Finite Group Databasebeta

← C3×#10591632 #13158634

Subgroups — C3×#10591632 #13158634

Order 1728.

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

normal subgroups only

This group has 598 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×#64156C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3C32×#835C212yesmaximal, complemented
4C3×#63564C212yesmaximal
5C3×#63555C211yesmaximal, characteristic, complemented
6C3×#4794—32nomaximal
7#10591632C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#627—21no—
11C3×#3752—21no—
12C3×#3752—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17#64156C2×C311yescharacteristic
18C3×#947—32no—
19C3×#947—61no—
20C3×#888—32no—
21#63919C2×C311yescharacteristic, complemented
22C32×#103—32no—
23C3×#835—22no—
24C3×#835D312yescomplemented
25C3×#835C2×C312yescomplemented
26C3×#821—32no—
27C3×#815—32no—
28C3×#809—32no—
29#63564C2×C312yes—
30#63555C2×C311yescharacteristic, complemented
31C22×C3×C3⋊D3—21no—
32C33×Q8—22no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×C4⋊(C3⋊D3)—22no—
38C2×C32×Dic3—22no—
39C3×#627—42no—
40C32×C3⋊C8—22no—
41C3×#489—91no—
42#4794—32no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D312yes—
47C2×C32×Q8C22×C311yescharacteristic
48C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
49C2×C3×D12—32no—
50C2×C3×D12—61no—
51C2×C3×Dic6—62no—
52C2×#627—21no—
53C2×C32×C8—32no—
54C32×C4⋊C4—32no—
55#3752—21no—
56#3752—21no—
57C4×C3⋊Dic3C22×C311yescharacteristic
58C3×(C2×D3)⋊C4—62no—
59C3×(C2×D3)⋊C4—62no—
60C3×(C2×D3)⋊C4—62no—
61C3×(C2×D3)⋊C4—62no—
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3C2×D312yes—
64C3×C4⋊Dic3C22×C312yes—
65C3×Dic3⋊C4—62no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—22no—
69C2×C3×C3⋊C8C2×D312yes—
70C2×C3×C3⋊C8C22×C312yes—
71#3724—21no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—41no—
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3—41no—
76C33×C4—41no—
77C33×C4#4112yes—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3—21no—
80C32×Dic3—42no—
81C3×C4⋊Q8—91no—
82C3×#124—91no—
83#947—32no—
84#947—61no—
85#888—32no—
86C3×#103—32no—
87C3×#103—32no—
88C3×#103—62no—
89C3×#102—92no—
90C3×#97—91no—
91#835C3×D312yescomplemented
92#821—32no—
93#815—32no—
94#809—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—
105C2×C32×C4—21no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—22no—
110C2×C32×C4—41no—
111C2×C32×C4—42no—
112C2×C32×C4—42no—
113C2×C32×C4—62no—
114C2×C32×C4C22×D312yes—
115C2×C32×C4C23×C311yescharacteristic
116C2×C3⋊Dic3C3×D411yescharacteristic
117C2×C3⋊Dic3C3×D411yescharacteristic
118C4⋊(C3⋊D3)—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×D12—62no—
127C3×D12—62no—
128C3×D12—122no—
129C3×Dic6—124no—
130#627—42no—
131C32×C8—62no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—42no—
135#489—91noSylow
136C2×C33#12411yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C2×C33#13711yescharacteristic
139C3×C3⋊D3—81no—
140C2×C3×Q8—21no—
141C2×C3×Q8—22no—
142C2×C3×Q8—41no—
143C2×C3×Q8—181no—
144C2×C3×Q8D3211yescharacteristic
145C2×C3×Q8C2×C3×D312yes—
146C2×C3×D4—91no—
147C2×D12—32no—
148C2×D12—61no—
149C2×Dic6—62no—
150C2×C3×C8—32no—
151C2×C3×C8—32no—
152C2×C3×C8—62no—
153C3×C4⋊C4—32no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—62no—
156C3×C4⋊C4—181no—
157C3×C22⋊C4—181no—
158C3×C22⋊C4—181no—
159C3×C42—91no—
160(C2×D3)⋊C4—62no—
161(C2×D3)⋊C4—62no—
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—41no—
175C2×C3⋊D3—41no—
176C2×C3⋊D3—41no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179C2×C3×D3—122no—
180C2×C3×D3—122no—
181C2×C3×D3—122no—
182C2×C3×D3—122no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—42no—
188C32×C4—44no—
189C32×C4—44no—
190C32×C4—42no—
191C32×C4—41no—
192C32×C4—122no—
193C32×C4#23512yes—
194C32×C4#23712yes—
195C32×C4C3×#4112yes—
196C3⋊Dic3—21no—
197C3⋊Dic3—21no—
198C3×Dic3—42no—
199C3×Dic3—42no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—82no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C4⋊Q8—91no—
206#124—91no—
207#103—32no—
208#102—92no—
209#97—91no—
210C33#48911yescharacteristic, complemented, Sylow
211C23×C3—181no—
212C22×D3—62no—
213C22×D3—62no—
214C3×Q8—24no—
215C3×Q8—22no—
216C3×Q8—22no—
217C3×Q8—22no—
218C3×Q8—42no—
219C3×Q8—42no—
220C3×Q8—44no—
221C3×Q8—362no—
222C3×D4—182no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4—122no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×C3×C4C22×C3×D312yes—
238C2×Dic3—22no—
239C2×Dic3—32no—
240C2×Dic3—32no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243D12—62no—
244D12—62no—
245D12—122no—
246Dic6—124no—
247C3×C8—62no—
248C3×C8—62no—
249C3×C8—122no—
250C3⋊C8—22no—
251C2×C32—22no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32#88512yes—
261C2×C32#91412yes—
262C2×C32#91812yes—
263C2×C32C3×#12411yescharacteristic
264C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
265C2×C32C3×#13711yescharacteristic
266C3⋊D3—81no—
267C3×D3—242no—
268C3×D3—242no—
269C2×Q8—181no—
270C2×Q8C3×D3211yescharacteristic
271C2×D4—91no—
272C2×C8—32no—
273C4⋊C4—32no—
274C4⋊C4—181no—
275C22⋊C4—181no—
276C22⋊C4—181no—
277C42—91no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3#381111yescharacteristic
285C22×C3C3×D3×D412yes—
286C2×D3—122no—
287C2×D3—122no—
288C2×D3—122no—
289C2×D3—122no—
290C2×D3—122no—
291C2×D3—122no—
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—42no—
296C3×C4—44no—
297C3×C4—44no—
298C3×C4—42no—
299C3×C4—41no—
300C3×C4—42no—
301C3×C4—122no—
302C3×C4—122no—
303C3×C4—181no—
304C3×C4—181no—
305C3×C4—242no—
306C3×C4#380412yes—
307C3×C4C3×#23512yes—
308C3×C4C3×#23712yes—
309Dic3—42no—
310Dic3—62no—
311Dic3—62no—
312Dic3—121no—
313Dic3—121no—
314C32—21no—
315C32—22no—
316C32—41no—
317C32#479412yescomplemented
318C32C3×#48911yescharacteristic, complemented
319C23—181no—
320Q8—22no—
321Q8—362no—
322D4—182no—
323C2×C4—21no—
324C2×C4—62no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4C2×C3×D3211yescharacteristic
328C8—62no—
329C2×C3—21no—
330C2×C3—22no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—21no—
334C2×C3—22no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—721no—
339C2×C3#6394111yescharacteristic
340C2×C3#6394511yescharacteristic
341C2×C3#6395311yescharacteristic
342C2×C3C3×#88512yes—
343C2×C3C3×#91412yes—
344C2×C3C3×#91812yes—
345D3—242no—
346D3—242no—
347C22—361no—
348C22—361no—
349C22—361no—
350C22C3×#381111yescharacteristic
351C4—41no—
352C4—122no—
353C4—181no—
354C4—181no—
355C4C3×#380412yes—
356C3—21no—
357C3—22no—
358C3—41no—
359C3#1059163211yescharacteristic, direct product factor
360C3C3×#479412yescomplemented
361C2—721no—
362C2C3×#6394111yescharacteristic
363C2C3×#6394511yescharacteristic
364C2C3×#6395311yescharacteristic
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×#489
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. C3×#63919C2semidirect
3. C32×#835C2semidirect
5. C3×#63555C2semidirect
7. #10591632C3direct (N × H)
21. #63919C2×C3semidirect
24. C3×#835D3semidirect
25. C3×#835C2×C3semidirect
30. #63555C2×C3semidirect
91. #835C3×D3semidirect
210. C33#489semidirect
317. C32#4794semidirect
318. C32C3×#489semidirect
359. C3#10591632direct (N × H)
360. C3C3×#4794semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C3×#639192
C32×#8352
C3×#635642
C3×#635552
C3×#47943
#105916323