Finite Group Databasebeta

← C3×#10590993 #13158002

Subgroups — C3×#10590993 #13158002

Order 1728.

558 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C3×#64116C211yesmaximal, characteristic
2C2×C3×#3718C211yesmaximal, characteristic, complemented
3C3×#63833C212yesmaximal
4C32×#824C212yesmaximal, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6C3×#4755—32nomaximal
7#10590993C311yesmaximal, characteristic, direct product factor
8C3×#3828C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C33×M16C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C3×#3743—21no—
13C3×C8×C3⋊D3—21no—
14C32×#208C2212yes—
15C3×#3719—22no—
16C3×#3718—21no—
17C3×#3718—21no—
18C3×#3717—22no—
19#64116C2×C311yescharacteristic
20C2×C3×#216—32no—
21C3×#942—32no—
22C3×#909—32no—
23C3×#907—32no—
24C3×#907—61no—
25C2×#3718C2×C311yescharacteristic, complemented
26#63833C2×C312yes—
27C32×#101—32no—
28C3×#824—22no—
29C3×#824D312yescomplemented
30C3×#824C2×C312yescomplemented
31C3×#822—32no—
32#63578C2×C311yescharacteristic, complemented
33C2×C33×C4C2311yescharacteristic
34C3×#638—21no—
35C3×C4⋊(C3⋊D3)D411yescharacteristic
36C3×C4×C3⋊D3—21no—
37C3×#634D411yescharacteristic
38C2×C32×Dic3—22no—
39C32×Dic6—22no—
40C32×Dic6—22no—
41C33×C8—21no—
42C3×C3⋊(C3⋊C8)D411yescharacteristic
43C3×C3⋊(C3⋊C8)D411yescharacteristic
44C32×C3⋊C8—22no—
45C3×#473—91no—
46#4755—32no—
47C2×C32×Q8—32no—
48#3828C22×C311yescharacteristic
49C3×#235—32no—
50C3×#235—61no—
51C2×C3×Dic6—22no—
52C2×C3×Dic6C2×D312yes—
53C2×C3×Dic6C22×C312yes—
54C32×M16—21no—
55C32×M16—22no—
56C32×M16—32no—
57C32×M16—41no—
58C32×M16C2×D312yes—
59C32×M16C22×C311yescharacteristic
60C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
61#3743—21no—
62C8×C3⋊D3—21no—
63C3×#216—62no—
64C3×#216—62no—
65C3×#216—62no—
66C3×#215—62no—
67C3×#208—22no—
68C3×#208C2×D312yes—
69C3×#208C22×C312yes—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—61no—
72C3×Dic12—62no—
73C3×#204—62no—
74C3×ΓL(1,𝔽25)—62no—
75C3×ΓL(1,𝔽25)—62no—
76C3×D3×C8—62no—
77C3×D3×C8—62no—
78#3719—22no—
79#3718—21no—
80#3718—21no—
81#3717—22no—
82C22×C33#4111yescharacteristic
83C2×C3×C3⋊D3—21no—
84C33×C4C2×D411yescharacteristic
85C33×C4C2×D411yescharacteristic
86C3×C3⋊Dic3—21no—
87C32×Dic3—42no—
88C3×#137—92no—
89C2×C3×Q16—91no—
90C3×#131—91no—
91C2×#216—32no—
92#942—32no—
93#909—32no—
94#907—32no—
95#907—61no—
96C3×#108—91no—
97C3×#101—32no—
98C3×#101—32no—
99C3×#101—62no—
100#824C3×D312yescomplemented
101#822—32no—
102C32×Q8—62no—
103C32×Q8—62no—
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—41no—
107C2×C32×C4—62no—
108C2×C32×C4C22×D312yes—
109C2×C32×C4C23×C311yescharacteristic
110#638—21no—
111C4⋊(C3⋊D3)C3×D411yescharacteristic
112C4×C3⋊D3—21no—
113#634C3×D411yescharacteristic
114C3×#67—62no—
115C3×#67—62no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C3×D12—32no—
120C3×D12—61no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×Dic6—22no—
124C3×Dic6—22no—
125C3×Dic6—22no—
126C3×Dic6—22no—
127C3×Dic6—32no—
128C3×Dic6—42no—
129C3×Dic6—42no—
130C3×Dic6—61no—
131C32×C8—21no—
132C32×C8—22no—
133C32×C8—22no—
134C32×C8—42no—
135C32×C8—42no—
136C32×C8—62no—
137C3⋊(C3⋊C8)C3×D411yescharacteristic
138C3⋊(C3⋊C8)C3×D411yescharacteristic
139C3×C3⋊C8—22no—
140C3×C3⋊C8—22no—
141C3×C3⋊C8—32no—
142C3×C3⋊C8—32no—
143C3×C3⋊C8—42no—
144C3×C3⋊C8—61no—
145C3×C3⋊C8—61no—
146#473—91noSylow
147C2×C33—21no—
148C2×C33#12111yescharacteristic
149C3×C3⋊D3—41no—
150C3×#41—91no—
151C2×C3×Q8—32no—
152C2×C3×Q8—32no—
153C2×C3×Q8—62no—
154#235—32no—
155#235—61no—
156C2×Dic6C2×C3×D312yes—
157C3×Q16—182no—
158C3×Q16—181no—
159C3×Q16—181no—
160C3×SD16—182no—
161C3×M16—21no—
162C3×M16—22no—
163C3×M16—32no—
164C3×M16—32no—
165C3×M16—41no—
166C3×M16—62no—
167C3×M16—181no—
168C3×M16D3211yescharacteristic
169C3×M16C2×C3×D312yes—
170C2×C3×C8—91no—
171C2×C3×C8—181no—
172#216—62no—
173#216—62no—
174#216—62no—
175#215—62no—
176#208C2×C3×D312yes—
177C2×C3⋊C8—32no—
178C2×C3⋊C8—61no—
179Dic12—62no—
180#204—62no—
181ΓL(1,𝔽25)—62no—
182ΓL(1,𝔽25)—62no—
183D3×C8—62no—
184D3×C8—62no—
185C22×C32—21no—
186C22×C32—22no—
187C22×C32—41no—
188C22×C32#23512yes—
189C22×C32C3×#4111yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—22no—
196C32×C4—22no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—122no—
200C32×C4D3×D412yes—
201C32×C4D3×D412yes—
202C32×C4C2×C3×D411yescharacteristic
203C32×C4C2×C3×D411yescharacteristic
204C3⋊Dic3—21no—
205C3×Dic3—44no—
206C3×Dic3—42no—
207C3×Dic3—42no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210#137—92no—
211C2×Q16—91no—
212#131—91no—
213#108—91no—
214#101—32no—
215C33#47311yescharacteristic, complemented, Sylow
216C3×Q8—62no—
217C3×Q8—62no—
218C3×Q8—62no—
219C3×Q8—62no—
220C3×Q8—91no—
221C3×Q8—122no—
222C3×Q8—122no—
223C3×D4—91no—
224C3×D4—181no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—64no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×C3×C4C22×C3×D312yes—
234#67—62no—
235#67—62no—
236C2×Dic3—22no—
237D12—32no—
238D12—61no—
239C4×D3—62no—
240C4×D3—62no—
241Dic6—22no—
242Dic6—22no—
243Dic6—32no—
244Dic6—61no—
245C3×C8—22no—
246C3×C8—21no—
247C3×C8—22no—
248C3×C8—42no—
249C3×C8—42no—
250C3×C8—62no—
251C3×C8—62no—
252C3×C8—91no—
253C3×C8—91no—
254C3×C8—122no—
255C3⋊C8—22no—
256C3⋊C8—32no—
257C3⋊C8—32no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—22no—
263C2×C32—21no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—81no—
268C2×C32#88412yes—
269C2×C32C3×#12111yescharacteristic
270C3⋊D3—41no—
271C3×D3—122no—
272C3×D3—122no—
273#41—91no—
274C2×Q8—32no—
275Q16—182no—
276Q16—181no—
277Q16—181no—
278SD16—182no—
279M16—32no—
280M16—181no—
281M16C3×D3211yescharacteristic
282C2×C8—91no—
283C2×C8—181no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—41no—
287C22×C3—181no—
288C22×C3#379811yescharacteristic
289C22×C3C3×#23512yes—
290C2×D3—62no—
291C2×D3—62no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—22no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—124no—
299C3×C4—122no—
300C3×C4—122no—
301C3×C4—181no—
302C3×C4#380211yescharacteristic
303C3×C4#381111yescharacteristic
304C3×C4C3×D3×D412yes—
305C3×C4C3×D3×D412yes—
306Dic3—42no—
307Dic3—62no—
308Dic3—62no—
309C32—21no—
310C32—22no—
311C32—41no—
312C32#475512yescomplemented
313C32C3×#47311yescharacteristic, complemented
314Q8—62no—
315Q8—62no—
316Q8—91no—
317D4—91no—
318D4—181no—
319C2×C4—62no—
320C2×C4—181no—
321C2×C4C2×C3×D3211yescharacteristic
322C8—21no—
323C8—62no—
324C8—91no—
325C8—91no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—41no—
331C2×C3—42no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3—361no—
335C2×C3#6391211yescharacteristic
336C2×C3C3×#88412yes—
337D3—122no—
338D3—122no—
339C22—181no—
340C22C3×#379811yescharacteristic
341C4—122no—
342C4—181no—
343C4C3×#380211yescharacteristic
344C4C3×#381111yescharacteristic
345C3—21no—
346C3—22no—
347C3—41no—
348C3#1059099311yescharacteristic, direct product factor
349C3C3×#475512yescomplemented
350C2—21no—
351C2—361no—
352C2C3×#6391211yescharacteristic
subgroupG/H
centerC2×C3#63912
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33#121
prime radicalC3×#3828C22
hypercenterC3×M16D32
hypocenterC32C3×#473
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3718C2semidirect
4. C32×#824C2semidirect
5. C3×#63578C2semidirect
7. #10590993C3direct (N × H)
25. C2×#3718C2×C3semidirect
29. C3×#824D3semidirect
30. C3×#824C2×C3semidirect
32. #63578C2×C3semidirect
100. #824C3×D3semidirect
215. C33#473semidirect
312. C32#4755semidirect
313. C32C3×#473semidirect
348. C3#10590993direct (N × H)
349. C3C3×#4755semidirect

Maximal subgroups

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

subgroupindex
C3×#641162
C2×C3×#37182
C3×#638332
C32×#8242
C3×#635782
C3×#47553
#105909933