Finite Group Databasebeta

← #13159437

Subgroups — #13159437

Order 1728.

1172 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 178 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×#89018C211yesmaximal, characteristic
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#88787C211yesmaximal, characteristic, complemented
4#11722179C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592502—31nomaximal
9#10590609—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6292—21no—
15C2×#6289—21no—
16#89018—22no—
17#89018C414yescomplemented
18C33×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3750—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88787—22no—
24C2×#3832—31no—
25C2×#3799—34no—
26C2×#3743—31no—
27C4⋊C4×C3⋊D3D311yescharacteristic, complemented
28C3×D3×C4⋊C4—34no—
29C2×#3711—34no—
30C32×#103—31no—
31#63662—31no—
32#63648—31no—
33#63636—31no—
34#63636—34no—
35#63636—64no—
36C3×#810D314yescomplemented
37#63565—34no—
38#63564—34no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4×C3⋊D3—42no—
48C3×C4×C3⋊D3C2×C412yes—
49C3×C4×C3⋊D3D412yes—
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6C2×C412yescomplemented
53#6292—24no—
54#6289—24no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C32×C3⋊C8—21no—
57#4966—91no—
58#4803—94no—
59C2×C32×Q8—31no—
60#3832—34no—
61#3832—62no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—61no—
65C2×C4⋊(C3⋊D3)—64no—
66C2×C4×C3⋊D3—21no—
67C2×C4×C3⋊D3—61no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×C4×D3—34no—
70C2×C3×C4×D3—64no—
71C2×C3×Dic6C2×D314yes—
72C2×#626—64no—
73C2×#624—64no—
74#3799—316no—
75#3799—68no—
76C2×C32×C8—31no—
77C32×C4⋊C4—24no—
78C32×C4⋊C4C2×D314yes—
79C32×C4⋊C4C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3750—21no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85#3743—62no—
86C3×C4⋊Dic3—34no—
87C3×Dic3⋊C4—64no—
88C2×C3×C3⋊C8C2×D314yes—
89#3711—68no—
90C22×C33C2×D411yescharacteristic
91C2×C3⋊(C3⋊D3)—22no—
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C3×C3⋊D3—21no—
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3C22⋊C412yes—
97C33×C4—41no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3×C3⋊Dic3—41no—
101C3×C3⋊Dic3C22⋊C412yes—
102C32×Dic3—22no—
103C2×#239—94no—
104C2×#235—91no—
105C2×C3×C4⋊C4—91no—
106C2×#208—91no—
107C2×ΓL(1,𝔽25)—94no—
108D3×C4⋊C4—34no—
109C3×#103—34no—
110#835—94no—
111#821—94no—
112#810—91no—
113#810C3⋊D311yescharacteristic, complemented
114#809—94no—
115#809—91no—
116#809—91no—
117#809—184no—
118C22×C3⋊D3—61no—
119C22×C3⋊D3—64no—
120C22×C3⋊D3—124no—
121C22×C3⋊D3#6711yescharacteristic
122C22×C3×D3—34no—
123C32×Q8—32no—
124C32×Q8—61no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—44no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3#6711yescharacteristic
134C4⋊(C3⋊D3)—32no—
135C4⋊(C3⋊D3)—38no—
136C4⋊(C3⋊D3)—68no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—124no—
141C4×C3⋊D3—42no—
142C4×C3⋊D3—64no—
143C4×C3⋊D3C4×D312yes—
144C4×C3⋊D3D1212yes—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—64no—
148C3×C4×D3—38no—
149C3×C4×D3—38no—
150C3×C4×D3—128no—
151C3×Dic6—24no—
152C3×Dic6C4×D318yescomplemented
153#626—616no—
154#624—616no—
155C32×C8—61no—
156C3⋊(C3⋊C8)—61no—
157C3⋊(C3⋊C8)—64no—
158C3⋊(C3⋊C8)—124no—
159C3×C3⋊C8—24no—
160#417—271noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C2×C33#13711yescharacteristic
164C3⋊(C3⋊D3)—42no—
165C3×C3⋊D3—22no—
166C2×C3×Q8—34no—
167C22×C3×C4—91no—
168C22×C3×C4—181no—
169C2×#67—181no—
170#239—916no—
171#239—188no—
172#235—94no—
173#235—182no—
174C2×D12—94no—
175C2×D12—91no—
176C2×D12—184no—
177C2×D12—184no—
178C2×C4×D3—34no—
179C2×C4×D3—64no—
180C2×C4×D3—181no—
181C2×C4×D3—184no—
182C2×Dic6C2×C3⋊D311yescharacteristic
183C2×C3×C8—34no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—91no—
186C3×C4⋊C4—181no—
187C3×C4⋊C4D3214yes—
188C3×C4⋊C4C2×C3⋊D311yescharacteristic
189C4⋊Dic3—34no—
190Dic3⋊C4—64no—
191#208—182no—
192C2×C3⋊C8—94no—
193C2×C3⋊C8—91no—
194C2×C3⋊C8—184no—
195C2×C3⋊C8C2×C3⋊D311yescharacteristic
196ΓL(1,𝔽25)—188no—
197C22×C32—24no—
198C22×C32D3×D414yes—
199C22×C32C2×#6711yescharacteristic
200C2×C3⋊D3—21no—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—62no—
203C2×C3⋊D3—68no—
204C2×C3⋊D3—128no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—121no—
207C2×C3⋊D3—124no—
208C2×C3⋊D3—124no—
209C2×C3⋊D3—244no—
210C2×C3⋊D3—244no—
211C2×C3⋊D3(C2×D3)⋊C412yes—
212C2×C3×D3—38no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—41no—
218C32×C4—44no—
219C32×C4—62no—
220C32×C4—84no—
221C32×C4C2×C4×D314yes—
222C32×C4C2×C4×D311yescharacteristic
223C32×C4C2×D1211yescharacteristic
224C32×C4D3×D414yes—
225C3⋊Dic3—41no—
226C3⋊Dic3(C2×D3)⋊C412yes—
227C3×Dic3—28no—
228C3×Dic3—38no—
229C3×Dic3—124no—
230C2×#41—271no—
231C2×M16—271no—
232C2×C4⋊C4—91no—
233#103—31no—
234#103—271no—
235#102—271no—
236#102—271no—
237C33#41711yescharacteristic, complemented, Sylow
238C23×C3—91no—
239C22×D3—34no—
240C22×D3—184no—
241C22×D3—181no—
242C22×D3—364no—
243C3×Q8—38no—
244C3×Q8—64no—
245C2×C3×C4—24no—
246C2×C3×C4—21no—
247C2×C3×C4—24no—
248C2×C3×C4—44no—
249C2×C3×C4—64no—
250C2×C3×C4—91no—
251C2×C3×C4—92no—
252C2×C3×C4—92no—
253C2×C3×C4—181no—
254C2×C3×C4—362no—
255C2×C3×C4C2×D3214yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257#67—184no—
258C2×Dic3—21no—
259C2×Dic3—34no—
260C2×Dic3—64no—
261D12—98no—
262D12—92no—
263D12—188no—
264D12—1816no—
265D12—184no—
266D12—181no—
267D12—364no—
268C4×D3—38no—
269C4×D3—38no—
270C4×D3—128no—
271C4×D3—184no—
272C4×D3—1816no—
273Dic6—21no—
274Dic6C4×C3⋊D312yescomplemented
275C3×C8—64no—
276C3⋊C8—21no—
277C3⋊C8—184no—
278C3⋊C8—181no—
279C3⋊C8—364no—
280C2×C32—24no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32D3×C22⋊C414yes—
284C2×C32#91414yes—
285C2×C32#91814yes—
286C2×C32C2×(C2×D3)⋊C411yescharacteristic
287C2×C32#93211yescharacteristic
288C2×C32#94211yescharacteristic
289C3⋊D3—22no—
290C3⋊D3—122no—
291C3⋊D3—128no—
292C3⋊D3—248no—
293C3×D3—68no—
294#41—274no—
295#41—542no—
296C2×Q8—31no—
297C2×D4—271no—
298C2×D4—541no—
299C22×C4—91no—
300C22×C4—181no—
301C22×C4—541no—
302M16—542no—
303C2×C8—31no—
304C2×C8—271no—
305C4⋊C4—91no—
306C4⋊C4—181no—
307C4⋊C4D3×C3⋊D311yescharacteristic
308C22×C3—24no—
309C22×C3—92no—
310C22×C3—181no—
311C22×C3—181no—
312C22×C3D3×#6714yes—
313C22×C3D4×C3⋊D311yescharacteristic
314C2×D3—38no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—182no—
318C2×D3—188no—
319C2×D3—368no—
320C2×D3—361no—
321C2×D3—361no—
322C2×D3—364no—
323C2×D3—364no—
324C2×D3—724no—
325C2×D3—724no—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—41no—
329C3×C4—44no—
330C3×C4—68no—
331C3×C4—84no—
332C3×C4—92no—
333C3×C4—361no—
334C3×C4C4×D3214yes—
335C3×C4D3×D1214yes—
336C3×C4C2×C4×C3⋊D311yescharacteristic
337C3×C4D4×C3⋊D311yescharacteristic
338Dic3—22no—
339Dic3—38no—
340Dic3—124no—
341C32—24no—
342C32#480314yescomplemented
343C32#496611yescharacteristic, complemented
344C23—91no—
345C23—541no—
346Q8—32no—
347Q8—61no—
348D4—272no—
349D4—544no—
350D4—541no—
351C2×C4—21no—
352C2×C4—61no—
353C2×C4—91no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—181no—
357C2×C4—362no—
358C2×C4—544no—
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—61no—
361C8—541no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—182no—
366C2×C3D3×(C2×D3)⋊C414yes—
367C2×C3#6393414yes—
368C2×C3#6394814yes—
369C2×C3C22⋊C4×C3⋊D311yescharacteristic
370C2×C3#6412311yescharacteristic
371C2×C3#6412711yescharacteristic
372D3—68no—
373D3—368no—
374D3—362no—
375D3—728no—
376C22—92no—
377C22—181no—
378C22—181no—
379C22—542no—
380C22—1081no—
381C22—1081no—
382C22C3⋊D3×#6711yescharacteristic
383C4—41no—
384C4—62no—
385C4—92no—
386C4—361no—
387C4C4×D3×C3⋊D311yescharacteristic
388C4C3⋊D3×D1211yescharacteristic
389C3—24no—
390C3#1059060914yescomplemented
391C3#1059250211yescharacteristic, complemented
392C2—182no—
393C2—1082no—
394C2C3⋊D3×(C2×D3)⋊C411yescharacteristic
395C2#1172356211yescharacteristic
396C2#1172358611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89018C2
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. C3×C4⋊C4×C3⋊D3C2semidirect
3. C2×#88787C2semidirect
4. #11722179C2semidirect
5. C32×#810C2semidirect
6. #11722037C2semidirect
7. #11722036C2semidirect
17. #89018C4semidirect
27. C4⋊C4×C3⋊D3D3semidirect
36. C3×#810D3semidirect
44. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
52. C32×Dic6C2×C4semidirect
113. #810C3⋊D3semidirect
152. C3×Dic6C4×D3semidirect
237. C33#417semidirect
274. Dic6C4×C3⋊D3semidirect
342. C32#4803semidirect
343. C32#4966semidirect
390. C3#10590609semidirect
391. C3#10592502semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C3×C4⋊C4×C3⋊D32
C2×#887872
#117221792
C32×#8102
#117220372
#117220362
#105925023
#105906093