Finite Group Databasebeta

← #13156250

Subgroups — #13156250

Order 1728.

558 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#11723349C211yesmaximal, characteristic
2C2×C3×#3718C211yesmaximal, characteristic, complemented
3C3×#63827C212yesmaximal
4#11722081C211yesmaximal, characteristic, complemented
5#11722028C212yesmaximal, complemented
6#10591683—31nomaximal
7#10591641—32nomaximal
8C32×#235C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10#88855C2211yescharacteristic
11C3×#3748C2212yes—
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13#88786—21no—
14D3×C3⋊(C3⋊C8)—21no—
15C3×#3718—22no—
16C3×#3718—21no—
17C3×#3718—21no—
18C3×#3715—22no—
19#64157—31no—
20C3×#951—32no—
21C2×C3×#216—32no—
22C3×#942—32no—
23C2×#3718D311yescharacteristic, complemented
24#63827D312yes—
25#63814—32no—
26#63814—61no—
27#63663—32no—
28#63578—31no—
29#63577—32no—
30#63560—32no—
31#63560—32no—
32#63560—62no—
33C2×C33×C4C2311yescharacteristic
34C32×#67—21no—
35C2×C32×Dic3—22no—
36C32×D12D411yescharacteristic
37C32×C4×D3—21no—
38C32×Dic6—22no—
39C32×Dic6—22no—
40C32×Dic6D411yescharacteristic
41C3⋊(C3⋊(C3⋊C8))—21no—
42C3×C3⋊(C3⋊C8)—22no—
43C3×C3⋊(C3⋊C8)D411yescharacteristic
44C3×C3⋊(C3⋊C8)D411yescharacteristic
45#5204—92no—
46#5192—91no—
47C32×#41—31no—
48C2×C32×Q8—32no—
49C3×#235—21no—
50C3×#235C2×D312yes—
51C2×C3×Dic6—22no—
52C2×C3×Dic6C2×D312yes—
53C2×C3×Dic6C2×D312yes—
54#3748—32no—
55#3748—31no—
56#3748—61no—
57#3748—61no—
58#3748—62no—
59#3748—121no—
60#3748C2×D312yes—
61C2×C3⋊(C3⋊C8)—61no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×#216—62no—
64C3×#216—62no—
65C3×#216—62no—
66C3×#216—62no—
67C3×#215—62no—
68C3×#214—62no—
69C3×#208—32no—
70C3×#208—32no—
71C3×#208—62no—
72C2×C3×C3⋊C8—32no—
73C2×C3×C3⋊C8—61no—
74#3718—22no—
75#3718—21no—
76#3718—21no—
77#3715—22no—
78#3711—62no—
79#3711—62no—
80D3×C3⋊C8—62no—
81D3×C3⋊C8—62no—
82C22×C33#4111yescharacteristic
83C2×C32×D3—21no—
84C33×C4C2×D411yescharacteristic
85C33×C4C2×D411yescharacteristic
86C32×Dic3—21no—
87C32×Dic3—42no—
88C3×#137—92no—
89C2×C3×Q16—91no—
90#951—32no—
91#948—92no—
92#948—181no—
93C2×#216—32no—
94#942—32no—
95#901—91no—
96#836—92no—
97#836—92no—
98#836—182no—
99#824—92no—
100#822—92no—
101#819—91no—
102C32×Q8—31no—
103C32×Q8—62no—
104C32×Q8—62no—
105C32×D4—31no—
106C32×D4—61no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—41no—
110C2×C32×C4—61no—
111C2×C32×C4—62no—
112C2×C32×C4C22×D312yes—
113C2×C32×C4C22×D311yescharacteristic
114C3×#67—22no—
115C3×#67—22no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C3×D12—21no—
120C3×D12#6712yes—
121C3×C4×D3—22no—
122C3×C4×D3—22no—
123C3×Dic6—22no—
124C3×Dic6—22no—
125C3×Dic6—22no—
126C3×Dic6—22no—
127C3×Dic6—21no—
128C3×Dic6—42no—
129C3×Dic6—42no—
130C3×Dic6#6712yes—
131C3⋊(C3⋊C8)—22no—
132C3⋊(C3⋊C8)—61no—
133C3⋊(C3⋊C8)—62no—
134C3⋊(C3⋊C8)—62no—
135C3⋊(C3⋊C8)—122no—
136C3⋊(C3⋊C8)—122no—
137C3⋊(C3⋊C8)#6711yescharacteristic
138C3⋊(C3⋊C8)#6711yescharacteristic
139C3×C3⋊C8—32no—
140C3×C3⋊C8—32no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—122no—
146#473—271noSylow
147C2×C33—21no—
148C2×C33#12111yescharacteristic
149C32×D3—41no—
150C3×#41—32no—
151C3×#41—61no—
152C2×C3×Q8—32no—
153C2×C3×Q8—32no—
154C2×C3×Q8—62no—
155#235D3211yescharacteristic
156C2×Dic6D3212yes—
157C3×Q16—181no—
158C3×Q16—181no—
159C3×Q16—182no—
160C3×SD16—182no—
161C3×M16—92no—
162C2×C3×C8—91no—
163#216—62no—
164#216—62no—
165#216—62no—
166#216—62no—
167#215—62no—
168#214—62no—
169#208—32no—
170#208—32no—
171#208—62no—
172#208—92no—
173#208—91no—
174#208—181no—
175#208—182no—
176#208—182no—
177#208—182no—
178#208—361no—
179C2×C3⋊C8—32no—
180C2×C3⋊C8—61no—
181C2×C3⋊C8—182no—
182C2×C3⋊C8—182no—
183ΓL(1,𝔽25)—181no—
184D3×C8—181no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—41no—
188C22×C32—61no—
189C22×C32#23711yescharacteristic
190C22×C32#23712yes—
191C2×C3×D3—22no—
192C2×C3×D3—22no—
193C32×C4—21no—
194C32×C4—22no—
195C32×C4—22no—
196C32×C4—21no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—61no—
200C32×C4—122no—
201C32×C4D3×D411yescharacteristic
202C32×C4D3×D412yes—
203C32×C4C2×#6712yes—
204C32×C4C2×#6711yescharacteristic
205C3×Dic3—22no—
206C3×Dic3—22no—
207C3×Dic3—44no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210#137—92no—
211C2×Q16—91no—
212#131—271no—
213#108—271no—
214#101—272no—
215C33#47311yescharacteristic, complemented, Sylow
216C3×Q8—32no—
217C3×Q8—61no—
218C3×Q8—62no—
219C3×Q8—62no—
220C3×Q8—62no—
221C3×Q8—62no—
222C3×Q8—122no—
223C3×Q8—122no—
224C3×D4—32no—
225C3×D4—61no—
226C3×D4—62no—
227C3×D4—62no—
228C2×C3×C4—21no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—64no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×C3×C4C2×D3212yes—
238#67—21no—
239C2×Dic3—22no—
240D12#62511yescharacteristic
241C4×D3—21no—
242Dic6—22no—
243Dic6—22no—
244Dic6#62511yescharacteristic
245C3×C8—91no—
246C3×C8—91no—
247C3×C8—182no—
248C3⋊C8—32no—
249C3⋊C8—32no—
250C3⋊C8—61no—
251C3⋊C8—61no—
252C3⋊C8—62no—
253C3⋊C8—62no—
254C3⋊C8—122no—
255C3⋊C8—182no—
256C3⋊C8—182no—
257C3⋊C8—181no—
258C3⋊C8—362no—
259C3⋊C8—362no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—21no—
264C2×C32—41no—
265C2×C32—42no—
266C2×C32—41no—
267C2×C32—81no—
268C2×C32—121no—
269C2×C32#93811yescharacteristic
270C2×C32#93912yes—
271C3×D3—42no—
272C3×D3—42no—
273#41—31no—
274C2×Q8—32no—
275Q16—181no—
276Q16—181no—
277Q16—182no—
278SD16—182no—
279M16—92no—
280M16—271no—
281M16—541no—
282C2×C8—91no—
283C2×C8—541no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—41no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3#380512yes—
290C22×C3#380511yescharacteristic
291C2×D3—21no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—22no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—62no—
299C3×C4—62no—
300C3×C4—124no—
301C3×C4—122no—
302C3×C4—122no—
303C3×C4C2×#62511yescharacteristic
304C3×C4D3×#6712yes—
305C3×C4D3×#6712yes—
306C3×C4#381111yescharacteristic
307Dic3—21no—
308Dic3—42no—
309C32—22no—
310C32—21no—
311C32—41no—
312C32#519211yescharacteristic, complemented
313C32#520412yescomplemented
314Q8—31no—
315Q8—62no—
316Q8—62no—
317D4—31no—
318D4—61no—
319C2×C4—61no—
320C2×C4—62no—
321C2×C4D3311yescharacteristic
322C8—91no—
323C8—91no—
324C8—182no—
325C8—541no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—41no—
331C2×C3—42no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3—122no—
335C2×C3—122no—
336C2×C3#6397012yes—
337C2×C3#6398011yescharacteristic
338D3—41no—
339C22—61no—
340C22#8896311yescharacteristic
341C4—61no—
342C4—122no—
343C4D3×#62511yescharacteristic
344C4#8895211yescharacteristic
345C3—21no—
346C3—22no—
347C3—41no—
348C3#1059164112yescomplemented
349C3#1059168311yescharacteristic, complemented
350C2—21no—
351C2—121no—
352C2#1172295311yescharacteristic
subgroupG/H
centerC2#11722953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
prime radicalC32×#235C22
hypocenterC33#473
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3718C2semidirect
4. #11722081C2semidirect
5. #11722028C2semidirect
23. C2×#3718D3semidirect
215. C33#473semidirect
312. C32#5192semidirect
313. C32#5204semidirect
348. C3#10591641semidirect
349. C3#10591683semidirect

Maximal subgroups

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

subgroupindex
#117233492
C2×C3×#37182
C3×#638272
#117220812
#117220282
#105916833
#105916413