—124no—
287C3×C4—21no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—32no—
293C3×C4—32no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—62no—
297C3×C4—61no—
298C3×C4—61no—
299C3×C4—62no—
300C3×C4—182no—
301C3×C4—361no—
302C3×C4#371712yes—
303C3×C4#379911yescharacteristic
304C3×C4D3×D1211yescharacteristic
305Dic3—62no—
306Dic3—121no—
307Dic3—182no—
308Dic3—361no—
309C32—21no—
310C32#484811yescharacteristic, complemented
311C32#502711yescharacteristic, complemented
312C23—182no—
313Q8—92no—
314Q8—181no—
315D4—182no—
316D4—182no—
317D4—362no—
318D4—362no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4#620011yescharacteristic
322C2×C4#620011yescharacteristic
323C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C8—182no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—61no—
332C2×C3—61no—
333C2×C3—61no—
334C2×C3—722no—
335C2×C3#6382511yescharacteristic
336C2×C3C2×#371711yescharacteristic
337C2×C3#6390911yescharacteristic
338D3—242no—
339D3—242no—
340D3—244no—
341C22—362no—
342C22—362no—
343C22—362no—
344C22C2×#620011yescharacteristic
345C4—21no—
346C4—182no—
347C4—361no—
348C4#8843712yescomplemented
349C4#8865211yescharacteristic
350C4#8865411yescharacteristic
351C3—31no—
352C3—31no—
353C3—61no—
354C3#1059036711yescharacteristic
355C2—722no—
356C2#1172106811yescharacteristic
357C2C2×#8843711yescharacteristic
358C2#1172115611yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#11721720C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
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. C4×#634⋊C3C2semidirect
3. C2×#88437C2semidirect
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. #11720324C2semidirect
18. C4×#634D3semidirect
25. C4⋊(C3⋊(C3⋊C8))D3semidirect
40. #634⋊C3D4semidirect
74. C4×H(3,𝔽3)SD16semidirect
110. #634D12semidirect
186. C32×C4#204semidirect
187. C32×C4#215semidirect
201. H(3,𝔽3)#460semidirect
310. C32#4848semidirect
311. C32#5027semidirect
348. C4#88437semidirect

Maximal subgroups

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

subgroupindex
#117217202
C4×#634⋊C32
C2×#884372
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203242
#105903773
#105903643