Finite Group Databasebeta

← #13161051

Subgroups — #13161051

Order 1728.

2060 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 2060 conjugacy classes of subgroups — too many to draw. Only its 158 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×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723358C212yesmaximal, complemented
3C3×#63660C211yesmaximal, characteristic, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5#11722027C212yesmaximal, complemented
6C3⋊D3×#100—31nomaximal
7#10591503—34nomaximal
8C23×C3⋊(C3⋊D3)C412yes—
9C2×C33×D4C2211yescharacteristic
10C2×#6312C412yes—
11D4×C3⋊(C3⋊D3)—22no—
12D4×C3⋊(C3⋊D3)—22no—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C3×#3748C2212yescomplemented
16C32×#208C2212yescomplemented
17#88788—22no—
18#88785—22no—
19C2×D4×C3⋊D3—31no—
20C2×D4×C3⋊D3—34no—
21C2×D4×C3⋊D3—64no—
22M16×C3⋊D3—32no—
23#63818—38no—
24C32×#100—31no—
25#63660D311yescharacteristic, complemented
26#63650—32no—
27C3×#833—34no—
28C3×#833D314yescomplemented
29#63559—38no—
30C23×C33C2×C412yes—
31C22×C3⋊(C3⋊D3)—22no—
32C22×C3⋊(C3⋊D3)—22no—
33C22×C3⋊(C3⋊D3)—44no—
34C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
35C22×C3⋊(C3⋊D3)C2×C412yes—
36C33×D4—22no—
37C2×C33×C4C2311yescharacteristic
38#6312—24no—
39C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—22no—
41C4×C3⋊(C3⋊D3)D412yes—
42C4×C3⋊(C3⋊D3)D412yes—
43C3×C3⋊(C3⋊C8)—22no—
44C32×C3⋊C8—22no—
45D3×#100—94no—
46D3×#100—91no—
47C23×C3⋊D3—32no—
48C23×C3⋊D3—38no—
49C23×C3⋊D3—68no—
50C2×C32×D4—24no—
51C2×C32×D4C2×D314yes—
52C2×C32×D4C2×D311yescharacteristic
53C2×#638—32no—
54C2×#638—38no—
55C2×#638—68no—
56D4×C3⋊D3—62no—
57D4×C3⋊D3—62no—
58D4×C3⋊D3—68no—
59D4×C3⋊D3—68no—
60D4×C3⋊D3—616no—
61D4×C3⋊D3—616no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—34no—
67C2×C4×C3⋊D3—64no—
68C32×M16—32no—
69#3748C2×D312yescomplemented
70#3743—62no—
71C8×C3⋊D3—62no—
72C3×#208—38no—
73C3×#208C2×D318yescomplemented
74#3712—68no—
75#3710—68no—
76C22×C33—22no—
77C22×C33—42no—
78C22×C33C22×C411yescharacteristic
79C2×C3⋊(C3⋊D3)—22no—
80C2×C3⋊(C3⋊D3)—22no—
81C2×C3⋊(C3⋊D3)—44no—
82C2×C3⋊(C3⋊D3)—44no—
83C2×C3⋊(C3⋊D3)—42no—
84C2×C3⋊(C3⋊D3)C22⋊C411yescharacteristic
85C2×C3⋊(C3⋊D3)C22⋊C411yescharacteristic
86C33×C4C2×D412yes—
87C3⋊(C3⋊Dic3)C22⋊C412yes—
88C2×D3×D4—94no—
89C2×D3×D4—91no—
90C2×D3×D4—184no—
91D3×M16—98no—
92D3×M16—92no—
93C3×#100—34no—
94C3×#100—91no—
95#833—34no—
96#833C3⋊D311yescharacteristic, complemented
97#823—98no—
98#823—92no—
99C23×C32—28no—
100C23×C32C4×D318yes—
101C23×C32C4×D312yes—
102C22×C3⋊D3—31no—
103C22×C3⋊D3—32no—
104C22×C3⋊D3—34no—
105C22×C3⋊D3—38no—
106C22×C3⋊D3—64no—
107C22×C3⋊D3—68no—
108C22×C3⋊D3—62no—
109C22×C3⋊D3—62no—
110C22×C3⋊D3—68no—
111C22×C3⋊D3—68no—
112C22×C3⋊D3—616no—
113C22×C3⋊D3—616no—
114C22×C3⋊D3—1232no—
115C22×C3⋊D3—1216no—
116C22×C3⋊D3—124no—
117C32×D4—216no—
118C32×D4—28no—
119C32×D4—22no—
120C2×C32×C4—24no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123#638—64no—
124#638—616no—
125#638—632no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—64no—
129C4⋊(C3⋊D3)—616no—
130C4⋊(C3⋊D3)—68no—
131C4⋊(C3⋊D3)—62no—
132C4×C3⋊D3—32no—
133C4×C3⋊D3—32no—
134C4×C3⋊D3—38no—
135C4×C3⋊D3—38no—
136C4×C3⋊D3—68no—
137C4×C3⋊D3—68no—
138C32×C8—62no—
139C3⋊(C3⋊C8)—22no—
140C3×C3⋊C8—28no—
141C3×C3⋊C8—68no—
142C2×#100—271noSylow
143C2×C33—21no—
144C2×C33—42no—
145C2×C33C2×C22⋊C411yescharacteristic
146C3⋊(C3⋊D3)—21no—
147C3⋊(C3⋊D3)—42no—
148C3⋊(C3⋊D3)#10012yescomplemented
149C23×D3—98no—
150C23×D3—92no—
151C23×D3—188no—
152C2×C3×D4—24no—
153C2×C3×D4D3214yes—
154C2×C3×D4C2×C3⋊D311yescharacteristic
155C2×#67—98no—
156C2×#67—92no—
157C2×#67—188no—
158D3×D4—188no—
159D3×D4—188no—
160D3×D4—182no—
161D3×D4—182no—
162D3×D4—1816no—
163D3×D4—1816no—
164C2×D12—94no—
165C2×D12—91no—
166C2×D12—184no—
167C2×C4×D3—94no—
168C2×C4×D3—91no—
169C2×C4×D3—184no—
170C3×M16—38no—
171C3×M16—92no—
172#208—38no—
173#208C2×C3⋊D312yescomplemented
174ΓL(1,𝔽25)—182no—
175ΓL(1,𝔽25)—188no—
176D3×C8—182no—
177D3×C8—188no—
178C22×C32—216no—
179C22×C32—28no—
180C22×C32—22no—
181C22×C32—24no—
182C22×C32—42no—
183C22×C32—48no—
184C22×C32—416no—
185C22×C32C2×C4×D314yes—
186C22×C32C2×C4×D311yescharacteristic
187C2×C3⋊D3—31no—
188C2×C3⋊D3—31no—
189C2×C3⋊D3—34no—
190C2×C3⋊D3—34no—
191C2×C3⋊D3—64no—
192C2×C3⋊D3—64no—
193C2×C3⋊D3—62no—
194C2×C3⋊D3—68no—
195C2×C3⋊D3—616no—
196C2×C3⋊D3—616no—
197C2×C3⋊D3—68no—
198C2×C3⋊D3—62no—
199C2×C3⋊D3—124no—
200C2×C3⋊D3—1216no—
201C2×C3⋊D3—1232no—
202C2×C3⋊D3—1232no—
203C2×C3⋊D3—1216no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—1216no—
206C2×C3⋊D3—128no—
207C2×C3⋊D3—122no—
208C32×C4—28no—
209C32×C4D3×D418yes—
210C32×C4D3×D412yes—
211C3⋊Dic3—32no—
212C3⋊Dic3—38no—
213C3⋊Dic3—68no—
214C22×D4—271no—
215C2×M16—272no—
216#100—31no—
217#100—91no—
218#100—272no—
219C33C2×#10011yescharacteristic, complemented, Sylow
220C23×C3—28no—
221C23×C3#62418yes—
222C23×C3C4×C3⋊D312yes—
223C22×D3—94no—
224C22×D3—91no—
225C22×D3—98no—
226C22×D3—92no—
227C22×D3—184no—
228C22×D3—188no—
229C22×D3—182no—
230C22×D3—188no—
231C22×D3—182no—
232C22×D3—188no—
233C22×D3—1816no—
234C22×D3—1816no—
235C22×D3—3632no—
236C22×D3—3616no—
237C22×D3—364no—
238C3×D4—216no—
239C3×D4—28no—
240C3×D4—22no—
241C2×C3×C4—24no—
242C2×C3×C4C2×D3214yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244#67—184no—
245#67—1816no—
246#67—1832no—
247C2×Dic3—94no—
248C2×Dic3—91no—
249C2×Dic3—184no—
250D12—1816no—
251D12—188no—
252D12—182no—
253C4×D3—98no—
254C4×D3—98no—
255C4×D3—92no—
256C4×D3—92no—
257C4×D3—188no—
258C4×D3—188no—
259C3×C8—68no—
260C3×C8—182no—
261C3⋊C8—22no—
262C3⋊C8—68no—
263C2×C32—28no—
264C2×C32—24no—
265C2×C32—21no—
266C2×C32—24no—
267C2×C32—416no—
268C2×C32—48no—
269C2×C32—42no—
270C2×C32D3×C22⋊C414yes—
271C2×C32D3×C22⋊C411yescharacteristic
272C3⋊D3—32no—
273C3⋊D3—38no—
274C3⋊D3—68no—
275C3⋊D3—68no—
276C3⋊D3—64no—
277C3⋊D3—61no—
278C3⋊D3—1216no—
279C3⋊D3—128no—
280C3⋊D3—122no—
281C24—272no—
282C2×D4—272no—
283C2×D4—271no—
284C2×D4—542no—
285C2×D4—542no—
286C2×D4D3×C3⋊D311yescharacteristic
287C22×C4—271no—
288M16—32no—
289M16—92no—
290M16—542no—
291C2×C8—542no—
292C22×C3—216no—
293C22×C3—22no—
294C22×C3—28no—
295C22×C3—24no—
296C22×C3—48no—
297C22×C3—42no—
298C22×C3—416no—
299C22×C3C2×#62414yes—
300C22×C3C2×C4×C3⋊D311yescharacteristic
301C2×D3—91no—
302C2×D3—94no—
303C2×D3—91no—
304C2×D3—94no—
305C2×D3—184no—
306C2×D3—184no—
307C2×D3—188no—
308C2×D3—182no—
309C2×D3—1816no—
310C2×D3—1816no—
311C2×D3—182no—
312C2×D3—188no—
313C2×D3—3616no—
314C2×D3—364no—
315C2×D3—3632no—
316C2×D3—3632no—
317C2×D3—3616no—
318C2×D3—364no—
319C2×D3—3616no—
320C2×D3—368no—
321C2×D3—362no—
322C3×C4—28no—
323C3×C4#381118yes—
324C3×C4D4×C3⋊D312yes—
325Dic3—92no—
326Dic3—98no—
327Dic3—188no—
328C32—24no—
329C32D3×#10014yescomplemented
330C32D3×#10011yescharacteristic, complemented
331C23—271no—
332C23—272no—
333C23—542no—
334C23—542no—
335C23—1084no—
336C23#628912yes—
337D4—22no—
338D4—544no—
339D4—542no—
340C2×C4—271no—
341C2×C4—272no—
342C2×C4—272no—
343C2×C4C2×D3×C3⋊D311yescharacteristic
344C8—62no—
345C8—182no—
346C2×C3—28no—
347C2×C3—24no—
348C2×C3—21no—
349C2×C3—24no—
350C2×C3—416no—
351C2×C3—48no—
352C2×C3—42no—
353C2×C3#6397414yes—
354C2×C3C22⋊C4×C3⋊D311yescharacteristic
355D3—98no—
356D3—92no—
357D3—188no—
358D3—188no—
359D3—184no—
360D3—181no—
361D3—3616no—
362D3—368no—
363D3—362no—
364C22—22no—
365C22—42no—
366C22—271no—
367C22—271no—
368C22—542no—
369C22—542no—
370C22—1082no—
371C22—1084no—
372C22—1084no—
373C22C2×#628911yescharacteristic
374C4—272no—
375C4#8903912yes—
376C3—24no—
377C3#1059150314yescomplemented
378C3C3⋊D3×#10011yescharacteristic, complemented
379C2—21no—
380C2—42no—
381C2—272no—
382C2—541no—
383C2—1082no—
384C2#1172362711yescharacteristic
subgroupG/H
centerC2#11723627
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×D4×C3⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#100
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723358C2semidirect
3. C3×#63660C2semidirect
4. C32×#833C2semidirect
5. #11722027C2semidirect
15. C3×#3748C22semidirect
16. C32×#208C22semidirect
25. #63660D3semidirect
28. C3×#833D3semidirect
69. #3748C2×D3semidirect
73. C3×#208C2×D3semidirect
96. #833C3⋊D3semidirect
148. C3⋊(C3⋊D3)#100semidirect
173. #208C2×C3⋊D3semidirect
219. C33C2×#100semidirect
329. C32D3×#100semidirect
330. C32D3×#100semidirect
377. C3#10591503semidirect
378. C3C3⋊D3×#100semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
#117233582
C3×#636602
C32×#8332
#117220272
C3⋊D3×#1003
#105915033