Finite Group Databasebeta

← #13161052

Subgroups — #13161052

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×#89078C211yesmaximal, characteristic
2#11723358C212yesmaximal, complemented
3C3×#63660C211yesmaximal, characteristic, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5#11722030C212yesmaximal, complemented
6#10592444—31nomaximal
7#10591504—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×#6312C412yes—
10C22×C3⋊(C3⋊Dic3)C412yes—
11#89078—22no—
12#89078—22no—
13C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×#6308C2211yescharacteristic
15C3×#3748C2212yes—
16C32×#208C2212yes—
17#88788—22no—
18#88785—22no—
19C2×#3830—31no—
20C2×#3830—34no—
21C2×#3830—64no—
22M16×C3⋊D3—32no—
23#63818—38no—
24C32×#100—31no—
25#63660D311yescharacteristic, complemented
26#63651—32no—
27C3×#833—34no—
28C3×#833D314yescomplemented
29#63561—38no—
30C23×C33C2×C412yes—
31C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34#6312—24no—
35C2×C3⋊(C3⋊Dic3)—22no—
36C2×C3⋊(C3⋊Dic3)—22no—
37C2×C3⋊(C3⋊Dic3)C2×C412yes—
38C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
39C4×C3⋊(C3⋊D3)D412yes—
40C4×C3⋊(C3⋊D3)D412yes—
41#6308—22no—
42C3×C3⋊(C3⋊C8)—22no—
43C32×C3⋊C8—22no—
44#4745—94no—
45#4745—91no—
46C2×C32×D4—24no—
47C2×C32×D4C2×D314yes—
48C2×C32×D4C2×D311yescharacteristic
49C2×#638—32no—
50C2×#638—38no—
51C2×#638—68no—
52C22×C3⋊Dic3—32no—
53C22×C3⋊Dic3—38no—
54C22×C3⋊Dic3—68no—
55#3830—62no—
56#3830—62no—
57#3830—68no—
58#3830—68no—
59#3830—616no—
60#3830—616no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—64no—
64C2×#634—31no—
65C2×#634—34no—
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×C33—22no—
76C22×C33—42no—
77C22×C33C22×C411yescharacteristic
78C2×C3⋊(C3⋊D3)—41no—
79C2×C3⋊(C3⋊D3)C22⋊C412yes—
80C33×C4C2×D412yes—
81C3⋊(C3⋊Dic3)—22no—
82C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
83C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
84C2×#237—94no—
85C2×#237—91no—
86C2×#237—184no—
87D3×M16—98no—
88D3×M16—92no—
89C3×#100—34no—
90C3×#100—91no—
91#833—34no—
92#833C3⋊D311yescharacteristic, complemented
93#824—98no—
94#824—92no—
95C23×C32—28no—
96C23×C32C4×D318yes—
97C23×C32C4×D312yes—
98C22×C3⋊D3—31no—
99C22×C3⋊D3—34no—
100C22×C3⋊D3—64no—
101C32×D4—216no—
102C32×D4—28no—
103C32×D4—22no—
104C2×C32×C4—24no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D311yescharacteristic
107#638—64no—
108#638—616no—
109#638—632no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—32no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—38no—
114C2×C3⋊Dic3—68no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—62no—
117C2×C3⋊Dic3—62no—
118C2×C3⋊Dic3—68no—
119C2×C3⋊Dic3—68no—
120C2×C3⋊Dic3—616no—
121C2×C3⋊Dic3—616no—
122C4×C3⋊D3—32no—
123C4×C3⋊D3—32no—
124C4×C3⋊D3—38no—
125C4×C3⋊D3—38no—
126C4×C3⋊D3—68no—
127C4×C3⋊D3—68no—
128#634—616no—
129#634—68no—
130#634—62no—
131C32×C8—62no—
132C3⋊(C3⋊C8)—22no—
133C3×C3⋊C8—28no—
134C3×C3⋊C8—68no—
135#413—271noSylow
136C2×C33—21no—
137C2×C33—42no—
138C2×C33C2×C22⋊C411yescharacteristic
139C3⋊(C3⋊D3)—22no—
140C2×C3×D4—24no—
141C2×C3×D4D3214yes—
142C2×C3×D4C2×C3⋊D311yescharacteristic
143C2×#67—98no—
144C2×#67—92no—
145C2×#67—188no—
146C22×Dic3—98no—
147C22×Dic3—92no—
148C22×Dic3—188no—
149#237—188no—
150#237—188no—
151#237—182no—
152#237—182no—
153#237—1816no—
154#237—1816no—
155C2×C4×D3—94no—
156C2×C4×D3—91no—
157C2×C4×D3—184no—
158C2×Dic6—94no—
159C2×Dic6—91no—
160C2×Dic6—184no—
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—216no—
170C22×C32—28no—
171C22×C32—22no—
172C22×C32—24no—
173C22×C32—42no—
174C22×C32—48no—
175C22×C32—416no—
176C22×C32C2×C4×D314yes—
177C22×C32C2×C4×D311yescharacteristic
178C2×C3⋊D3—32no—
179C2×C3⋊D3—38no—
180C2×C3⋊D3—68no—
181C2×C3⋊D3—121no—
182C2×C3⋊D3—124no—
183C2×C3⋊D3—128no—
184C32×C4—28no—
185C32×C4D3×D418yes—
186C32×C4D3×D412yes—
187C3⋊Dic3—31no—
188C3⋊Dic3—31no—
189C3⋊Dic3—34no—
190C3⋊Dic3—34no—
191C3⋊Dic3—64no—
192C3⋊Dic3—64no—
193C3⋊Dic3—616no—
194C3⋊Dic3—68no—
195C3⋊Dic3—62no—
196C2×#41—271no—
197C2×M16—272no—
198#101—272no—
199#100—31no—
200#100—91no—
201C33#41311yescharacteristic, complemented, Sylow
202C23×C3—28no—
203C23×C3#62418yes—
204C23×C3C4×C3⋊D312yes—
205C22×D3—94no—
206C22×D3—91no—
207C22×D3—184no—
208C3×D4—216no—
209C3×D4—28no—
210C3×D4—22no—
211C2×C3×C4—24no—
212C2×C3×C4C2×D3214yes—
213C2×C3×C4C22×C3⋊D311yescharacteristic
214#67—184no—
215#67—1816no—
216#67—1832no—
217C2×Dic3—94no—
218C2×Dic3—91no—
219C2×Dic3—98no—
220C2×Dic3—92no—
221C2×Dic3—188no—
222C2×Dic3—184no—
223C2×Dic3—182no—
224C2×Dic3—182no—
225C2×Dic3—188no—
226C2×Dic3—188no—
227C2×Dic3—1816no—
228C2×Dic3—1816no—
229C4×D3—98no—
230C4×D3—98no—
231C4×D3—92no—
232C4×D3—92no—
233C4×D3—188no—
234C4×D3—188no—
235Dic6—1816no—
236Dic6—188no—
237Dic6—182no—
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×C32—416no—
247C2×C32—48no—
248C2×C32—42no—
249C2×C32D3×C22⋊C414yes—
250C2×C32D3×C22⋊C411yescharacteristic
251C3⋊D3—62no—
252C3⋊D3—68no—
253C3⋊D3—128no—
254#41—542no—
255#41—542no—
256C2×Q8—271no—
257C2×D4—272no—
258C2×D4D3×C3⋊D311yescharacteristic
259C22×C4—271no—
260C22×C4—272no—
261M16—32no—
262M16—92no—
263M16—542no—
264C2×C8—542no—
265C22×C3—216no—
266C22×C3—22no—
267C22×C3—28no—
268C22×C3—24no—
269C22×C3—48no—
270C22×C3—42no—
271C22×C3—416no—
272C22×C3C2×#62414yes—
273C22×C3C2×C4×C3⋊D311yescharacteristic
274C2×D3—92no—
275C2×D3—98no—
276C2×D3—188no—
277C2×D3—364no—
278C2×D3—361no—
279C2×D3—368no—
280C3×C4—28no—
281C3×C4#381118yes—
282C3×C4D4×C3⋊D312yes—
283Dic3—91no—
284Dic3—91no—
285Dic3—94no—
286Dic3—94no—
287Dic3—184no—
288Dic3—184no—
289Dic3—1816no—
290Dic3—182no—
291Dic3—188no—
292C32—24no—
293C32#474514yescomplemented
294C32#474511yescharacteristic, complemented
295C23—271no—
296C23#628912yes—
297Q8—542no—
298D4—22no—
299D4—544no—
300C2×C4—271no—
301C2×C4—272no—
302C2×C4—272no—
303C2×C4—272no—
304C2×C4—542no—
305C2×C4—542no—
306C2×C4C2×D3×C3⋊D311yescharacteristic
307C8—62no—
308C8—182no—
309C2×C3—28no—
310C2×C3—24no—
311C2×C3—21no—
312C2×C3—24no—
313C2×C3—416no—
314C2×C3—48no—
315C2×C3—42no—
316C2×C3#6397414yes—
317C2×C3C22⋊C4×C3⋊D311yescharacteristic
318D3—188no—
319D3—182no—
320D3—368no—
321C22—22no—
322C22—42no—
323C22—272no—
324C22—1081no—
325C22C2×#628911yescharacteristic
326C4—271no—
327C4—271no—
328C4—542no—
329C4#8903912yes—
330C3—24no—
331C3#1059150414yescomplemented
332C3#1059244411yescharacteristic, complemented
333C2—21no—
334C2—42no—
335C2—542no—
336C2#1172362711yescharacteristic
subgroupG/H
centerC2#11723627
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×#89078C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#413
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. #11722030C2semidirect
25. #63660D3semidirect
28. C3×#833D3semidirect
92. #833C3⋊D3semidirect
201. C33#413semidirect
293. C32#4745semidirect
294. C32#4745semidirect
331. C3#10591504semidirect
332. C3#10592444semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
#117233582
C3×#636602
C32×#8332
#117220302
#105924443
#105915043