Finite Group Databasebeta

← #13161278

Subgroups — #13161278

Order 1728.

1500 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1500 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×Q8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723358C212yesmaximal
3C3×#63663C211yesmaximal, characteristic, complemented
4C32×#836C211yesmaximal, characteristic, complemented
5#11722030C212yesmaximal, complemented
6C3⋊D3×#101—31nomaximal
7#10591639—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9Q8×C3⋊(C3⋊D3)—22no—
10Q8×C3⋊(C3⋊D3)—22no—
11C2×C4×C3⋊(C3⋊D3)C412yes—
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×#6308C412yes—
14C2×#6308C2211yescharacteristic
15C3×#3748C2212yes—
16C32×#208C2212yes—
17#88788—22no—
18#88785—22no—
19C2×Q8×C3⋊D3—31no—
20C2×Q8×C3⋊D3—34no—
21C2×Q8×C3⋊D3—64no—
22M16×C3⋊D3—32no—
23#63818—38no—
24C32×#101—31no—
25#63663D311yescharacteristic, complemented
26#63651—32no—
27C3×#836—34no—
28C3×#836D314yescomplemented
29#63561—38no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C33×Q8—22no—
32C2×C33×C4C2×C412yes—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)C2×C412yes—
35C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
36C4×C3⋊(C3⋊D3)—22no—
37C4×C3⋊(C3⋊D3)—22no—
38C4×C3⋊(C3⋊D3)D412yes—
39C4×C3⋊(C3⋊D3)D412yes—
40#6308—22no—
41#6308—24no—
42C3×C3⋊(C3⋊C8)—22no—
43C32×C3⋊C8—22no—
44D3×#101—94no—
45D3×#101—91no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C2×D311yescharacteristic
49Q8×C3⋊D3—62no—
50Q8×C3⋊D3—62no—
51Q8×C3⋊D3—68no—
52Q8×C3⋊D3—68no—
53Q8×C3⋊D3—616no—
54Q8×C3⋊D3—616no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—34no—
57C2×C4×C3⋊D3—32no—
58C2×C4×C3⋊D3—38no—
59C2×C4×C3⋊D3—68no—
60C2×C4×C3⋊D3—64no—
61C2×#634—32no—
62C2×#634—38no—
63C2×#634—31no—
64C2×#634—34no—
65C2×#634—68no—
66C2×#634—64no—
67C32×M16—32no—
68#3748C2×D312yes—
69#3743—62no—
70C8×C3⋊D3—62no—
71C3×#208—38no—
72C3×#208C2×D318yes—
73#3712—68no—
74#3710—68no—
75C22×C33C22×C411yescharacteristic
76C2×C3⋊(C3⋊D3)—22no—
77C2×C3⋊(C3⋊D3)C22⋊C411yescharacteristic
78C2×C3⋊(C3⋊D3)C22⋊C411yescharacteristic
79C33×C4—22no—
80C33×C4C2×D412yes—
81C3⋊(C3⋊Dic3)—22no—
82C3⋊(C3⋊Dic3)C22⋊C412yes—
83C2×D3×Q8—94no—
84C2×D3×Q8—91no—
85C2×D3×Q8—184no—
86D3×M16—98no—
87D3×M16—92no—
88C3×#101—34no—
89C3×#101—91no—
90#836—34no—
91#836C3⋊D311yescharacteristic, complemented
92#824—98no—
93#824—92no—
94C22×C3⋊D3—31no—
95C22×C3⋊D3—34no—
96C22×C3⋊D3—64no—
97C32×Q8—216no—
98C32×Q8—28no—
99C32×Q8—22no—
100C2×C32×C4—28no—
101C2×C32×C4—24no—
102C2×C32×C4C4×D318yes—
103C2×C32×C4C4×D312yes—
104C2×C32×C4C22×D314yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—32no—
108C2×C3⋊Dic3—34no—
109C2×C3⋊Dic3—38no—
110C2×C3⋊Dic3—64no—
111C2×C3⋊Dic3—68no—
112C4×C3⋊D3—32no—
113C4×C3⋊D3—32no—
114C4×C3⋊D3—38no—
115C4×C3⋊D3—38no—
116C4×C3⋊D3—68no—
117C4×C3⋊D3—68no—
118C4×C3⋊D3—62no—
119C4×C3⋊D3—62no—
120C4×C3⋊D3—68no—
121C4×C3⋊D3—68no—
122C4×C3⋊D3—616no—
123C4×C3⋊D3—616no—
124#634—64no—
125#634—616no—
126#634—632no—
127#634—616no—
128#634—68no—
129#634—62no—
130C32×C8—62no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—28no—
133C3×C3⋊C8—68no—
134C2×#101—271noSylow
135C2×C33—21no—
136C2×C33C2×C22⋊C411yescharacteristic
137C3⋊(C3⋊D3)—21no—
138C3⋊(C3⋊D3)#10112yescomplemented
139C2×C3×Q8—24no—
140C2×C3×Q8D3214yes—
141C2×C3×Q8C2×C3⋊D311yescharacteristic
142D3×Q8—188no—
143D3×Q8—188no—
144D3×Q8—182no—
145D3×Q8—182no—
146D3×Q8—1816no—
147D3×Q8—1816no—
148C2×C4×D3—94no—
149C2×C4×D3—91no—
150C2×C4×D3—98no—
151C2×C4×D3—92no—
152C2×C4×D3—184no—
153C2×C4×D3—188no—
154C2×Dic6—98no—
155C2×Dic6—92no—
156C2×Dic6—94no—
157C2×Dic6—91no—
158C2×Dic6—188no—
159C2×Dic6—184no—
160C3×M16—38no—
161C3×M16—92no—
162#208—38no—
163#208C2×C3⋊D312yes—
164ΓL(1,𝔽25)—182no—
165ΓL(1,𝔽25)—188no—
166D3×C8—182no—
167D3×C8—188no—
168C22×C32—24no—
169C22×C32C2×C4×D314yes—
170C22×C32C2×C4×D311yescharacteristic
171C2×C3⋊D3—31no—
172C2×C3⋊D3—31no—
173C2×C3⋊D3—34no—
174C2×C3⋊D3—34no—
175C2×C3⋊D3—64no—
176C2×C3⋊D3—64no—
177C2×C3⋊D3—62no—
178C2×C3⋊D3—68no—
179C2×C3⋊D3—616no—
180C32×C4—216no—
181C32×C4—28no—
182C32×C4—22no—
183C32×C4—28no—
184C32×C4D3×D418yes—
185C32×C4D3×D412yes—
186C3⋊Dic3—32no—
187C3⋊Dic3—38no—
188C3⋊Dic3—68no—
189C3⋊Dic3—616no—
190C3⋊Dic3—68no—
191C3⋊Dic3—62no—
192C22×Q8—271no—
193C2×M16—272no—
194#101—31no—
195#101—91no—
196#101—272no—
197C33C2×#10111yescharacteristic, complemented, Sylow
198C22×D3—94no—
199C22×D3—91no—
200C22×D3—184no—
201C3×Q8—216no—
202C3×Q8—28no—
203C3×Q8—22no—
204C2×C3×C4—28no—
205C2×C3×C4—24no—
206C2×C3×C4#62418yes—
207C2×C3×C4C4×C3⋊D312yes—
208C2×C3×C4C2×D3214yes—
209C2×C3×C4C22×C3⋊D311yescharacteristic
210C2×Dic3—94no—
211C2×Dic3—91no—
212C2×Dic3—98no—
213C2×Dic3—92no—
214C2×Dic3—184no—
215C2×Dic3—188no—
216C4×D3—98no—
217C4×D3—98no—
218C4×D3—92no—
219C4×D3—92no—
220C4×D3—188no—
221C4×D3—188no—
222C4×D3—182no—
223C4×D3—182no—
224C4×D3—188no—
225C4×D3—188no—
226C4×D3—1816no—
227C4×D3—1816no—
228Dic6—184no—
229Dic6—1816no—
230Dic6—1832no—
231Dic6—1816no—
232Dic6—188no—
233Dic6—182no—
234C3×C8—68no—
235C3×C8—182no—
236C3⋊C8—22no—
237C3⋊C8—68no—
238C2×C32—28no—
239C2×C32—24no—
240C2×C32—21no—
241C2×C32—24no—
242C2×C32D3×C22⋊C414yes—
243C2×C32D3×C22⋊C411yescharacteristic
244C3⋊D3—32no—
245C3⋊D3—38no—
246C3⋊D3—68no—
247C3⋊D3—68no—
248C3⋊D3—64no—
249C3⋊D3—61no—
250C2×Q8—272no—
251C2×Q8—271no—
252C2×Q8—542no—
253C2×Q8—542no—
254C2×Q8D3×C3⋊D311yescharacteristic
255C22×C4—271no—
256C22×C4—272no—
257M16—32no—
258M16—92no—
259M16—542no—
260C2×C8—542no—
261C22×C3—24no—
262C22×C3C2×#62414yes—
263C22×C3C2×C4×C3⋊D311yescharacteristic
264C2×D3—91no—
265C2×D3—94no—
266C2×D3—91no—
267C2×D3—94no—
268C2×D3—184no—
269C2×D3—184no—
270C2×D3—188no—
271C2×D3—182no—
272C2×D3—1816no—
273C3×C4—216no—
274C3×C4—22no—
275C3×C4—28no—
276C3×C4—28no—
277C3×C4#381118yes—
278C3×C4D4×C3⋊D312yes—
279Dic3—92no—
280Dic3—98no—
281Dic3—188no—
282Dic3—1816no—
283Dic3—182no—
284Dic3—188no—
285C32—24no—
286C32D3×#10114yescomplemented
287C32D3×#10111yescharacteristic, complemented
288C23—271no—
289Q8—22no—
290Q8—544no—
291Q8—542no—
292C2×C4—271no—
293C2×C4—272no—
294C2×C4—272no—
295C2×C4—272no—
296C2×C4—542no—
297C2×C4—542no—
298C2×C4#628912yes—
299C2×C4C2×D3×C3⋊D311yescharacteristic
300C8—62no—
301C8—182no—
302C2×C3—28no—
303C2×C3—24no—
304C2×C3—21no—
305C2×C3—24no—
306C2×C3#6397414yes—
307C2×C3C22⋊C4×C3⋊D311yescharacteristic
308D3—98no—
309D3—92no—
310D3—188no—
311D3—188no—
312D3—184no—
313D3—181no—
314C22—271no—
315C22—271no—
316C22—542no—
317C22C2×#628911yescharacteristic
318C4—22no—
319C4—272no—
320C4—542no—
321C4#8903912yes—
322C3—24no—
323C3#1059163914yescomplemented
324C3C3⋊D3×#10111yescharacteristic, complemented
325C2—21no—
326C2—272no—
327C2—541no—
328C2#1172362711yescharacteristic
subgroupG/H
centerC2#11723627
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33C2×C22⋊C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C2×#101
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63663C2semidirect
4. C32×#836C2semidirect
5. #11722030C2semidirect
25. #63663D3semidirect
28. C3×#836D3semidirect
91. #836C3⋊D3semidirect
138. C3⋊(C3⋊D3)#101semidirect
197. C33C2×#101semidirect
286. C32D3×#101semidirect
287. C32D3×#101semidirect
323. C3#10591639semidirect
324. C3C3⋊D3×#101semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊(C3⋊D3)2
#117233582
C3×#636632
C32×#8362
#117220302
C3⋊D3×#1013
#105916393