Finite Group Databasebeta

← #13161279

Subgroups — #13161279

Order 1728.

1556 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1556 conjugacy classes of subgroups — too many to draw. Only its 156 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×#89080C211yesmaximal, characteristic
2#11723358C212yesmaximal, complemented
3C3×#63663C211yesmaximal, characteristic, complemented
4C32×#836C211yesmaximal, characteristic, complemented
5#11722027C212yesmaximal, complemented
6#10592450—31nomaximal
7#10591640—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9#89080—22no—
10#89080—22no—
11C2×C4⋊(C3⋊(C3⋊D3))C412yes—
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×C4×C3⋊(C3⋊D3)C412yes—
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C3×#3748C2212yes—
16C32×#208C2212yes—
17#88788—22no—
18#88785—22no—
19C2×#3832—31no—
20C2×#3832—34no—
21C2×#3832—64no—
22M16×C3⋊D3—32no—
23#63818—38no—
24C32×#101—31no—
25#63663D311yescharacteristic, complemented
26#63650—32no—
27C3×#836—34no—
28C3×#836D314yescomplemented
29#63559—38no—
30C22×C3⋊(C3⋊D3)C2×C412yes—
31C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
32C33×Q8—22no—
33C2×C33×C4C2×C412yes—
34C2×C33×C4C2311yescharacteristic
35C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
36C4⋊(C3⋊(C3⋊D3))—24no—
37C4⋊(C3⋊(C3⋊D3))—22no—
38C4×C3⋊(C3⋊D3)—22no—
39C4×C3⋊(C3⋊D3)—22no—
40C4×C3⋊(C3⋊D3)D412yes—
41C4×C3⋊(C3⋊D3)D412yes—
42C3×C3⋊(C3⋊C8)—22no—
43C32×C3⋊C8—22no—
44#4751—94no—
45#4751—91no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C2×D311yescharacteristic
49#3832—62no—
50#3832—62no—
51#3832—68no—
52#3832—68no—
53#3832—616no—
54#3832—616no—
55C2×C4⋊(C3⋊D3)—32no—
56C2×C4⋊(C3⋊D3)—38no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—68no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—32no—
64C2×C4×C3⋊D3—38no—
65C2×C4×C3⋊D3—68no—
66C2×C4×C3⋊D3—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)—41no—
78C2×C3⋊(C3⋊D3)—42no—
79C2×C3⋊(C3⋊D3)C22⋊C412yes—
80C33×C4—22no—
81C33×C4C2×D412yes—
82C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
83C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
84C2×#239—94no—
85C2×#239—91no—
86C2×#239—184no—
87D3×M16—98no—
88D3×M16—92no—
89C3×#101—34no—
90C3×#101—91no—
91#836—34no—
92#836C3⋊D311yescharacteristic, complemented
93#823—98no—
94#823—92no—
95C22×C3⋊D3—31no—
96C22×C3⋊D3—32no—
97C22×C3⋊D3—34no—
98C22×C3⋊D3—38no—
99C22×C3⋊D3—64no—
100C22×C3⋊D3—68no—
101C32×Q8—216no—
102C32×Q8—28no—
103C32×Q8—22no—
104C2×C32×C4—28no—
105C2×C32×C4—24no—
106C2×C32×C4C4×D318yes—
107C2×C32×C4C4×D312yes—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—34no—
112C2×C3⋊Dic3—64no—
113C4⋊(C3⋊D3)—64no—
114C4⋊(C3⋊D3)—616no—
115C4⋊(C3⋊D3)—632no—
116C4⋊(C3⋊D3)—616no—
117C4⋊(C3⋊D3)—68no—
118C4⋊(C3⋊D3)—62no—
119C4×C3⋊D3—32no—
120C4×C3⋊D3—32no—
121C4×C3⋊D3—38no—
122C4×C3⋊D3—38no—
123C4×C3⋊D3—68no—
124C4×C3⋊D3—68no—
125C4×C3⋊D3—62no—
126C4×C3⋊D3—62no—
127C4×C3⋊D3—68no—
128C4×C3⋊D3—68no—
129C4×C3⋊D3—616no—
130C4×C3⋊D3—616no—
131C32×C8—62no—
132C3⋊(C3⋊C8)—22no—
133C3×C3⋊C8—28no—
134C3×C3⋊C8—68no—
135#413—271noSylow
136C2×C33—21no—
137C2×C33C2×C22⋊C411yescharacteristic
138C3⋊(C3⋊D3)—22no—
139C3⋊(C3⋊D3)—42no—
140C2×C3×Q8—24no—
141C2×C3×Q8D3214yes—
142C2×C3×Q8C2×C3⋊D311yescharacteristic
143#239—188no—
144#239—188no—
145#239—182no—
146#239—182no—
147#239—1816no—
148#239—1816no—
149C2×D12—98no—
150C2×D12—92no—
151C2×D12—94no—
152C2×D12—91no—
153C2×D12—188no—
154C2×D12—184no—
155C2×C4×D3—94no—
156C2×C4×D3—91no—
157C2×C4×D3—98no—
158C2×C4×D3—92no—
159C2×C4×D3—184no—
160C2×C4×D3—188no—
161C3×M16—38no—
162C3×M16—92no—
163#208—38no—
164#208C2×C3⋊D312yes—
165ΓL(1,𝔽25)—182no—
166ΓL(1,𝔽25)—188no—
167D3×C8—182no—
168D3×C8—188no—
169C22×C32—24no—
170C22×C32C2×C4×D314yes—
171C22×C32C2×C4×D311yescharacteristic
172C2×C3⋊D3—32no—
173C2×C3⋊D3—38no—
174C2×C3⋊D3—68no—
175C2×C3⋊D3—616no—
176C2×C3⋊D3—68no—
177C2×C3⋊D3—62no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—124no—
180C2×C3⋊D3—128no—
181C2×C3⋊D3—1216no—
182C2×C3⋊D3—128no—
183C2×C3⋊D3—122no—
184C32×C4—216no—
185C32×C4—28no—
186C32×C4—22no—
187C32×C4—28no—
188C32×C4D3×D418yes—
189C32×C4D3×D412yes—
190C3⋊Dic3—31no—
191C3⋊Dic3—31no—
192C3⋊Dic3—34no—
193C3⋊Dic3—34no—
194C3⋊Dic3—64no—
195C3⋊Dic3—64no—
196C2×#41—271no—
197C2×M16—272no—
198#101—31no—
199#101—91no—
200#100—272no—
201C33#41311yescharacteristic, complemented, Sylow
202C22×D3—94no—
203C22×D3—91no—
204C22×D3—98no—
205C22×D3—92no—
206C22×D3—184no—
207C22×D3—188no—
208C3×Q8—216no—
209C3×Q8—28no—
210C3×Q8—22no—
211C2×C3×C4—28no—
212C2×C3×C4—24no—
213C2×C3×C4#62418yes—
214C2×C3×C4C4×C3⋊D312yes—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—94no—
218C2×Dic3—91no—
219C2×Dic3—184no—
220D12—184no—
221D12—1816no—
222D12—1832no—
223D12—1816no—
224D12—188no—
225D12—182no—
226C4×D3—98no—
227C4×D3—98no—
228C4×D3—92no—
229C4×D3—92no—
230C4×D3—188no—
231C4×D3—188no—
232C4×D3—182no—
233C4×D3—182no—
234C4×D3—188no—
235C4×D3—188no—
236C4×D3—1816no—
237C4×D3—1816no—
238C3×C8—68no—
239C3×C8—182no—
240C3⋊C8—22no—
241C3⋊C8—68no—
242C2×C32—28no—
243C2×C32—24no—
244C2×C32—21no—
245C2×C32—24no—
246C2×C32D3×C22⋊C414yes—
247C2×C32D3×C22⋊C411yescharacteristic
248C3⋊D3—62no—
249C3⋊D3—68no—
250C3⋊D3—128no—
251C3⋊D3—1216no—
252C3⋊D3—128no—
253C3⋊D3—122no—
254#41—542no—
255#41—542no—
256C2×Q8D3×C3⋊D311yescharacteristic
257C2×D4—272no—
258C2×D4—271no—
259C22×C4—271no—
260C22×C4—272no—
261M16—32no—
262M16—92no—
263M16—542no—
264C2×C8—542no—
265C22×C3—24no—
266C22×C3C2×#62414yes—
267C22×C3C2×C4×C3⋊D311yescharacteristic
268C2×D3—92no—
269C2×D3—98no—
270C2×D3—188no—
271C2×D3—1816no—
272C2×D3—182no—
273C2×D3—188no—
274C2×D3—364no—
275C2×D3—361no—
276C2×D3—368no—
277C2×D3—3616no—
278C2×D3—368no—
279C2×D3—362no—
280C3×C4—216no—
281C3×C4—22no—
282C3×C4—28no—
283C3×C4—28no—
284C3×C4#381118yes—
285C3×C4D4×C3⋊D312yes—
286Dic3—91no—
287Dic3—91no—
288Dic3—94no—
289Dic3—94no—
290Dic3—184no—
291Dic3—184no—
292C32—24no—
293C32#475114yescomplemented
294C32#475111yescharacteristic, complemented
295C23—271no—
296C23—272no—
297Q8—22no—
298D4—544no—
299D4—542no—
300C2×C4—271no—
301C2×C4—272no—
302C2×C4—272no—
303C2×C4—542no—
304C2×C4—542no—
305C2×C4#628912yes—
306C2×C4C2×D3×C3⋊D311yescharacteristic
307C8—62no—
308C8—182no—
309C2×C3—28no—
310C2×C3—24no—
311C2×C3—21no—
312C2×C3—24no—
313C2×C3#6397414yes—
314C2×C3C22⋊C4×C3⋊D311yescharacteristic
315D3—188no—
316D3—182no—
317D3—368no—
318D3—3616no—
319D3—368no—
320D3—362no—
321C22—272no—
322C22—542no—
323C22—1081no—
324C22—1082no—
325C22C2×#628911yescharacteristic
326C4—22no—
327C4—271no—
328C4—271no—
329C4#8903912yes—
330C3—24no—
331C3#1059164014yescomplemented
332C3#1059245011yescharacteristic, complemented
333C2—21no—
334C2—542no—
335C2—1082no—
336C2#1172362711yescharacteristic
subgroupG/H
centerC2#11723627
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33C2×C22⋊C4
prime radicalC2×#89080C2
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#413
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
2. #11723358C2semidirect
3. C3×#63663C2semidirect
4. C32×#836C2semidirect
5. #11722027C2semidirect
25. #63663D3semidirect
28. C3×#836D3semidirect
92. #836C3⋊D3semidirect
201. C33#413semidirect
293. C32#4751semidirect
294. C32#4751semidirect
331. C3#10591640semidirect
332. C3#10592450semidirect

Maximal subgroups

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

subgroupindex
C2×#890802
#117233582
C3×#636632
C32×#8362
#117220272
#105924503
#105916403