Finite Group Databasebeta

← #13137396

Subgroups — #13137396

Order 1728.

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

normal subgroups only

This group has 528 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#11721385C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#88432C212yesmaximal, complemented
4C4⋊#6180C211yesmaximal, characteristic, complemented
5#11720318C212yesmaximal, complemented
6#10590355—31nomaximal
7#10590349—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)—21no—
9C2×C4⋊((C3⋊D3)⋊C3)—21no—
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2212yes—
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#88432—24no—
18#64083D311yescharacteristic, complemented
19C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
20C3×#874—31no—
21C3×C4×D12—31no—
22C2×#3714—32no—
23C2×#3714—32no—
24C3×C4⋊(C3⋊C8)—31no—
25C3×C4⋊(C3⋊C8)—31no—
26C3×C4⋊(C3⋊C8)—61no—
27#63564—32no—
28#63563—32no—
29C22×(C3⋊D3)⋊C3—21no—
30C22×(C3⋊D3)⋊C3—22no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3—21no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36C4⋊((C3⋊D3)⋊C3)—21no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)—42no—
39C4⋊((C3⋊D3)⋊C3)—42no—
40C4⋊((C3⋊D3)⋊C3)D412yescomplemented
41C4×(C3⋊D3)⋊C3—42no—
42#6180—22no—
43#5072—91no—
44#5015—91no—
45#4843—91no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)C2×D312yes—
49C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
50C2×C4×C3⋊D3—21no—
51C2×C3×D12—31no—
52C2×C3×D12—32no—
53C2×C3×D12—61no—
54C2×C3×D12—61no—
55C2×C3×C4×D3—61no—
56C32×C42—21no—
57C32×C42C2×D311yescharacteristic
58C32×C42C2×D311yescharacteristic
59#3752—21no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C3×(C2×D3)⋊C4—61no—
62C3×C4⋊Dic3—31no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—62no—
66#3714—64no—
67#3714—64no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C2×(C3⋊D3)⋊C3—22no—
70C2×(C3⋊D3)⋊C3—41no—
71C2×(C3⋊D3)⋊C3—41no—
72C2×(C3⋊D3)⋊C3—42no—
73C2×(C3⋊D3)⋊C3—42no—
74C2×(C3⋊D3)⋊C3—42no—
75C4×H(3,𝔽3)—21no—
76C4×H(3,𝔽3)D812yescomplemented
77C4×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)#4111yescharacteristic
79(C3⋊Dic3)⋊C3—41no—
80C3×#127—91no—
81C3×C4×D4—91no—
82C2×#213—92no—
83C2×#213—92no—
84C2×D24—92no—
85#874—31no—
86#874—91no—
87C4×D12—31no—
88C4×D12—91no—
89C3×C4⋊C8—91no—
90#832—92no—
91#821—92no—
92#809—92no—
93C4⋊(C3⋊C8)—31no—
94C4⋊(C3⋊C8)—31no—
95C4⋊(C3⋊C8)—61no—
96C22×C3⋊D3—21no—
97C22×C3⋊D3—22no—
98C22×C3×D3—61no—
99C22×C3×D3—62no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4#6711yescharacteristic
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4#6711yescharacteristic
106C2×C32×C4#6711yescharacteristic
107C2×C32×C4C22×D311yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C4⋊(C3⋊D3)—21no—
111C4⋊(C3⋊D3)—22no—
112C4⋊(C3⋊D3)—22no—
113C4⋊(C3⋊D3)—42no—
114C4⋊(C3⋊D3)—42no—
115C4⋊(C3⋊D3)D1212yescomplemented
116C4×C3⋊D3—42no—
117C2×C3×Dic3—61no—
118C3×D12—32no—
119C3×D12—62no—
120C3×D12—62no—
121C3×D12—61no—
122C3×D12—122no—
123C3×D12—122no—
124C3×C4×D3—122no—
125C3×C3⋊C8—62no—
126C3×C3⋊C8—62no—
127C3×C3⋊C8—122no—
128#459—271noSylow
129C2×H(3,𝔽3)#12111yescharacteristic
130C2×H(3,𝔽3)C2×D811yescharacteristic
131C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
132(C3⋊D3)⋊C3—42no—
133(C3⋊D3)⋊C3—82no—
134C2×C3×D4—91no—
135C2×C3×D4—92no—
136C2×C3×D4—181no—
137C2×C3×D4—181no—
138C22×C3×C4—181no—
139C2×D12—31no—
140C2×D12—32no—
141C2×D12—61no—
142C2×D12—61no—
143C2×D12—91no—
144C2×D12—92no—
145C2×D12—181no—
146C2×D12—181no—
147C2×C4×D3—61no—
148C2×C4×D3—181no—
149C2×C3×C8—92no—
150C3×C4⋊C4—91no—
151C3×C22⋊C4—181no—
152C3×C42—31no—
153C3×C42—31no—
154C3×C42—61no—
155C3×C42D3211yescharacteristic
156#213—184no—
157#213—184no—
158(C2×D3)⋊C4—61no—
159(C2×D3)⋊C4—181no—
160C4⋊Dic3—31no—
161C4⋊Dic3—91no—
162C2×C3⋊C8—32no—
163C2×C3⋊C8—32no—
164C2×C3⋊C8—62no—
165D24—184no—
166C22×C32—21no—
167C22×C32C2×#6711yescharacteristic
168C22×C32C2×#6711yescharacteristic
169C2×C3⋊D3—22no—
170C2×C3⋊D3—41no—
171C2×C3⋊D3—41no—
172C2×C3⋊D3—42no—
173C2×C3⋊D3—42no—
174C2×C3⋊D3—42no—
175C2×C3×D3—62no—
176C2×C3×D3—121no—
177C2×C3×D3—121no—
178C2×C3×D3—122no—
179C2×C3×D3—122no—
180C2×C3×D3—122no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4#21312yescomplemented
188C32×C4#21312yescomplemented
189C32×C4C2×D1211yescharacteristic
190C32×C4#23511yescharacteristic
191C32×C4D3×D411yescharacteristic
192C32×C4#23711yescharacteristic
193C3⋊Dic3—41no—
194C3×Dic3—121no—
195C2×D8—272no—
196#127—91no—
197C4×D4—91no—
198C4⋊C8—91no—
199#102—272no—
200H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
201C23×C3—181no—
202C23×C3—182no—
203C22×D3—61no—
204C22×D3—62no—
205C22×D3—181no—
206C22×D3—182no—
207C3×D4—92no—
208C3×D4—181no—
209C3×D4—182no—
210C3×D4—182no—
211C3×D4—362no—
212C3×D4—362no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—181no—
223C2×C3×C4—362no—
224C2×C3×C4#62511yescharacteristic
225C2×C3×C4#62511yescharacteristic
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—61no—
228C2×Dic3—181no—
229D12—32no—
230D12—61no—
231D12—62no—
232D12—62no—
233D12—92no—
234D12—122no—
235D12—122no—
236D12—181no—
237D12—182no—
238D12—182no—
239D12—362no—
240D12—362no—
241C4×D3—122no—
242C4×D3—362no—
243C3×C8—182no—
244C3⋊C8—62no—
245C3⋊C8—62no—
246C3⋊C8—122no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32#93011yescharacteristic
251C2×C32C2×#21311yescharacteristic
252C2×C32C2×#21311yescharacteristic
253C2×C32#93211yescharacteristic
254C2×C32#93811yescharacteristic
255C2×C32#94911yescharacteristic
256C3⋊D3—42no—
257C3⋊D3—82no—
258C3×D3—122no—
259C3×D3—242no—
260C2×D4—91no—
261C2×D4—92no—
262C2×D4—181no—
263C2×D4—181no—
264C22×C4—181no—
265D8—544no—
266C2×C8—92no—
267C4⋊C4—91no—
268C22⋊C4—181no—
269C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—61no—
273C22×C3—182no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3—362no—
277C22×C3—362no—
278C22×C3—362no—
279C22×C3C2×#62511yescharacteristic
280C2×D3—62no—
281C2×D3—121no—
282C2×D3—121no—
283C2×D3—122no—
284C2×D3—122no—
285C2×D3—122no—
286C2×D3—182no—
287C2×D3—361no—
288C2×D3—361no—
289C2×D3—362no—
290C2×D3—362no—
291C2×D3—362no—
292C3×C4—21no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—32no—
298C3×C4—32no—
299C3×C4—61no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—61no—
303C3×C4—62no—
304C3×C4—121no—
305C3×C4—361no—
306C3×C4#371312yes—
307C3×C4#379511yescharacteristic
308C3×C4D3×D1211yescharacteristic
309Dic3—121no—
310Dic3—361no—
311C32—21no—
312C32#501511yescharacteristic, complemented
313C32#507211yescharacteristic, complemented
314C23—181no—
315C23—182no—
316D4—92no—
317D4—181no—
318D4—182no—
319D4—182no—
320D4—362no—
321D4—362no—
322C2×C4—181no—
323C2×C4—362no—
324C2×C4#619811yescharacteristic
325C2×C4#619811yescharacteristic
326C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
327C8—182no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—61no—
337C2×C3—362no—
338C2×C3—722no—
339C2×C3C2×#371311yescharacteristic
340C2×C3#6382311yescharacteristic
341C2×C3#6390811yescharacteristic
342D3—122no—
343D3—242no—
344D3—362no—
345D3—722no—
346C22—182no—
347C22—361no—
348C22—361no—
349C22—362no—
350C22—362no—
351C22—362no—
352C22C2×#619811yescharacteristic
353C4—21no—
354C4—361no—
355C4#8843212yescomplemented
356C4#8864811yescharacteristic
357C4#8865411yescharacteristic
358C3—31no—
359C3—31no—
360C3—61no—
361C3#1059034711yescharacteristic
362C2—362no—
363C2—722no—
364C2C2×#8843211yescharacteristic
365C2#1172106111yescharacteristic
366C2#1172114111yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#459
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
1. #11721385C2semidirect
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C2×#88432C2semidirect
4. C4⋊#6180C2semidirect
5. #11720318C2semidirect
18. #64083D3semidirect
19. C4×C4⋊(C3⋊D3)D3semidirect
40. C4⋊((C3⋊D3)⋊C3)D4semidirect
76. C4×H(3,𝔽3)D8semidirect
115. C4⋊(C3⋊D3)D12semidirect
187. C32×C4#213semidirect
188. C32×C4#213semidirect
200. H(3,𝔽3)#459semidirect
312. C32#5015semidirect
313. C32#5072semidirect
355. C4#88432semidirect

Maximal subgroups

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

subgroupindex
#117213852
C4×C4⋊((C3⋊D3)⋊C3)2
C2×#884322
C4⋊#61802
#117203182
#105903553
#105903493