Finite Group Databasebeta

← #10592542⋊C3 #13142064

Subgroups — #10592542⋊C3 #13142064

Order 1728.

528 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3745⋊C3C212yesmaximal, complemented
2#11721385C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
5#63648⋊C3C212yesmaximal, complemented
6#10592542C311yesmaximal, characteristic, complemented
7C3×#4843—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—
13C2×C8×H(3,𝔽3)C2212yes—
14C42×H(3,𝔽3)C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3745⋊C3—24no—
18C2×#3745C2×C312yescomplemented
19#64083C2×C311yescharacteristic, complemented
20C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
21C2×C3×D24—32no—
22C3×#874—31no—
23C3×C4×D12—31no—
24C32×C4⋊C8—21no—
25C32×C4⋊C8D311yescharacteristic, complemented
26C32×C4⋊C8C2×C311yescharacteristic, complemented
27#63648C2×C312yescomplemented
28C3×#821—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—
42C8×H(3,𝔽3)—22no—
43C3×#459—91no—
44#4843—31no—
45#4843—91no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)C22×C312yes—
49C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
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—
56C2×C32×C8—22no—
57C2×C32×C8C2×D312yes—
58C2×C32×C8C22×C312yes—
59C32×C42—21no—
60C32×C42C2×D311yescharacteristic
61C32×C42C22×C311yescharacteristic
62#3752—21no—
63C4⋊(C3⋊Dic3)C22×C311yescharacteristic
64#3745—24no—
65C3×(C2×D3)⋊C4—61no—
66C3×C4⋊Dic3—31no—
67C3×D24—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—
80C2×C3×D8—92no—
81C3×#127—91no—
82C3×C4×D4—91no—
83C2×D24—32no—
84C2×D24—92no—
85#874—31no—
86#874—91no—
87C4×D12—31no—
88C4×D12—91no—
89C3×C4⋊C8—31no—
90C3×C4⋊C8—31no—
91C3×C4⋊C8—61no—
92C3×C4⋊C8C3×D311yescharacteristic
93C3×#102—92no—
94#821—32no—
95#821—92no—
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×C4D1211yescharacteristic
104C2×C32×C4D1211yescharacteristic
105C2×C32×C4C3×D411yescharacteristic
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C22×D311yescharacteristic
108C2×C32×C4C23×C311yescharacteristic
109C2×C3⋊Dic3—21no—
110C4⋊(C3⋊D3)—22no—
111C4⋊(C3⋊D3)—21no—
112C4⋊(C3⋊D3)—22no—
113C4⋊(C3⋊D3)—42no—
114C4⋊(C3⋊D3)—42no—
115C4⋊(C3⋊D3)C3×D412yescomplemented
116C4×C3⋊D3—42no—
117C2×C3×Dic3—61no—
118C3×D12—32no—
119C3×D12—62no—
120C3×D12—61no—
121C3×D12—62no—
122C3×D12—122no—
123C3×D12—122no—
124C3×C4×D3—122no—
125C32×C8—22no—
126C32×C8—22no—
127C32×C8—42no—
128#459—91noSylow
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—
149C3×D8—184no—
150C2×C3×C8—32no—
151C2×C3×C8—32no—
152C2×C3×C8—62no—
153C2×C3×C8C2×C3×D312yes—
154C3×C4⋊C4—91no—
155C3×C22⋊C4—181no—
156C3×C42—31no—
157C3×C42—31no—
158C3×C42—61no—
159C3×C42C2×C3×D311yescharacteristic
160(C2×D3)⋊C4—61no—
161(C2×D3)⋊C4—181no—
162C4⋊Dic3—31no—
163C4⋊Dic3—91no—
164D24—64no—
165D24—184no—
166C22×C32—21no—
167C22×C32C2×D1211yescharacteristic
168C22×C32C2×C3×D411yescharacteristic
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×C4D2412yescomplemented
188C32×C4C3×D812yescomplemented
189C32×C4D3×D411yescharacteristic
190C32×C4#23911yescharacteristic
191C32×C4C2×C3×D411yescharacteristic
192C32×C4C3×#4111yescharacteristic
193C3⋊Dic3—41no—
194C3×Dic3—121no—
195C2×D8—92no—
196#127—91no—
197C4×D4—91no—
198C4⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
199#102—92no—
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×C4C3×D1211yescharacteristic
225C2×C3×C4C3×D1211yescharacteristic
226C2×C3×C4C22×C3×D311yescharacteristic
227C2×Dic3—61no—
228C2×Dic3—181no—
229D12—32no—
230D12—62no—
231D12—61no—
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—22no—
244C3×C8—62no—
245C3×C8—62no—
246C3×C8—122no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32#89511yescharacteristic
251C2×C32C2×D2411yescharacteristic
252C2×C32#90811yescharacteristic
253C2×C32C3×#12111yescharacteristic
254C2×C32C2×C3×D811yescharacteristic
255C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
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—184no—
266C2×C8C2×(C3⋊D3)⋊C312yes—
267C4⋊C4—91no—
268C22⋊C4—181no—
269C42C2×(C3⋊D3)⋊C311yescharacteristic
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×C3×D1211yescharacteristic
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×C4C3×D2412yes—
307C3×C4C3×D3×D411yescharacteristic
308C3×C4C3×#23911yescharacteristic
309Dic3—121no—
310Dic3—361no—
311C32—21no—
312C32#484311yescharacteristic, complemented
313C32C3×#45911yescharacteristic, complemented
314C23—181no—
315C23—182no—
316D4—92no—
317D4—182no—
318D4—181no—
319D4—182no—
320D4—362no—
321D4—362no—
322C2×C4—181no—
323C2×C4—362no—
324C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
325C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
326C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
327C8—22no—
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×C3C3×#89511yescharacteristic
340C2×C3C2×C3×D2411yescharacteristic
341C2×C3C3×#90811yescharacteristic
342D3—122no—
343D3—242no—
344D3—362no—
345D3—722no—
346C22—182no—
347C22—361no—
348C22—361no—
349C22—362no—
350C22—362no—
351C22—362no—
352C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
353C4—21no—
354C4—361no—
355C4#3745⋊C312yescomplemented
356C4D4×(C3⋊D3)⋊C311yescharacteristic
357C4#8872211yescharacteristic
358C3—31no—
359C3—31no—
360C3—61no—
361C3C3×#484311yescharacteristic
362C2—362no—
363C2—722no—
364C2#64104⋊C311yescharacteristic
365C2C2×#3745⋊C311yescharacteristic
366C2#64117⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×D12
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#459
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#3745⋊C3C2semidirect
2. #11721385C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. H(3,𝔽3)×C4⋊C8C2semidirect
5. #63648⋊C3C2semidirect
6. #10592542C3semidirect
18. C2×#3745C2×C3semidirect
19. #64083C2×C3semidirect
20. C4×C4⋊(C3⋊D3)C2×C3semidirect
25. C32×C4⋊C8D3semidirect
26. C32×C4⋊C8C2×C3semidirect
27. #63648C2×C3semidirect
40. C4⋊((C3⋊D3)⋊C3)D4semidirect
76. C4×H(3,𝔽3)D8semidirect
115. C4⋊(C3⋊D3)C3×D4semidirect
187. C32×C4D24semidirect
188. C32×C4C3×D8semidirect
198. C4⋊C8(C3⋊D3)⋊C3semidirect
200. H(3,𝔽3)#459semidirect
312. C32#4843semidirect
313. C32C3×#459semidirect
355. C4#3745⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117213852
C4×C4⋊((C3⋊D3)⋊C3)2
H(3,𝔽3)×C4⋊C82
#63648⋊C32
#105925423
C3×#48433