Finite Group Databasebeta

← C2×#1007⋊C9 #13164617

Subgroups — C2×#1007⋊C9 #13164617

Order 1728.

791 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 791 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C22×C3×Q8⋊C9C211yesmaximal, characteristic, complemented
2#1007⋊C9C214yesmaximal, direct product factor
3C2×D3×Q8⋊C9C211yesmaximal, characteristic, complemented
4C2×C3⋊(#41⋊C9)C211yesmaximal, characteristic, complemented
5C2×C3×#1007C311yesmaximal, characteristic
6C2×2−1+4⋊C9—31nomaximal
7C2×C9×#67—41nomaximal
8C2×C3×Q8⋊C9C2211yescharacteristic
9C2×C3×Q8⋊C9C2212yescomplemented
10D3×Q8⋊C9C2212yescomplemented
11C3⋊(#41⋊C9)C2212yescomplemented
12C22×C32×Q8C2×C311yescharacteristic
13C3×#1007—34no—
14C3×#1007C2×C314yes—
15C2×C3×#239—31no—
16C2×C3×#239C2×C311yescharacteristic
17C2×C3×D3×Q8—31no—
18C2×C3×D3×Q8C2×C311yescharacteristic
19C2×C3×#235—31no—
20C2×C3×#235—31no—
212−1+4⋊C9—34no—
22C2×#41⋊C9—31no—
23C22×Q8⋊C9—21no—
24C22×Q8⋊C9—31no—
25C22×Q8⋊C9D311yescharacteristic, complemented
26C23×C3×C9—41no—
27C22×D3×C9—41no—
28C9×#67—44no—
29C2×C9×Dic3—41no—
30C3×Q8⋊C9C2311yescharacteristic
31C2×C3×2−1+4—31no—
32C2×#1007C911yescharacteristic, complemented
33C2×C32×Q8—31no—
34C2×C32×Q8—32no—
35C2×C32×Q8C22×C311yescharacteristic
36C2×C32×Q8C22×C312yes—
37C22×C32×C4—31no—
38C2×C3×#67—31no—
39C2×C3×#67A411yescharacteristic
40C3×#239—34no—
41C3×#239—34no—
42C3×#239—32no—
43C3×#239C22×C312yes—
44C3×D3×Q8—34no—
45C3×D3×Q8—32no—
46C3×D3×Q8—34no—
47C3×D3×Q8C22×C312yes—
48C3×#235—34no—
49C3×#235—34no—
50C3×#235—34no—
51C3×#235—34no—
52C2×C3×D12—31no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—32no—
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—31no—
57C2×C3×Dic6—31no—
58C2×C3×Dic6—31no—
59C2×D4×C9—121no—
60#41⋊C9—32no—
61C2×Q8⋊C9—21no—
62C2×Q8⋊C9—22no—
63C2×Q8⋊C9—32no—
64C2×Q8⋊C9C2×D311yescharacteristic
65C2×Q8⋊C9C2×D312yescomplemented
66C22×C3×C9—41no—
67C22×C3×C9—42no—
68C22×C3×C9—82no—
69C2×D3×C9—42no—
70C2×D3×C9—82no—
71C9×Dic3—42no—
72C3×2−1+4—34no—
73C3×2−1+4—94no—
74C2×C3×#41—31no—
75C2×C3×#41—91no—
76C2×C3×#41—91no—
77C2×C3×#41—91no—
78C22×C3×Q8—21no—
79C22×C3×Q8—31no—
80C22×C3×Q8—91no—
81C22×C3×Q8C2×C911yescharacteristic, complemented
82C22×C3×Q8C3×D311yescharacteristic
83#1007—34no—
84#1007C2×C914yescomplemented
85C2×#239—31no—
86C2×#239C2×C911yescharacteristic, complemented
87C2×D3×Q8—31no—
88C2×D3×Q8C2×C911yescharacteristic, complemented
89C2×#235—31no—
90C2×#235—31no—
91C23×C32C2×A411yescharacteristic
92C22×C3×D3—31no—
93C22×C3×D3C2×A411yescharacteristic
94C32×Q8—31no—
95C32×Q8—32no—
96C32×Q8—32no—
97C32×Q8C23×C311yescharacteristic
98C2×C32×C4—31no—
99C2×C32×C4—31no—
100C2×C32×C4—32no—
101C2×C32×C4—32no—
102C3×#67—34no—
103C3×#67C2×A414yes—
104C2×C3×Dic3—31no—
105C2×C3×Dic3C2×A411yescharacteristic
106C3×D12—32no—
107C3×D12—32no—
108C3×D12—34no—
109C3×C4×D3—38no—
110C3×C4×D3—32no—
111C3×C4×D3—32no—
112C3×C4×D3—32no—
113C3×C4×D3—32no—
114C3×Dic6—32no—
115C3×Dic6—34no—
116C3×Dic6—32no—
117C23×C9—41no—
118C23×C9—81no—
119C23×C9—121no—
120D4×C9—124no—
121C2×C4×C9—121no—
122Q8⋊C9—21no—
123Q8⋊C9C22×D311yescharacteristic
124C2×2−1+4—31noSylow
125C2×C3×C9—42no—
126C2×C3×C9—41no—
127C2×C3×C9—82no—
128D3×C9—82no—
129C3×#41—32no—
130C3×#41—94no—
131C3×#41—94no—
132C3×#41—94no—
133C3×#41—94no—
134C3×#41—94no—
135C3×#41—94no—
136C3×#41—92no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—31no—
140C2×C3×Q8—32no—
141C2×C3×Q8—31no—
142C2×C3×Q8—32no—
143C2×C3×Q8—32no—
144C2×C3×Q8—61no—
145C2×C3×Q8—62no—
146C2×C3×Q8—91no—
147C2×C3×Q8—91no—
148C2×C3×Q8—94no—
149C2×C3×Q8—94no—
150C2×C3×Q8—92no—
151C2×C3×Q8C22×C911yescharacteristic
152C2×C3×Q8C22×C912yescomplemented
153C2×C3×Q8C2×C3×D311yescharacteristic
154C2×C3×Q8C2×C3×D312yes—
155C2×C3×D4—31no—
156C2×C3×D4—91no—
157C2×C3×D4—91no—
158C2×C3×D4—91no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—61no—
162C22×C3×C4—91no—
163C22×C3×C4—91no—
164C22×C3×C4—92no—
165C2×#67—31no—
166C2×#67C22⋊C911yescharacteristic
167#239—34no—
168#239—34no—
169#239—32no—
170#239C22×C912yescomplemented
171D3×Q8—34no—
172D3×Q8—32no—
173D3×Q8—34no—
174D3×Q8C22×C912yescomplemented
175#235—34no—
176#235—34no—
177#235—34no—
178#235—34no—
179C2×D12—31no—
180C2×D12—31no—
181C2×C4×D3—32no—
182C2×C4×D3—31no—
183C2×C4×D3—31no—
184C2×Dic6—31no—
185C2×Dic6—31no—
186C22×C32—22no—
187C22×C32C22×A412yes—
188C22×C32C22×A411yescharacteristic
189C2×C3×D3—22no—
190C2×C3×D3—32no—
191C2×C3×D3—62no—
192C2×C3×D3C22×A412yes—
193C32×C4—31no—
194C32×C4—32no—
195C32×C4—31no—
196C3×Dic3—32no—
197C3×Dic3C22×A412yes—
198C22×C9—42no—
199C22×C9—41no—
200C22×C9—82no—
201C22×C9—84no—
202C22×C9—82no—
203C22×C9—81no—
204C22×C9—122no—
205C22×C9—242no—
206C4×C9—122no—
2072−1+4—34no—
2082−1+4—94no—
209C2×#41—31no—
210C2×#41—91no—
211C2×#41—91no—
212C2×#41—91no—
213C22×Q8—31no—
214C22×Q8—91no—
215C22×Q8D3×C911yescharacteristic, complemented
216C3×C9—41noSylow
217C23×C3—21no—
218C23×C3—31no—
219C23×C3—91no—
220C23×C3C2×C22⋊C911yescharacteristic
221C23×C3D3×A411yescharacteristic
222C22×D3—31no—
223C22×D3C2×C22⋊C911yescharacteristic
224C3×Q8—21no—
225C3×Q8—31no—
226C3×Q8—32no—
227C3×Q8—31no—
228C3×Q8—32no—
229C3×Q8—32no—
230C3×Q8—32no—
231C3×Q8—61no—
232C3×Q8—62no—
233C3×Q8—62no—
234C3×Q8—94no—
235C3×Q8—92no—
236C3×Q8—92no—
237C3×Q8C23×C911yescharacteristic
238C3×Q8C22×C3×D311yescharacteristic
239C3×D4—34no—
240C3×D4—94no—
241C3×D4—92no—
242C3×D4—92no—
243C3×D4—94no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—32no—
252C2×C3×C4—32no—
253C2×C3×C4—61no—
254C2×C3×C4—62no—
255C2×C3×C4—61no—
256C2×C3×C4—62no—
257C2×C3×C4—92no—
258C2×C3×C4—92no—
259C2×C3×C4—92no—
260C2×C3×C4—92no—
261C2×C3×C4—98no—
262C2×C3×C4—91no—
263#67—34no—
264#67C2×C22⋊C914yes—
265C2×Dic3—31no—
266C2×Dic3C2×C22⋊C911yescharacteristic
267D12—32no—
268D12—32no—
269D12—34no—
270C4×D3—38no—
271C4×D3—32no—
272C4×D3—32no—
273C4×D3—32no—
274C4×D3—32no—
275Dic6—32no—
276Dic6—34no—
277Dic6—32no—
278C2×C32—22no—
279C2×C322−1+4⋊C312yes—
280C2×C32C23×A411yescharacteristic
281C3×D3—22no—
282C3×D3—62no—
283C2×C9—42no—
284C2×C9—41no—
285C2×C9—82no—
286C2×C9—84no—
287C2×C9—82no—
288C2×C9—81no—
289C2×C9—242no—
290#41—32no—
291#41—94no—
292#41—94no—
293#41—94no—
294#41—94no—
295#41—94no—
296#41—94no—
297#41—92no—
298C2×Q8—31no—
299C2×Q8—32no—
300C2×Q8—32no—
301C2×Q8—91no—
302C2×Q8—91no—
303C2×Q8—94no—
304C2×Q8—94no—
305C2×Q8—92no—
306C2×Q8C2×D3×C911yescharacteristic
307C2×Q8C2×D3×C912yescomplemented
308C2×D4—31no—
309C2×D4—91no—
310C2×D4—91no—
311C2×D4—91no—
312C22×C4—31no—
313C22×C4—91no—
314C22×C4—91no—
315C22×C4—92no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—32no—
322C22×C3—62no—
323C22×C3—92no—
324C22×C3—182no—
325C22×C3C22×C22⋊C912yes—
326C22×C3C22×C22⋊C911yescharacteristic
327C22×C3C2×D3×A412yes—
328C22×C3C2×D3×A411yescharacteristic
329C2×D3—22no—
330C2×D3—32no—
331C2×D3—62no—
332C2×D3C22×C22⋊C912yes—
333C3×C4—31no—
334C3×C4—32no—
335C3×C4—31no—
336C3×C4—32no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—32no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—62no—
343C3×C4—92no—
344Dic3—32no—
345Dic3C22×C22⋊C912yes—
346C32C2×2−1+4⋊C311yescharacteristic
347C9—41no—
348C9—81no—
349C23—31no—
350C23—91no—
351C23D3×C22⋊C911yescharacteristic
352Q8—31no—
353Q8—32no—
354Q8—32no—
355Q8—94no—
356Q8—92no—
357Q8—92no—
358Q8C22×D3×C911yescharacteristic
359D4—34no—
360D4—94no—
361D4—92no—
362D4—92no—
363D4—94no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—32no—
367C2×C4—32no—
368C2×C4—31no—
369C2×C4—92no—
370C2×C4—92no—
371C2×C4—92no—
372C2×C4—92no—
373C2×C4—98no—
374C2×C4—91no—
375C2×C3—24no—
376C2×C3—22no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—62no—
381C2×C3—182no—
382C2×C32−1+4⋊C912yescomplemented
383C2×C3C23×C22⋊C911yescharacteristic
384C2×C3#1007⋊C312yes—
385C2×C3C22×D3×A411yescharacteristic
386D3—22no—
387D3—62no—
388C22—22no—
389C22—32no—
390C22—62no—
391C22—92no—
392C22—182no—
393C22C2×D3×C22⋊C912yes—
394C22C2×D3×C22⋊C911yescharacteristic
395C4—31no—
396C4—31no—
397C4—32no—
398C4—32no—
399C4—92no—
400C3—21no—
401C3C2×2−1+4⋊C911yescharacteristic, complemented
402C3C2×#1007⋊C311yescharacteristic
403C2—22no—
404C2—62no—
405C2—182no—
406C2#1007⋊C912yesdirect product factor
407C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×A4
commutator, hypocenterC3×Q8C23×C9
FrattiniC2×C3C22×D3×A4
FittingC22×C32×Q8C2×C3
socleC22×C32C22×A4
prime radicalC2×C3×#1007C3
cosocleC32×Q8C23×C3
hypercenterC23×C3D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q8⋊C9C2semidirect
2. #1007⋊C9C2direct (N × H)
3. C2×D3×Q8⋊C9C2semidirect
4. C2×C3⋊(#41⋊C9)C2semidirect
9. C2×C3×Q8⋊C9C22semidirect
10. D3×Q8⋊C9C22semidirect
11. C3⋊(#41⋊C9)C22semidirect
25. C22×Q8⋊C9D3semidirect
32. C2×#1007C9semidirect
65. C2×Q8⋊C9C2×D3semidirect
81. C22×C3×Q8C2×C9semidirect
84. #1007C2×C9semidirect
86. C2×#239C2×C9semidirect
88. C2×D3×Q8C2×C9semidirect
152. C2×C3×Q8C22×C9semidirect
170. #239C22×C9semidirect
174. D3×Q8C22×C9semidirect
215. C22×Q8D3×C9semidirect
307. C2×Q8C2×D3×C9semidirect
382. C2×C32−1+4⋊C9semidirect
401. C3C2×2−1+4⋊C9semidirect
406. C2#1007⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8⋊C92
#1007⋊C92
C2×D3×Q8⋊C92
C2×C3⋊(#41⋊C9)2
C2×C3×#10073
C2×2−1+4⋊C93
C2×C9×#674