Finite Group Databasebeta

← C3×#10590554 #13157537

Subgroups — C3×#10590554 #13157537

Order 1728.

622 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 116 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×#64099C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic
3C32×#809C212yesmaximal, complemented
4C3×#63563C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6C3×#4764—32nomaximal
7#10590554C311yesmaximal, characteristic, direct product factor
8C2×C32×D12C2212yes—
9C2×C3×#625—21no—
10C2×C3×#625—21no—
11C33×C4⋊C4C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×#3750—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2212yes—
17#64099C2×C311yescharacteristic, complemented
18C3×#940—32no—
19C3×#890—32no—
20C3×#890—61no—
21#63914C2×C311yescharacteristic
22C32×#102—32no—
23C3×#832—32no—
24C3×#821—32no—
25C3×#815—32no—
26C3×#809—22no—
27C3×#809D312yescomplemented
28C3×#809C2×C312yescomplemented
29#63563C2×C312yescomplemented
30#63555C2×C311yescharacteristic, complemented
31C22×C32×D3—22no—
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C32×D12—22no—
38C32×D12—22no—
39C3×#625—42no—
40C3×#625—42no—
41C32×C3⋊C8—22no—
42C3×#490—91no—
43#4764—32no—
44C2×C32×D4—32no—
45C2×C3×#67—62no—
46C2×C3×#67—62no—
47C2×C3×D12—22no—
48C2×C3×D12C2×D312yes—
49C2×C3×D12C22×C312yes—
50C2×#625—21no—
51C2×#625—21no—
52C2×C32×C8—32no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4—22no—
55C32×C4⋊C4—41no—
56C32×C4⋊C4C2×D312yes—
57C32×C4⋊C4C22×C311yescharacteristic
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C22×C311yescharacteristic
60#3750—21no—
61C4×C3⋊Dic3C22×C311yescharacteristic
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—62no—
64C3×C4⋊Dic3—61no—
65C3×C4⋊Dic3—62no—
66C3×Dic3⋊C4—62no—
67C3×Dic3⋊C4—62no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8C2×D312yes—
72C2×C3×C3⋊C8C22×C312yes—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—42no—
75C2×C32×D3—42no—
76C2×C32×D3—42no—
77C33×C4—41no—
78C33×C4#4112yes—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3—41no—
82C3×#127—91no—
83C3×#125—91no—
84#940—32no—
85#890—32no—
86#890—61no—
87C3×#102—32no—
88C3×#102—32no—
89C3×#102—62no—
90C3×#102—92no—
91C3×#97—91no—
92#832—32no—
93#821—32no—
94#815—32no—
95#809C3×D312yescomplemented
96C23×C32—62no—
97C22×C3×D3—22no—
98C22×C3×D3—22no—
99C22×C3×D3—42no—
100C32×D4—62no—
101C32×D4—62no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—21no—
106C2×C32×C4—22no—
107C2×C32×C4—41no—
108C2×C32×C4—42no—
109C2×C32×C4—42no—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C23×C311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3C3×D411yescharacteristic
114C2×C3⋊Dic3C3×D411yescharacteristic
115C3×#67—124no—
116C3×#67—124no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C3×D12—24no—
124C3×D12—22no—
125C3×D12—22no—
126C3×D12—22no—
127C3×D12—22no—
128C3×D12—42no—
129#625—42no—
130#625—42no—
131C32×C8—62no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—42no—
135#490—91noSylow
136C2×C33#12411yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C32×D3—82no—
140C2×C3×D4—32no—
141C2×C3×D4—32no—
142C2×C3×D4—62no—
143C2×C3×D4—181no—
144C2×C3×D4—181no—
145C2×#67—62no—
146C2×#67—62no—
147C2×D12C2×C3×D312yes—
148C2×C3×C8—32no—
149C2×C3×C8—32no—
150C2×C3×C8—62no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—181no—
156C3×C4⋊C4—181no—
157C3×C4⋊C4D3211yescharacteristic
158C3×C4⋊C4C2×C3×D312yes—
159C3×C42—91no—
160C4⋊Dic3—32no—
161C4⋊Dic3—61no—
162C4⋊Dic3—62no—
163C4⋊Dic3—62no—
164Dic3⋊C4—62no—
165Dic3⋊C4—62no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8C2×C3×D312yes—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—41no—
172C22×C32—122no—
173C22×C32—122no—
174C22×C32—122no—
175C22×C32D3×D412yes—
176C22×C32C2×C3×D411yescharacteristic
177C2×C3×D3—44no—
178C2×C3×D3—42no—
179C2×C3×D3—42no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C2×C3×D3—42no—
184C2×C3×D3—82no—
185C2×C3×D3—82no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—42no—
190C32×C4—41no—
191C32×C4—42no—
192C32×C4—42no—
193C32×C4—82no—
194C32×C4—82no—
195C32×C4#23512yes—
196C32×C4#23712yes—
197C32×C4C3×#4112yes—
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3⋊Dic3—41no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—122no—
206C3×Dic3—122no—
207#127—91no—
208#125—91no—
209#102—32no—
210#102—92no—
211#97—91no—
212C33#49011yescharacteristic, complemented, Sylow
213C23×C3—62no—
214C23×C3—62no—
215C23×C3—122no—
216C22×D3—22no—
217C3×D4—64no—
218C3×D4—62no—
219C3×D4—62no—
220C3×D4—62no—
221C3×D4—62no—
222C3×D4—122no—
223C3×D4—362no—
224C3×D4—362no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×C3×C4C22×C3×D312yes—
238#67—124no—
239#67—124no—
240C2×Dic3—32no—
241C2×Dic3—32no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246D12—22no—
247D12—22no—
248C3×C8—62no—
249C3×C8—62no—
250C3×C8—122no—
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—242no—
262C2×C32#88512yes—
263C2×C32#91112yes—
264C2×C32#91412yes—
265C2×C32C3×#12411yescharacteristic
266C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
267C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
268C3×D3—84no—
269C3×D3—82no—
270C3×D3—82no—
271C2×D4—32no—
272C2×D4—181no—
273C2×D4—181no—
274C2×C8—32no—
275C4⋊C4—91no—
276C4⋊C4—181no—
277C4⋊C4—181no—
278C4⋊C4C3×D3211yescharacteristic
279C42—91no—
280C22×C3—21no—
281C22×C3—22no—
282C22×C3—41no—
283C22×C3—124no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3—122no—
288C22×C3—122no—
289C22×C3—122no—
290C22×C3—242no—
291C22×C3—242no—
292C22×C3#381111yescharacteristic
293C22×C3C3×D3×D412yes—
294C2×D3—42no—
295C2×D3—42no—
296C2×D3—42no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—42no—
301C3×C4—42no—
302C3×C4—41no—
303C3×C4—42no—
304C3×C4—82no—
305C3×C4—82no—
306C3×C4—181no—
307C3×C4—181no—
308C3×C4—361no—
309C3×C4#379512yes—
310C3×C4C3×#23512yes—
311C3×C4C3×#23712yes—
312Dic3—62no—
313Dic3—62no—
314Dic3—121no—
315Dic3—121no—
316Dic3—122no—
317Dic3—122no—
318C32—21no—
319C32—22no—
320C32—41no—
321C32#476412yescomplemented
322C32C3×#49011yescharacteristic, complemented
323C23—62no—
324D4—62no—
325D4—62no—
326D4—362no—
327D4—362no—
328C2×C4—21no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4C2×C3×D3211yescharacteristic
333C8—62no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—22no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—244no—
344C2×C3—242no—
345C2×C3—242no—
346C2×C3#6391311yescharacteristic
347C2×C3#6392711yescharacteristic
348C2×C3#6394111yescharacteristic
349C2×C3C3×#88512yes—
350C2×C3C3×#91112yes—
351C2×C3C3×#91412yes—
352D3—82no—
353C22—122no—
354C22—122no—
355C22—122no—
356C22C3×#381111yescharacteristic
357C4—41no—
358C4—181no—
359C4—181no—
360C4—361no—
361C4C3×#379512yes—
362C3—22no—
363C3—21no—
364C3—41no—
365C3#1059055411yescharacteristic, direct product factor
366C3C3×#476412yescomplemented
367C2—242no—
368C2C3×#6391311yescharacteristic
369C2C3×#6392711yescharacteristic
370C2C3×#6394111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#490
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
1. C3×#64099C2semidirect
3. C32×#809C2semidirect
4. C3×#63563C2semidirect
5. C3×#63555C2semidirect
7. #10590554C3direct (N × H)
17. #64099C2×C3semidirect
27. C3×#809D3semidirect
28. C3×#809C2×C3semidirect
29. #63563C2×C3semidirect
30. #63555C2×C3semidirect
95. #809C3×D3semidirect
212. C33#490semidirect
321. C32#4764semidirect
322. C32C3×#490semidirect
365. C3#10590554direct (N × H)
366. C3C3×#4764semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
C3×#639142
C32×#8092
C3×#635632
C3×#635552
C3×#47643
#105905543