Finite Group Databasebeta

← C3×#10590430 #13157550

Subgroups — C3×#10590430 #13157550

Order 1728.

624 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 124 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×#64102C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3C32×#807C212yesmaximal, complemented
4C3×#63572C211yesmaximal, characteristic, complemented
5C3×#63564C212yesmaximal
6C3×#4890—32nomaximal
7#10590430C311yesmaximal, characteristic, direct product factor
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×C4×C3⋊D3—21no—
10C2×C3×#627—21no—
11C33×C4⋊C4C2211yescharacteristic
12C3×#3752—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17#64102C2×C311yescharacteristic
18C3×#893—32no—
19C3×#893—61no—
20C3×#888—32no—
21#63919C2×C311yescharacteristic, complemented
22C32×#107—32no—
23C3×#821—32no—
24C3×Dic3⋊C8—32no—
25C3×#809—32no—
26C3×#807—22no—
27C3×#807D312yescomplemented
28C3×#807C2×C312yescomplemented
29#63572C2×C311yescharacteristic, complemented
30#63564C2×C312yes—
31C22×C3×C3⋊D3—21no—
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4⋊(C3⋊D3)—21no—
37C3×C4⋊(C3⋊D3)Q812yescomplemented
38C3×C4×C3⋊D3—42no—
39C2×C32×Dic3—22no—
40C3×#627—42no—
41C32×C3⋊C8—22no—
42C3×#474—91no—
43#4890—32no—
44C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
45C2×C4×C3⋊D3—21no—
46C2×C3×D12—32no—
47C2×C3×D12—61no—
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—62no—
50C2×C3×Dic6—62no—
51C2×#627—21no—
52C2×C32×C8—32no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4—22no—
55C32×C4⋊C4—32no—
56C32×C4⋊C4—41no—
57C32×C4⋊C4C2×D312yes—
58C32×C4⋊C4C22×C311yescharacteristic
59#3752—21no—
60C4×C3⋊Dic3C22×C311yescharacteristic
61C3×(C2×D3)⋊C4—62no—
62C3×(C2×D3)⋊C4—62no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C22×C312yes—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—22no—
70C2×C3×C3⋊C8C2×D312yes—
71C2×C3×C3⋊C8C22×C312yes—
72#3724—21no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—41no—
77C33×C4—41no—
78C33×C4C2×Q811yescharacteristic
79C33×C4#4111yescharacteristic
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3D812yes—
82C32×Dic3—42no—
83C3×C4⋊Q8—91no—
84C3×C4×D4—91no—
85#893—32no—
86#893—61no—
87#888—32no—
88C3×#107—32no—
89C3×#107—32no—
90C3×#107—62no—
91C3×C4⋊C8—91no—
92C3×#102—92no—
93#821—32no—
94Dic3⋊C8—32no—
95#809—32no—
96#807C3×D312yescomplemented
97C22×C3⋊D3—21no—
98C22×C3×D3—62no—
99C22×C3×D3—62no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4—41no—
106C2×C32×C4—42no—
107C2×C32×C4—42no—
108C2×C32×C4—62no—
109C2×C32×C4C22×D312yes—
110C2×C32×C4C23×C311yescharacteristic
111C2×C3⋊Dic3C3×D411yescharacteristic
112C2×C3⋊Dic3C3×D411yescharacteristic
113C4⋊(C3⋊D3)—21no—
114C4⋊(C3⋊D3)C3×Q812yescomplemented
115C4×C3⋊D3—42no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—42no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×D12—34no—
124C3×D12—62no—
125C3×D12—62no—
126C3×D12—62no—
127C3×C4×D3—124no—
128C3×C4×D3—124no—
129C3×Dic6—124no—
130#627—42no—
131C32×C8—62no—
132C3×C3⋊C8—24no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—22no—
135#474—91noSylow
136C2×C33C22⋊Q811yescharacteristic
137C2×C33C2×D811yescharacteristic
138C2×C33#13711yescharacteristic
139C3×C3⋊D3—42no—
140C2×C3×Q8—181no—
141C2×C3×D4—91no—
142C22×C3×C4—181no—
143C2×D12—32no—
144C2×D12—61no—
145C2×C4×D3—62no—
146C2×C4×D3—62no—
147C2×Dic6—62no—
148C2×C3×C8—32no—
149C2×C3×C8—32no—
150C2×C3×C8—62no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—32no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4D3211yescharacteristic
159C3×C4⋊C4C2×C3×D312yes—
160C3×C22⋊C4—181no—
161C3×C42—91no—
162(C2×D3)⋊C4—62no—
163(C2×D3)⋊C4—62no—
164C4⋊Dic3C2×C3×D312yes—
165Dic3⋊C4—62no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8C2×C3×D312yes—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—41no—
172C22×C32D3×D412yes—
173C22×C32C2×C3×D411yescharacteristic
174C2×C3⋊D3—22no—
175C2×C3⋊D3—41no—
176C2×C3⋊D3—41no—
177C2×C3×D3—64no—
178C2×C3×D3—64no—
179C2×C3×D3—122no—
180C2×C3×D3—122no—
181C2×C3×D3—122no—
182C2×C3×D3—122no—
183C32×C4—21no—
184C32×C4—21no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—41no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—42no—
191C32×C4—42no—
192C32×C4—82no—
193C32×C4—82no—
194C32×C4—122no—
195C32×C4#23512yes—
196C32×C4D3×Q812yes—
197C32×C4C2×C3×Q811yescharacteristic
198C32×C4C3×#4111yescharacteristic
199C3⋊Dic3—21no—
200C3⋊Dic3C3×D812yes—
201C3×Dic3—34no—
202C3×Dic3—42no—
203C3×Dic3—42no—
204C3×Dic3—62no—
205C3×Dic3—62no—
206C3×Dic3—82no—
207C3×Dic3—121no—
208C4⋊Q8—91no—
209C4×D4—91no—
210#107—32no—
211C4⋊C8—91no—
212#102—92no—
213C33#47411yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—62no—
216C22×D3—62no—
217C3×Q8—362no—
218C3×D4—92no—
219C3×D4—181no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—122no—
233C2×C3×C4—362no—
234C2×C3×C4C2×D3211yescharacteristic
235C2×C3×C4C22×C3×D312yes—
236C2×Dic3—22no—
237C2×Dic3—32no—
238C2×Dic3—32no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241D12—34no—
242D12—62no—
243D12—62no—
244D12—62no—
245C4×D3—124no—
246C4×D3—124no—
247Dic6—124no—
248C3×C8—64no—
249C3×C8—62no—
250C3×C8—62no—
251C3⋊C8—22no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32#89612yes—
262C2×C32D3×D812yes—
263C2×C32#91812yes—
264C2×C32C3×C22⋊Q811yescharacteristic
265C2×C32C2×C3×D811yescharacteristic
266C2×C32C3×#13711yescharacteristic
267C3⋊D3—42no—
268C3×D3—124no—
269C3×D3—124no—
270C2×Q8—181no—
271C2×D4—91no—
272C22×C4—181no—
273C2×C8—32no—
274C4⋊C4—32no—
275C4⋊C4—181no—
276C4⋊C4C3×D3211yescharacteristic
277C22⋊C4—181no—
278C42—91no—
279C22×C3—21no—
280C22×C3—22no—
281C22×C3—41no—
282C22×C3—182no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3#381111yescharacteristic
286C22×C3C3×D3×D412yes—
287C2×D3—64no—
288C2×D3—64no—
289C2×D3—122no—
290C2×D3—122no—
291C2×D3—122no—
292C2×D3—122no—
293C3×C4—21no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—42no—
298C3×C4—42no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4—82no—
303C3×C4—82no—
304C3×C4—92no—
305C3×C4—122no—
306C3×C4—122no—
307C3×C4—181no—
308C3×C4—242no—
309C3×C4#379711yescharacteristic
310C3×C4#379811yescharacteristic
311C3×C4C3×#23512yes—
312C3×C4C3×D3×Q812yes—
313Dic3—34no—
314Dic3—42no—
315Dic3—62no—
316Dic3—62no—
317Dic3—121no—
318C32—21no—
319C32—22no—
320C32—41no—
321C32#489012yescomplemented
322C32C3×#47411yescharacteristic, complemented
323C23—181no—
324Q8—362no—
325D4—92no—
326D4—181no—
327C2×C4—21no—
328C2×C4—62no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—362no—
332C2×C4C2×C3×D3211yescharacteristic
333C8—62no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—362no—
344C2×C3#6386511yescharacteristic
345C2×C3#6392611yescharacteristic
346C2×C3#6394511yescharacteristic
347C2×C3C3×#89612yes—
348C2×C3C3×D3×D812yes—
349C2×C3C3×#91812yes—
350D3—124no—
351D3—124no—
352C22—182no—
353C22—361no—
354C22—361no—
355C22C3×#381111yescharacteristic
356C4—41no—
357C4—92no—
358C4—122no—
359C4—181no—
360C4C3×#379711yescharacteristic
361C4C3×#379811yescharacteristic
362C3—21no—
363C3—22no—
364C3—41no—
365C3#1059043011yescharacteristic, direct product factor
366C3C3×#489012yescomplemented
367C2—362no—
368C2C3×#6386511yescharacteristic
369C2C3×#6392611yescharacteristic
370C2C3×#6394511yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#474
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#63919C2semidirect
3. C32×#807C2semidirect
4. C3×#63572C2semidirect
7. #10590430C3direct (N × H)
21. #63919C2×C3semidirect
27. C3×#807D3semidirect
28. C3×#807C2×C3semidirect
29. #63572C2×C3semidirect
37. C3×C4⋊(C3⋊D3)Q8semidirect
96. #807C3×D3semidirect
114. C4⋊(C3⋊D3)C3×Q8semidirect
213. C33#474semidirect
321. C32#4890semidirect
322. C32C3×#474semidirect
365. C3#10590430direct (N × H)
366. C3C3×#4890semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
C3×#639192
C32×#8072
C3×#635722
C3×#635642
C3×#48903
#105904303