Finite Group Databasebeta

← #13159444

Subgroups — #13159444

Order 1728.

1340 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 170 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×#89017C211yesmaximal, characteristic
2C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×#88788C211yesmaximal, characteristic, complemented
4C3×#63636C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592502—31nomaximal
9#10590612—34nomaximal
10C2×C4×C3⋊(C3⋊D3)—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6292—21no—
15C2×Dic3×C3⋊D3—21no—
16#89017—22no—
17#89017C414yescomplemented
18C33×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20#88857—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88788—22no—
24C2×#3832—31no—
25C2×#3796—34no—
26C2×#3743—31no—
27C4⋊C4×C3⋊D3—31no—
28C4⋊C4×C3⋊D3—34no—
29C4⋊C4×C3⋊D3—64no—
30C2×#3712—34no—
31C32×#103—31no—
32#63662—31no—
33#63648—31no—
34#63636D311yescharacteristic, complemented
35C3×#810D314yescomplemented
36C3×#809—34no—
37#63566—34no—
38#63563—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4×C3⋊(C3⋊D3)—42no—
46C4×C3⋊(C3⋊D3)C2×C412yes—
47C4×C3⋊(C3⋊D3)D412yes—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)C2×C412yescomplemented
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6C2×C412yescomplemented
53#6292—24no—
54Dic3×C3⋊D3—24no—
55C3×C3⋊(C3⋊C8)—21no—
56C32×C3⋊C8—21no—
57#4803—94no—
58#4771—91no—
59C2×C32×Q8—31no—
60#3832—34no—
61#3832—62no—
62C2×C4⋊(C3⋊D3)—61no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—34no—
66C2×C4×C3⋊D3—61no—
67C2×C4×C3⋊D3—64no—
68C2×C4×C3⋊D3—61no—
69C2×C4×C3⋊D3—64no—
70C2×C4×C3⋊D3—68no—
71C2×C3×D12—34no—
72C2×C3×Dic6C2×D314yes—
73C2×#626—64no—
74C2×D3×Dic3—64no—
75#3796—316no—
76#3796—68no—
77C2×C32×C8—31no—
78C32×C4⋊C4—24no—
79C32×C4⋊C4C2×D314yes—
80C32×C4⋊C4C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—34no—
83C4⋊(C3⋊Dic3)—64no—
84#3750—61no—
85#3750—64no—
86#3750—68no—
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88#3743—62no—
89C2×C3×C3⋊C8—34no—
90C2×C3×C3⋊C8C2×D314yes—
91#3712—68no—
92C22×C33C2×D411yescharacteristic
93C2×C3⋊(C3⋊D3)—21no—
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3⋊(C3⋊D3)C22⋊C412yes—
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—41no—
98C2×C3×C3⋊D3—41no—
99C33×C4—41no—
100C33×C4C22×C411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3⋊(C3⋊Dic3)—41no—
103C3⋊(C3⋊Dic3)C22⋊C412yes—
104C32×Dic3—22no—
105C2×#239—94no—
106C2×#237—91no—
107C2×ΓL(1,𝔽25)—94no—
108C2×ΓL(1,𝔽25)—91no—
109D3×C4⋊C4—94no—
110D3×C4⋊C4—91no—
111D3×C4⋊C4—184no—
112C3×#103—34no—
113C3×#102—91no—
114#835—94no—
115#832—91no—
116#821—94no—
117#816—91no—
118#810C3⋊D311yescharacteristic, complemented
119#809—34no—
120C22×C3⋊D3—21no—
121C22×C3⋊D3—31no—
122C22×C3⋊D3—34no—
123C22×C3⋊D3—64no—
124C22×C3×D3—64no—
125C32×Q8—32no—
126C32×Q8—61no—
127C2×C32×C4—28no—
128C2×C32×C4—24no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—61no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—31no—
135C2×C3⋊Dic3—34no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—68no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—61no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)C4×D312yescomplemented
143C4×C3⋊D3—32no—
144C4×C3⋊D3—32no—
145C4×C3⋊D3—38no—
146C4×C3⋊D3—38no—
147C4×C3⋊D3—68no—
148C4×C3⋊D3—68no—
149C4×C3⋊D3—64no—
150C4×C3⋊D3—1216no—
151C4×C3⋊D3—122no—
152C4×C3⋊D3—128no—
153C2×C3×Dic3—24no—
154C3×D12—38no—
155C3×D12—64no—
156C3×Dic6—24no—
157C3×Dic6C4×D318yescomplemented
158#626—616no—
159D3×Dic3—616no—
160C32×C8—61no—
161C3⋊(C3⋊C8)—21no—
162C3×C3⋊C8—24no—
163C3×C3⋊C8—64no—
164#417—271noSylow
165C2×C33C2×C22⋊C411yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C2×C33#13711yescharacteristic
168C3⋊(C3⋊D3)—22no—
169C3×C3⋊D3—42no—
170C2×C3×Q8—34no—
171C2×C3×D4—91no—
172C2×#67—181no—
173C22×Dic3—181no—
174#239—916no—
175#239—188no—
176#237—94no—
177#237—182no—
178C2×D12—34no—
179C2×D12—184no—
180C2×C4×D3—94no—
181C2×C4×D3—91no—
182C2×C4×D3—184no—
183C2×C4×D3—184no—
184C2×C4×D3—184no—
185C2×C4×D3—181no—
186C2×C4×D3—188no—
187C2×Dic6C2×C3⋊D311yescharacteristic
188C2×C3×C8—34no—
189C2×C3×C8—91no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4D3214yes—
192C3×C4⋊C4C2×C3⋊D311yescharacteristic
193C4⋊Dic3—94no—
194C4⋊Dic3—91no—
195C4⋊Dic3—184no—
196Dic3⋊C4—184no—
197Dic3⋊C4—181no—
198Dic3⋊C4—188no—
199C2×C3⋊C8—34no—
200C2×C3⋊C8C2×C3⋊D311yescharacteristic
201ΓL(1,𝔽25)—182no—
202ΓL(1,𝔽25)—188no—
203C22×C32—24no—
204C22×C32D3×D414yes—
205C22×C32D3×D411yescharacteristic
206C2×C3⋊D3—22no—
207C2×C3⋊D3—32no—
208C2×C3⋊D3—38no—
209C2×C3⋊D3—41no—
210C2×C3⋊D3—41no—
211C2×C3⋊D3—68no—
212C2×C3⋊D3—68no—
213C2×C3⋊D3—68no—
214C2×C3⋊D3—61no—
215C2×C3⋊D3—64no—
216C2×C3⋊D3—61no—
217C2×C3⋊D3—64no—
218C2×C3×D3—68no—
219C2×C3×D3—124no—
220C2×C3×D3—124no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—48no—
224C32×C4—41no—
225C32×C4—44no—
226C32×C4—62no—
227C32×C4C2×C4×D314yes—
228C32×C4C2×C4×D311yescharacteristic
229C32×C4D3×D414yes—
230C32×C4D3×D411yescharacteristic
231C3⋊Dic3—32no—
232C3⋊Dic3—38no—
233C3⋊Dic3—68no—
234C3⋊Dic3—128no—
235C3⋊Dic3—121no—
236C3⋊Dic3—124no—
237C3×Dic3—28no—
238C2×#41—271no—
239C2×M16—271no—
240C2×C4⋊C4—271no—
241#103—31no—
242#103—271no—
243#102—91no—
244#102—271no—
245C33#41711yescharacteristic, complemented, Sylow
246C23×C3—181no—
247C22×D3—64no—
248C22×D3—94no—
249C22×D3—91no—
250C22×D3—184no—
251C3×Q8—38no—
252C3×Q8—64no—
253C3×D4—92no—
254C3×D4—181no—
255C2×C3×C4—28no—
256C2×C3×C4—24no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—64no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262#67—184no—
263C2×Dic3—21no—
264C2×Dic3—94no—
265C2×Dic3—91no—
266C2×Dic3—184no—
267C2×Dic3—184no—
268C2×Dic3—188no—
269C2×Dic3—184no—
270C2×Dic3—181no—
271D12—38no—
272D12—64no—
273D12—1816no—
274C4×D3—98no—
275C4×D3—98no—
276C4×D3—92no—
277C4×D3—92no—
278C4×D3—188no—
279C4×D3—188no—
280C4×D3—1816no—
281C4×D3—3616no—
282C4×D3—362no—
283C4×D3—368no—
284Dic6—21no—
285Dic6C4×C3⋊D312yescomplemented
286C3×C8—64no—
287C3×C8—181no—
288C3⋊C8—21no—
289C3⋊C8—64no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32D3×C22⋊C414yes—
294C2×C32D3×C22⋊C411yescharacteristic
295C2×C32#91111yescharacteristic
296C2×C32#91414yes—
297C2×C32#91511yescharacteristic
298C2×C32#91814yes—
299C3⋊D3—42no—
300C3⋊D3—616no—
301C3⋊D3—62no—
302C3⋊D3—68no—
303C3×D3—128no—
304#41—274no—
305#41—542no—
306C2×Q8—31no—
307C2×D4—91no—
308C2×D4—541no—
309C22×C4—271no—
310C22×C4—541no—
311C22×C4—541no—
312M16—542no—
313C2×C8—31no—
314C2×C8—91no—
315C4⋊C4—271no—
316C4⋊C4—541no—
317C4⋊C4D3×C3⋊D311yescharacteristic
318C22×C3—24no—
319C22×C3—182no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3#381114yes—
323C22×C3D4×C3⋊D311yescharacteristic
324C2×D3—68no—
325C2×D3—92no—
326C2×D3—98no—
327C2×D3—124no—
328C2×D3—124no—
329C2×D3—188no—
330C2×D3—188no—
331C2×D3—188no—
332C2×D3—181no—
333C2×D3—181no—
334C2×D3—184no—
335C2×D3—184no—
336C3×C4—24no—
337C3×C4—24no—
338C3×C4—48no—
339C3×C4—41no—
340C3×C4—44no—
341C3×C4—68no—
342C3×C4C4×D3214yes—
343C3×C4#380214yes—
344C3×C4C2×C4×C3⋊D311yescharacteristic
345C3×C4D4×C3⋊D311yescharacteristic
346Dic3—22no—
347Dic3—92no—
348Dic3—98no—
349Dic3—188no—
350Dic3—368no—
351Dic3—361no—
352Dic3—364no—
353C32—24no—
354C32#477111yescharacteristic, complemented
355C32#480314yescomplemented
356C23—181no—
357C23—271no—
358Q8—32no—
359Q8—61no—
360D4—92no—
361D4—181no—
362D4—544no—
363C2×C4—21no—
364C2×C4—61no—
365C2×C4—271no—
366C2×C4—272no—
367C2×C4—272no—
368C2×C4—544no—
369C2×C4—541no—
370C2×C4—1082no—
371C2×C4C2×D3×C3⋊D311yescharacteristic
372C8—61no—
373C8—181no—
374C2×C3—24no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—362no—
378C2×C3#6392114yes—
379C2×C3#6393714yes—
380C2×C3#6395114yes—
381C2×C3C22⋊C4×C3⋊D311yescharacteristic
382C2×C3#6412311yescharacteristic
383C2×C3#6412711yescharacteristic
384D3—128no—
385D3—1816no—
386D3—188no—
387D3—182no—
388C22—182no—
389C22—272no—
390C22—361no—
391C22—361no—
392C22—541no—
393C22—541no—
394C22#8903911yescharacteristic
395C4—41no—
396C4—62no—
397C4—272no—
398C4—1081no—
399C4C4×D3×C3⋊D311yescharacteristic
400C4#8902611yescharacteristic
401C3—24no—
402C3#1059061214yescomplemented
403C3#1059250211yescharacteristic, complemented
404C2—362no—
405C2—542no—
406C2#1172353911yescharacteristic
407C2#1172356611yescharacteristic
408C2#1172359011yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89017C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
3. C2×#88788C2semidirect
4. C3×#63636C2semidirect
5. C32×#810C2semidirect
6. #11722038C2semidirect
7. #11722034C2semidirect
17. #89017C4semidirect
34. #63636D3semidirect
35. C3×#810D3semidirect
49. C3×C4⋊(C3⋊D3)C2×C4semidirect
52. C32×Dic6C2×C4semidirect
118. #810C3⋊D3semidirect
142. C4⋊(C3⋊D3)C4×D3semidirect
157. C3×Dic6C4×D3semidirect
245. C33#417semidirect
285. Dic6C4×C3⋊D3semidirect
354. C32#4771semidirect
355. C32#4803semidirect
402. C3#10590612semidirect
403. C3#10592502semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
C4⋊C4×C3⋊(C3⋊D3)2
C2×#887882
C3×#636362
C32×#8102
#117220382
#117220342
#105925023
#105906123