Finite Group Databasebeta

← C3×#10591629 #13158654

Subgroups — C3×#10591629 #13158654

Order 1728.

730 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 132 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×#64156C211yesmaximal, characteristic
2C2×C32×#216C212yesmaximal, complemented
3C3×#63873C211yesmaximal, characteristic, complemented
4C2×C3×#3717C212yesmaximal
5C3×#63555C211yesmaximal, characteristic, complemented
6C3×#5190—32nomaximal
7#10591629C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C32×Dic6C2212yes—
11C3×#3752—22no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×#216—24no—
14C2×C32×C3⋊C8C2212yes—
15C3×#3724—22no—
16C3×#3717—24no—
17C2×C32×Q16—32no—
18#64156C2×C311yescharacteristic
19C3×#947—32no—
20C3×#947—61no—
21C3×#944—32no—
22C2×C3×#216—22no—
23C2×C3×#216D312yescomplemented
24C2×C3×#216C2×C312yescomplemented
25C2×C3×#215—32no—
26C2×C3×#204—32no—
27#63873C2×C311yescharacteristic, complemented
28C2×#3717C2×C312yes—
29C3×#815—32no—
30#63555C2×C311yescharacteristic, complemented
31C22×C3×C3⋊D3—21no—
32C33×Q8—22no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3D412yes—
36C3×C4⋊(C3⋊D3)—22no—
37C2×C32×Dic3—22no—
38C32×Dic6—22no—
39C32×Dic6—22no—
40C32×C3⋊C8D414yes—
41C3×#497—91no—
42#5190—32no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—32no—
46C2×C32×Q8—41no—
47C2×C32×Q8C2×D312yes—
48C2×C32×Q8C22×C311yescharacteristic
49C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
50C2×C3×D12—32no—
51C2×C3×D12—61no—
52C2×C3×Dic6—22no—
53C2×C3×Dic6C2×D312yes—
54C2×C3×Dic6C22×C312yes—
55C32×Q16—64no—
56C2×C32×C8—32no—
57#3752—22no—
58C4×C3⋊Dic3C22×C311yescharacteristic
59C3×#216—28no—
60C3×#216—24no—
61C3×#216—24no—
62C3×#215—64no—
63C3×(C2×D3)⋊C4—64no—
64C3×(C2×D3)⋊C4—64no—
65C3×Dic3⋊C4—64no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—22no—
69C2×C3×C3⋊C8C2×D312yes—
70C2×C3×C3⋊C8C22×C312yes—
71C3×#204—64no—
72#3724—22no—
73#3717—24no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—42no—
77C33×C4—41no—
78C33×C4C2×D412yes—
79C3×C3⋊Dic3—22no—
80C32×Dic3—42no—
81C2×C3×Q16—32no—
82C2×C3×Q16—32no—
83C2×C3×Q16—62no—
84C2×C3×SD16—92no—
85C3×C4⋊Q8—91no—
86C3×#124—91no—
87#947—32no—
88#947—61no—
89#944—32no—
90C2×#216C3×D312yescomplemented
91C2×#215—32no—
92C2×#204—32no—
93C3×#97—91no—
94#815—32no—
95C22×C3⋊D3—21no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C32×Q8—22no—
99C32×Q8—24no—
100C32×Q8—22no—
101C32×Q8—22no—
102C32×Q8—42no—
103C32×Q8—42no—
104C32×Q8—44no—
105C32×Q8—62no—
106C32×Q8—62no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4—41no—
113C2×C32×C4—42no—
114C2×C32×C4—42no—
115C2×C32×C4—62no—
116C2×C32×C4C22×D312yes—
117C2×C32×C4C23×C311yescharacteristic
118C2×C3⋊Dic3C3×D412yes—
119C4⋊(C3⋊D3)—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—62no—
125C3×D12—62no—
126C3×D12—62no—
127C3×D12—122no—
128C3×Dic6—24no—
129C3×Dic6—22no—
130C3×Dic6—22no—
131C3×Dic6—22no—
132C3×Dic6—22no—
133C3×Dic6—42no—
134C32×C8—34no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8#6714yes—
137C3×C3⋊C8C3×D414yes—
138#497—91noSylow
139C2×C33#12711yescharacteristic
140C2×C33#13712yes—
141C3×C3⋊D3—81no—
142C2×C3×Q8—21no—
143C2×C3×Q8—22no—
144C2×C3×Q8—32no—
145C2×C3×Q8—32no—
146C2×C3×Q8—41no—
147C2×C3×Q8—62no—
148C2×C3×Q8D3211yescharacteristic
149C2×C3×Q8C2×C3×D312yes—
150C2×C3×D4—91no—
151C2×D12—32no—
152C2×D12—61no—
153C2×Dic6C2×C3×D312yes—
154C3×Q16—64no—
155C3×Q16—64no—
156C3×Q16—68no—
157C3×SD16—184no—
158C2×C3×C8—32no—
159C2×C3×C8—32no—
160C2×C3×C8—62no—
161C3×C4⋊C4—182no—
162C3×C22⋊C4—182no—
163C3×C42—91no—
164#216—24no—
165#215—64no—
166(C2×D3)⋊C4—64no—
167(C2×D3)⋊C4—64no—
168Dic3⋊C4—64no—
169C4×Dic3—32no—
170C4×Dic3—61no—
171C2×C3⋊C8C2×C3×D312yes—
172#204—64no—
173C22×C32—21no—
174C22×C32—22no—
175C22×C32—41no—
176C22×C32D3×D412yes—
177C22×C32C2×C3×D411yescharacteristic
178C2×C3⋊D3—41no—
179C2×C3⋊D3—42no—
180C2×C3×D3—122no—
181C2×C3×D3—124no—
182C2×C3×D3—124no—
183C2×C3×D3—122no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—42no—
188C32×C4—42no—
189C32×C4—44no—
190C32×C4—44no—
191C32×C4—42no—
192C32×C4—41no—
193C32×C4—122no—
194C32×C4D3×D412yes—
195C32×C4C2×#6712yes—
196C32×C4C2×C3×D412yes—
197C3⋊Dic3—22no—
198C3×Dic3—44no—
199C3×Dic3—42no—
200C3×Dic3—42no—
201C3×Dic3—64no—
202C3×Dic3—122no—
203C2×Q16—32no—
204C2×SD16—92no—
205C4⋊Q8—91no—
206#124—91no—
207#97—91no—
208C33#49711yescharacteristic, complemented, Sylow
209C23×C3—181no—
210C22×D3—62no—
211C22×D3—62no—
212C3×Q8—24no—
213C3×Q8—22no—
214C3×Q8—22no—
215C3×Q8—22no—
216C3×Q8—42no—
217C3×Q8—42no—
218C3×Q8—44no—
219C3×Q8—64no—
220C3×Q8—62no—
221C3×Q8—62no—
222C3×Q8—62no—
223C3×Q8—62no—
224C3×Q8—122no—
225C3×D4—182no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—92no—
237C2×C3×C4—122no—
238C2×C3×C4C2×D3211yescharacteristic
239C2×C3×C4C22×C3×D312yes—
240C2×Dic3—22no—
241C2×Dic3—34no—
242C2×Dic3—62no—
243D12—62no—
244D12—62no—
245D12—122no—
246Dic6—22no—
247Dic6—22no—
248C3×C8—34no—
249C3×C8—34no—
250C3×C8—64no—
251C3⋊C8C3×#6714yes—
252C2×C32—24no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—41no—
257C2×C32—42no—
258C2×C32#91814yes—
259C2×C32#94012yes—
260C2×C32C3×#12711yescharacteristic
261C2×C32C3×#13712yes—
262C3⋊D3—81no—
263C3×D3—242no—
264C3×D3—242no—
265C2×Q8—32no—
266C2×Q8C3×D3211yescharacteristic
267C2×D4—91no—
268Q16—64no—
269SD16—184no—
270C2×C8—32no—
271C4⋊C4—182no—
272C22⋊C4—182no—
273C42—91no—
274C22×C3—21no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—362no—
278C22×C3—361no—
279C22×C3#381111yescharacteristic
280C22×C3C3×D3×D412yes—
281C2×D3—122no—
282C2×D3—124no—
283C2×D3—124no—
284C2×D3—122no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—42no—
289C3×C4—44no—
290C3×C4—44no—
291C3×C4—42no—
292C3×C4—41no—
293C3×C4—42no—
294C3×C4—124no—
295C3×C4—122no—
296C3×C4—122no—
297C3×C4—182no—
298C3×C4D3×#6712yes—
299C3×C4C3×D3×D412yes—
300C3×C4C2×C3×#6712yes—
301Dic3—42no—
302Dic3—64no—
303Dic3—122no—
304C32—22no—
305C32—21no—
306C32—41no—
307C32#519012yescomplemented
308C32C3×#49711yescharacteristic, complemented
309C23—181no—
310Q8—22no—
311Q8—62no—
312Q8—62no—
313D4—182no—
314C2×C4—21no—
315C2×C4—62no—
316C2×C4—92no—
317C2×C4C2×C3×D3211yescharacteristic
318C8—34no—
319C2×C3—22no—
320C2×C3—24no—
321C2×C3—22no—
322C2×C3—21no—
323C2×C3—42no—
324C2×C3—41no—
325C2×C3—721no—
326C2×C3#6394512yes—
327C2×C3#6397911yescharacteristic
328C2×C3C3×#91814yes—
329C2×C3C3×#94012yes—
330D3—242no—
331D3—242no—
332C22—362no—
333C22—361no—
334C22C3×#381111yescharacteristic
335C4—41no—
336C4—122no—
337C4—182no—
338C4C3×D3×#6712yes—
339C3—21no—
340C3—22no—
341C3—41no—
342C3#1059162911yescharacteristic, direct product factor
343C3C3×#519012yescomplemented
344C2—721no—
345C2C3×#6394512yes—
346C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C3×Q8D32
hypocenterC32C3×#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×#216C2semidirect
3. C3×#63873C2semidirect
5. C3×#63555C2semidirect
7. #10591629C3direct (N × H)
23. C2×C3×#216D3semidirect
24. C2×C3×#216C2×C3semidirect
27. #63873C2×C3semidirect
30. #63555C2×C3semidirect
90. C2×#216C3×D3semidirect
208. C33#497semidirect
307. C32#5190semidirect
308. C32C3×#497semidirect
342. C3#10591629direct (N × H)
343. C3C3×#5190semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C32×#2162
C3×#638732
C2×C3×#37172
C3×#635552
C3×#51903
#105916293