Finite Group Databasebeta

← #10592714⋊C3 #13142749

Subgroups — #10592714⋊C3 #13142749

Order 1728.

584 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 112 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×D4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C212yesmaximal, complemented
3#11721385C211yesmaximal, characteristic, complemented
4#63636⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#10592714C311yesmaximal, characteristic, complemented
7C3×#5015—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)—21no—
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)C2211yescharacteristic
16#3753⋊C3—24no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C32×C4×D4—21no—
19C32×C4×D4D311yescharacteristic, complemented
20C32×C4×D4C2×C311yescharacteristic, complemented
21C2×#3753C2×C312yescomplemented
22#64083C2×C311yescharacteristic, complemented
23C2×C3×#213—32no—
24C3×#874—31no—
25#63636C2×C312yescomplemented
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×#809—32no—
28C3×C4⋊(C3⋊C8)—31no—
29C23×H(3,𝔽3)—21no—
30C22×(C3⋊D3)⋊C3—22no—
31D4×H(3,𝔽3)—21no—
32D4×H(3,𝔽3)D412yescomplemented
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—42no—
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C4⋊((C3⋊D3)⋊C3)—22no—
39C4⋊((C3⋊D3)⋊C3)—22no—
40C4⋊((C3⋊D3)⋊C3)—42no—
41C4⋊((C3⋊D3)⋊C3)—42no—
42(C3⋊(C3⋊C8))⋊C3—22no—
43C3×#459—91no—
44#5015—31no—
45#5015—91no—
46C2×C32×D4—21no—
47C2×C32×D4C2×D311yescharacteristic
48C2×C32×D4C22×C311yescharacteristic
49C22×C32×C4—22no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)C22×C312yes—
55C2×C3×D12—32no—
56C2×C3×D12—61no—
57C2×C3×D12—61no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C4⋊C4C22×C311yescharacteristic
61C32×C22⋊C4—22no—
62C32×C22⋊C4—21no—
63C32×C22⋊C4—21no—
64C32×C42—21no—
65C32×C42C2×D311yescharacteristic
66C32×C42C22×C311yescharacteristic
67#3753—24no—
68C2×C3⋊(C3⋊C8)C22×C312yes—
69C3×#213—64no—
70C2×C3×C3⋊C8—32no—
71C22×H(3,𝔽3)—22no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)—41no—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C2×(C3⋊D3)⋊C3—42no—
76C2×(C3⋊D3)⋊C3—42no—
77C2×(C3⋊D3)⋊C3—42no—
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)D812yescomplemented
81C4×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83C2×C3×D8—92no—
84C3×#127—91no—
85C3×C4×D4—31no—
86C3×C4×D4—31no—
87C3×C4×D4—61no—
88C3×C4×D4C3×D311yescharacteristic
89C2×#213—32no—
90C2×#213—92no—
91#874—31no—
92#874—91no—
93C3×C4⋊C8—91no—
94C3×#102—92no—
95#809—32no—
96#809—92no—
97C4⋊(C3⋊C8)—31no—
98C4⋊(C3⋊C8)—91no—
99C23×C32—22no—
100C23×C32—21no—
101C23×C32—21no—
102C22×C3⋊D3—22no—
103C22×C3×D3—62no—
104C32×D4—22no—
105C32×D4—21no—
106C32×D4—21no—
107C32×D4—22no—
108C32×D4D1212yescomplemented
109C32×D4C3×D412yescomplemented
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—44no—
117C2×C32×C4—42no—
118C2×C32×C4—42no—
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4#6711yescharacteristic
121C2×C32×C4C3×D411yescharacteristic
122C2×C32×C4C3×D411yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C4⋊(C3⋊D3)—22no—
126C4⋊(C3⋊D3)—22no—
127C4⋊(C3⋊D3)—42no—
128C4⋊(C3⋊D3)—42no—
129C3×D12—62no—
130C3×D12—62no—
131C3×D12—122no—
132C3×D12—122no—
133C3⋊(C3⋊C8)—22no—
134C3×C3⋊C8—62no—
135#459—91noSylow
136C2×H(3,𝔽3)—42no—
137C2×H(3,𝔽3)#12111yescharacteristic
138C2×H(3,𝔽3)C2×D811yescharacteristic
139C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
140(C3⋊D3)⋊C3—82no—
141C2×C3×D4—31no—
142C2×C3×D4—31no—
143C2×C3×D4—61no—
144C2×C3×D4—92no—
145C2×C3×D4—181no—
146C2×C3×D4—181no—
147C2×C3×D4C2×C3×D311yescharacteristic
148C22×C3×C4—21no—
149C22×C3×C4—61no—
150C22×C3×C4—61no—
151C22×C3×C4—62no—
152C2×D12—32no—
153C2×D12—61no—
154C2×D12—61no—
155C2×D12—92no—
156C2×D12—181no—
157C2×D12—181no—
158C3×D8—184no—
159C2×C3×C8—92no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—61no—
163C3×C4⋊C4C2×C3×D311yescharacteristic
164C3×C22⋊C4—21no—
165C3×C22⋊C4—61no—
166C3×C22⋊C4—61no—
167C3×C22⋊C4—62no—
168C3×C42—31no—
169C3×C42—31no—
170C3×C42—61no—
171C3×C42C2×C3×D311yescharacteristic
172#213—64no—
173#213—184no—
174C2×C3⋊C8—32no—
175C2×C3⋊C8—92no—
176C22×C32—24no—
177C22×C32—22no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—42no—
181C22×C32—42no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32C2×#6711yescharacteristic
187C22×C32C2×C3×D411yescharacteristic
188C2×C3⋊D3—42no—
189C2×C3⋊D3—42no—
190C2×C3⋊D3—42no—
191C2×C3×D3—122no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—22no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—41no—
200C32×C4—42no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4#21312yescomplemented
204C32×C4C3×D812yescomplemented
205C32×C4C2×D1211yescharacteristic
206C32×C4#23511yescharacteristic
207C32×C4C2×C3×D411yescharacteristic
208C32×C4C3×#4111yescharacteristic
209C2×D8—92no—
210#127—91no—
211C4×D4(C3⋊D3)⋊C311yescharacteristic, complemented
212C4⋊C8—91no—
213#102—92no—
214H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
215C23×C3—21no—
216C23×C3—61no—
217C23×C3—61no—
218C23×C3—62no—
219C23×C3—182no—
220C22×D3—62no—
221C22×D3—182no—
222C3×D4—21no—
223C3×D4—32no—
224C3×D4—32no—
225C3×D4—61no—
226C3×D4—61no—
227C3×D4—62no—
228C3×D4—62no—
229C3×D4—182no—
230C3×D4—182no—
231C3×D4—362no—
232C3×D4—362no—
233C3×D4C3×D1212yes—
234C2×C3×C4—21no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—42no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—62no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—122no—
249C2×C3×C4—122no—
250C2×C3×C4—124no—
251C2×C3×C4C3×#6711yescharacteristic
252C2×C3×C4C3×#6711yescharacteristic
253C2×C3×C4C22×C3×D311yescharacteristic
254D12—62no—
255D12—62no—
256D12—122no—
257D12—122no—
258D12—182no—
259D12—182no—
260D12—362no—
261D12—362no—
262C3×C8—182no—
263C3⋊C8—62no—
264C3⋊C8—182no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—44no—
269C2×C32—42no—
270C2×C32—42no—
271C2×C32#93011yescharacteristic
272C2×C32C2×#21311yescharacteristic
273C2×C32#94911yescharacteristic
274C2×C32C3×#12111yescharacteristic
275C2×C32C2×C3×D811yescharacteristic
276C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
277C3⋊D3—82no—
278C3×D3—242no—
279C2×D4—92no—
280C2×D4—181no—
281C2×D4—181no—
282C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
283C22×C4—21no—
284D8—184no—
285C2×C8—92no—
286C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
287C22⋊C4—21no—
288C42C2×(C3⋊D3)⋊C311yescharacteristic
289C22×C3—22no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—62no—
295C22×C3—62no—
296C22×C3—64no—
297C22×C3—61no—
298C22×C3—121no—
299C22×C3—121no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3—122no—
303C22×C3—122no—
304C22×C3—362no—
305C22×C3—362no—
306C22×C3—362no—
307C22×C3C2×C3×#6711yescharacteristic
308C2×D3—122no—
309C2×D3—122no—
310C2×D3—122no—
311C2×D3—362no—
312C2×D3—362no—
313C2×D3—362no—
314C3×C4—21no—
315C3×C4—31no—
316C3×C4—31no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—32no—
320C3×C4—32no—
321C3×C4—41no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—61no—
325C3×C4—61no—
326C3×C4—62no—
327C3×C4—121no—
328C3×C4—121no—
329C3×C4—121no—
330C3×C4—122no—
331C3×C4C3×#21312yes—
332C3×C4C2×C3×D1211yescharacteristic
333C3×C4C3×#23511yescharacteristic
334C32—21no—
335C32#501511yescharacteristic, complemented
336C32C3×#45911yescharacteristic, complemented
337C23—21no—
338C23—182no—
339D4—21no—
340D4—182no—
341D4—182no—
342D4—362no—
343D4—362no—
344D4C4⋊((C3⋊D3)⋊C3)12yescomplemented
345C2×C4—21no—
346C2×C4—42no—
347C2×C4#638⋊C311yescharacteristic
348C2×C4#638⋊C311yescharacteristic
349C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
350C8—182no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—42no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3—122no—
362C2×C3—122no—
363C2×C3—124no—
364C2×C3—722no—
365C2×C3C3×#93011yescharacteristic
366C2×C3C2×C3×#21311yescharacteristic
367C2×C3C3×#94911yescharacteristic
368D3—242no—
369D3—722no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—362no—
374C22—362no—
375C22—362no—
376C22C2×#638⋊C311yescharacteristic
377C4—21no—
378C4—41no—
379C4#3753⋊C312yescomplemented
380C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
381C4#3828⋊C311yescharacteristic
382C3—31no—
383C3—31no—
384C3—61no—
385C3C3×#501511yescharacteristic
386C2—42no—
387C2—722no—
388C2#64139⋊C311yescharacteristic
389C2C2×#3753⋊C311yescharacteristic
390C2#64158⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×#459
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×D4×H(3,𝔽3)C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721385C2semidirect
4. #63636⋊C3C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #10592714C3semidirect
19. C32×C4×D4D3semidirect
20. C32×C4×D4C2×C3semidirect
21. C2×#3753C2×C3semidirect
22. #64083C2×C3semidirect
25. #63636C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
32. D4×H(3,𝔽3)D4semidirect
80. C4×H(3,𝔽3)D8semidirect
108. C32×D4D12semidirect
109. C32×D4C3×D4semidirect
203. C32×C4#213semidirect
204. C32×C4C3×D8semidirect
211. C4×D4(C3⋊D3)⋊C3semidirect
214. H(3,𝔽3)#459semidirect
335. C32#5015semidirect
336. C32C3×#459semidirect
344. D4C4⋊((C3⋊D3)⋊C3)semidirect
379. C4#3753⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×D4×H(3,𝔽3)2
C2×#3753⋊C32
#117213852
#63636⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927143
C3×#50153