Finite Group Databasebeta

← #13137322

Subgroups — #13137322

Order 1728.

413 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721716C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
4#11720308C211yesmaximal, characteristic
5#11720308C211yesmaximal, characteristic
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590308—31nomaximal
9#10590270—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C42×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16#88540—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C4×C4⋊(C3⋊D3)D311yescharacteristic
22C3×C4×D12—31no—
23C3×#870—31no—
24C3×#870—31no—
25C3×#870—61no—
26C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
27C3×C4⋊(C3⋊C8)—31no—
28#63576—31no—
29#63575—31no—
30#63574—31no—
31#63573—31no—
32#63564—31no—
33#63564—31no—
34#63562—31no—
35#63562—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×#1084—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yes—
45C4×(C3⋊D3)⋊C3—42no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#5066—91no—
49#5009—91no—
50#4847—91no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C4×C3⋊D3—21no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—61no—
55C32×C42—21no—
56C32×C42C2×D311yescharacteristic
57C32×C42C2×D311yescharacteristic
58#3752—21no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—61no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—62no—
71C3×Dic3⋊C4—62no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—61no—
74C3×Dic3⋊C4—61no—
75C3×Dic3⋊C4—61no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—22no—
80C2×(C3⋊D3)⋊C3—41no—
81C2×(C3⋊D3)⋊C3—41no—
82C4×H(3,𝔽3)—21no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)C2×Q811yescharacteristic
85C4×H(3,𝔽3)#4111yescharacteristic
86#1084—41no—
87#1084—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×#125—91no—
90C3×C4×D4—91no—
91C4×D12—31no—
92C4×D12—91no—
93#870—31no—
94#870—31no—
95#870—61no—
96C3×C4⋊C8—91no—
97#832—91no—
98#832—91no—
99#821—91no—
100#821—91no—
101#818—91no—
102#817—91no—
103#809—91no—
104#809—91no—
105#808—91no—
106#808—91no—
107#807—91no—
108#807—91no—
109C4⋊(C3⋊C8)—31no—
110C4⋊(C3⋊C8)—91no—
111C22×C3⋊D3—21no—
112C22×C3×D3—61no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4D1211yescharacteristic
117C2×C32×C4D1211yescharacteristic
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C4⋊(C3⋊D3)—21no—
124C4⋊(C3⋊D3)Dic612yes—
125C4×C3⋊D3—42no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3×D12—32no—
134C3×D12—61no—
135C3×C4×D3—122no—
136C3⋊(C3⋊C8)—21no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140#478—271noSylow
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)#13511yescharacteristic
143C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
144(C3⋊D3)⋊C3—42no—
145C2×C3×D4—91no—
146C22×C3×C4—181no—
147C2×D12—31no—
148C2×D12—91no—
149C2×C4×D3—61no—
150C2×C4×D3—181no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—181no—
157C3×C4⋊C4—181no—
158C3×C22⋊C4—181no—
159C3×C42—31no—
160C3×C42—31no—
161C3×C42—61no—
162C3×C42D3211yescharacteristic
163(C2×D3)⋊C4—61no—
164(C2×D3)⋊C4—181no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172C4⋊Dic3—91no—
173Dic3⋊C4—61no—
174Dic3⋊C4—61no—
175Dic3⋊C4—61no—
176Dic3⋊C4—61no—
177Dic3⋊C4—62no—
178Dic3⋊C4—62no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32C2×D1211yescharacteristic
185C22×C32C2×#6711yescharacteristic
186C2×C3⋊D3—22no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—62no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4C2×Dic611yescharacteristic
201C32×C4#23511yescharacteristic
202C32×C4#23711yescharacteristic
203C32×C4D3×Q811yescharacteristic
204C3⋊Dic3—41no—
205C3×Dic3—122no—
206C3×Dic3—122no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212#125—91no—
213C4×D4—91no—
214#107—271no—
215#106—271no—
216C4⋊C8—91no—
217#102—271no—
218#102—271no—
219H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
220C23×C3—181no—
221C22×D3—61no—
222C22×D3—181no—
223C3×D4—92no—
224C3×D4—181no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—362no—
238C2×C3×C4#62611yescharacteristic
239C2×C3×C4#62611yescharacteristic
240C2×C3×C4C2×D3211yescharacteristic
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×Dic3—61no—
248C2×Dic3—181no—
249D12—32no—
250D12—61no—
251D12—92no—
252D12—181no—
253C4×D3—122no—
254C4×D3—362no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C3⋊C8—181no—
260C3⋊C8—181no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32#89711yescharacteristic
265C2×C32#90411yescharacteristic
266C2×C32#90811yescharacteristic
267C2×C32#92411yescharacteristic
268C2×C32#93211yescharacteristic
269C2×C32#95011yescharacteristic
270C3⋊D3—42no—
271C3×D3—122no—
272C2×D4—91no—
273C22×C4—181no—
274C2×C8—91no—
275C2×C8—91no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—91no—
279C4⋊C4—181no—
280C4⋊C4—181no—
281C22⋊C4—181no—
282C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3—182no—
287C22×C3—361no—
288C22×C3—361no—
289C22×C3C2×#62611yescharacteristic
290C2×D3—62no—
291C2×D3—121no—
292C2×D3—121no—
293C2×D3—182no—
294C2×D3—361no—
295C2×D3—361no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—61no—
305C3×C4—61no—
306C3×C4—61no—
307C3×C4—61no—
308C3×C4—121no—
309C3×C4—121no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4—361no—
313C3×C4D3×Dic611yescharacteristic
314C3×C4#379511yescharacteristic
315Dic3—121no—
316Dic3—121no—
317Dic3—121no—
318Dic3—121no—
319Dic3—122no—
320Dic3—122no—
321Dic3—121no—
322Dic3—361no—
323C32—21no—
324C32#484711yescharacteristic, complemented
325C32#500911yescharacteristic, complemented
326C23—181no—
327D4—92no—
328D4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—362no—
333C2×C4#620011yescharacteristic
334C2×C4#620011yescharacteristic
335C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C8—181no—
337C8—181no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—362no—
348C2×C3#6382911yescharacteristic
349C2×C3#6383011yescharacteristic
350C2×C3#6385711yescharacteristic
351D3—122no—
352D3—362no—
353C22—182no—
354C22—361no—
355C22—361no—
356C22C2×#620011yescharacteristic
357C4—21no—
358C4—21no—
359C4—361no—
360C4—361no—
361C4—361no—
362C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
363C4#8864811yescharacteristic
364C3—31no—
365C3—31no—
366C3—61no—
367C3#1059027111yescharacteristic
368C2—362no—
369C2#1172107111yescharacteristic
370C2#1172107311yescharacteristic
371C2#1172110711yescharacteristic
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⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#478
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. #11721716C2semidirect
3. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
26. C4⋊(C3⋊(C3⋊C8))D3semidirect
219. H(3,𝔽3)#478semidirect
324. C32#4847semidirect
325. C32#5009semidirect

Maximal subgroups

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

subgroupindex
#117217162
C4×C4⋊((C3⋊D3)⋊C3)2
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203082
#117203072
#117203062
#105903083
#105902703