Finite Group Databasebeta

← C3×#70987 #11798738

Subgroups — C3×#70987 #11798738

Order 1152.

569 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 82 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×#5683C211yesmaximal, characteristic, complemented
2C32×#454C211yesmaximal, characteristic, complemented
3C3×#5245C212yesmaximal, complemented
4C3×#4542C211yesmaximal, characteristic, complemented
5C3×#4477C212yesmaximal, complemented
6C3×#2202—31nomaximal
7#70987C311yesmaximal, characteristic, direct product factor
8C32×2−1+4C2211yescharacteristic, complemented
9C32×AGL(1,ℤ/8)—21no—
10C32×#124C2211yescharacteristic
11C3×#949—22no—
12C3×#947C2212yes—
13C3×#937—21no—
14C3×#885—22no—
15C3×#882—22no—
16C3×#875—21no—
17C32×#104—21no—
18C32×#101C2211yescharacteristic, complemented
19C3×#837—22no—
20C3×#836C2212yescomplemented
21C3×H(3,𝔽4)—31no—
22#5683C2×C311yescharacteristic, complemented
23C3×#454—21no—
24C3×#454—32no—
25C3×#454D311yescharacteristic, complemented
26C3×#454C2×C311yescharacteristic, complemented
27#5245C2×C312yescomplemented
28C3×#355—31no—
29C3×#355—32no—
30#4542C2×C311yescharacteristic, complemented
31#4477C2×C312yescomplemented
32C23×C3×D3—21no—
33C32×#41—21no—
34C32×#41—22no—
35C32×#41—41no—
36C2×C32×Q8—41no—
37C2×C32×Q8C2311yescharacteristic
38C2×C32×D4D411yescharacteristic
39C2×C3×#67—22no—
40C2×C3×D12—21no—
41C2×C3×D12D412yes—
42C2×C3×Dic6—21no—
43C32×SD16—41no—
44C32×D8—41no—
45C32×M16—21no—
46C32×C22⋊C4—22no—
47C32×C42D411yescharacteristic
48C3×#217—21no—
49C3×#215—42no—
50C3×#213—42no—
51C3×(C2×D3)⋊C4—22no—
52C3×(C2×D3)⋊C4—22no—
53C3×(C2×D3)⋊C4—22no—
54C3×C4×Dic3D412yes—
55C3×#208—22no—
56#2202—31noSylow
57C3×2−1+4—21no—
58C3×2−1+4C2×D311yescharacteristic, complemented
59C3×2−1+4C22×C311yescharacteristic, complemented
60C3×AGL(1,ℤ/8)—22no—
61C3×AGL(1,ℤ/8)—21no—
62C3×AGL(1,ℤ/8)—21no—
63C3×AGL(1,ℤ/8)—62no—
64C3×#124—21no—
65C3×#124—32no—
66C3×#124—62no—
67C3×#124—61no—
68C3×#124C2×D311yescharacteristic
69C3×#124C22×C311yescharacteristic
70C3×#120—62no—
71C3×#120—61no—
72#949—22no—
73#947C22×C312yes—
74#937—21no—
75#885—22no—
76#882—22no—
77#875—21no—
78C3×#104—22no—
79C3×#104—21no—
80C3×#104—21no—
81C3×#104—62no—
82C3×#101—21no—
83C3×#101—32no—
84C3×#101C2×D311yescharacteristic, complemented
85C3×#101C22×C311yescharacteristic, complemented
86#837—22no—
87#836C22×C312yescomplemented
88C23×C32—21no—
89C22×C3×D3—22no—
90C22×C3×D3—82no—
91C22×C3×D3—81no—
92C32×Q8—21no—
93C32×Q8—22no—
94C32×Q8—41no—
95C32×D4—21no—
96C32×D4—22no—
97C32×D4—41no—
98C32×D4—41no—
99C2×C32×C4—21no—
100C2×C32×C4—41no—
101C2×C32×C4—42no—
102C2×C32×C4C2×D412yes—
103C2×C32×C4C2×D411yescharacteristic
104C3×#67—82no—
105C2×C3×Dic3—22no—
106C3×D12—42no—
107C3×D12—81no—
108C3×Dic6—81no—
109C32×C8—41no—
110C3×C3⋊C8—42no—
111H(3,𝔽4)—31no—
112#454—32no—
113#454C3×D311yescharacteristic, complemented
114#355—31no—
115#355—32no—
116C24×C3—61no—
117C23×D3—21no—
118C3×#41—22no—
119C3×#41—22no—
120C3×#41—21no—
121C3×#41—22no—
122C3×#41—21no—
123C3×#41—42no—
124C3×#41—41no—
125C3×#41—41no—
126C3×#41—42no—
127C2×C3×Q8—21no—
128C2×C3×Q8—41no—
129C2×C3×Q8—41no—
130C2×C3×Q8—42no—
131C2×C3×Q8—61no—
132C2×C3×Q8C22×D311yescharacteristic
133C2×C3×Q8C23×C311yescharacteristic
134C2×C3×D4—21no—
135C2×C3×D4—32no—
136C2×C3×D4—61no—
137C2×C3×D4—62no—
138C2×C3×D4#6711yescharacteristic
139C2×C3×D4C3×D411yescharacteristic
140C2×#67—22no—
141C2×D12—21no—
142C2×D12C3×D412yes—
143C2×Dic6—21no—
144C3×SD16—41no—
145C3×SD16—41no—
146C3×SD16—42no—
147C3×SD16—122no—
148C3×D8—41no—
149C3×D8—41no—
150C3×D8—42no—
151C3×D8—122no—
152C3×M16—22no—
153C3×M16—21no—
154C3×M16—21no—
155C3×M16—62no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—22no—
158C3×C22⋊C4—42no—
159C3×C22⋊C4—62no—
160C3×C22⋊C4—61no—
161C3×C22⋊C4—62no—
162C3×C22⋊C4—62no—
163C3×C42—21no—
164C3×C42—32no—
165C3×C42#6711yescharacteristic
166C3×C42C3×D411yescharacteristic
167#217—21no—
168#215—42no—
169#213—42no—
170(C2×D3)⋊C4—22no—
171(C2×D3)⋊C4—22no—
172(C2×D3)⋊C4—22no—
173C4×Dic3C3×D412yes—
174#208—22no—
175C22×C32—41no—
176C22×C32—42no—
177C22×C32—41no—
178C22×C32#12011yescharacteristic
179C2×C3×D3—42no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—84no—
183C32×C4—21no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C3×Dic3—42no—
1882−1+4C2×C3×D311yescharacteristic, complemented
189AGL(1,ℤ/8)—21no—
190AGL(1,ℤ/8)—62no—
191#124—32no—
192#124—62no—
193#124—61no—
194#124C2×C3×D311yescharacteristic
195#120—62no—
196#120—61no—
197#104—21no—
198#104—62no—
199#101—32no—
200#101C2×C3×D311yescharacteristic, complemented
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—41no—
204C23×C3—62no—
205C23×C3—242no—
206C23×C3—241no—
207C22×D3—22no—
208C22×D3—82no—
209C22×D3—81no—
210C3×Q8—22no—
211C3×Q8—22no—
212C3×Q8—21no—
213C3×Q8—21no—
214C3×Q8—22no—
215C3×Q8—42no—
216C3×Q8—41no—
217C3×Q8—41no—
218C3×Q8—42no—
219C3×Q8—241no—
220C3×D4—22no—
221C3×D4—22no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—22no—
225C3×D4—42no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—42no—
229C3×D4—42no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—122no—
233C3×D4—241no—
234C3×D4—242no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—44no—
246C2×C3×C4—62no—
247C2×C3×C4D3×D412yes—
248C2×C3×C4C2×#6711yescharacteristic
249C2×C3×C4C2×C3×D412yes—
250C2×C3×C4C2×C3×D411yescharacteristic
251#67—82no—
252C2×Dic3—22no—
253D12—42no—
254D12—81no—
255Dic6—81no—
256C3×C8—42no—
257C3×C8—41no—
258C3×C8—41no—
259C3×C8—122no—
260C3⋊C8—42no—
261C2×C32—21no—
262C2×C32—81no—
263C2×C32—81no—
264C2×C32B(4,𝔽2)11yescharacteristic
265C3×D3—82no—
266C24—61no—
267#41—22no—
268#41—21no—
269#41—41no—
270C2×Q8—41no—
271C2×Q8—61no—
272C2×Q8C22×C3×D311yescharacteristic
273C2×D4—32no—
274C2×D4—61no—
275C2×D4—62no—
276C2×D4C3×#6711yescharacteristic
277SD16—41no—
278SD16—122no—
279D8—41no—
280D8—122no—
281M16—21no—
282M16—62no—
283C22⋊C4—22no—
284C22⋊C4—62no—
285C22⋊C4—61no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C42—32no—
289C42C3×#6711yescharacteristic
290C22×C3—21no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—81no—
299C22×C3—82no—
300C22×C3—122no—
301C22×C3—122no—
302C22×C3—122no—
303C22×C3—244no—
304C22×C3#93711yescharacteristic
305C22×C3C3×#12011yescharacteristic
306C2×D3—42no—
307C2×D3—42no—
308C2×D3—42no—
309C2×D3—84no—
310C3×C4—22no—
311C3×C4—21no—
312C3×C4—22no—
313C3×C4—22no—
314C3×C4—21no—
315C3×C4—42no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—42no—
319C3×C4—41no—
320C3×C4—41no—
321C3×C4—81no—
322C3×C4—122no—
323Dic3—42no—
324C32#220211yescharacteristic, complemented, Sylow
325C23—21no—
326C23—62no—
327C23—242no—
328C23—241no—
329Q8—21no—
330Q8—22no—
331Q8—41no—
332Q8—241no—
333D4—21no—
334D4—22no—
335D4—41no—
336D4—41no—
337D4—122no—
338D4—241no—
339D4—242no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—62no—
344C2×C4C3×D3×D412yes—
345C2×C4C2×C3×#6711yescharacteristic
346C8—41no—
347C8—122no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—41no—
352C2×C3—82no—
353C2×C3—81no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3—81no—
357C2×C3—161no—
358C2×C3—242no—
359C2×C3#503211yescharacteristic
360C2×C3C3×B(4,𝔽2)11yescharacteristic
361D3—82no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—122no—
366C22—122no—
367C22—122no—
368C22—244no—
369C22C3×#93711yescharacteristic
370C4—21no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—122no—
375C3—21no—
376C3#7098711yescharacteristic, direct product factor
377C3C3×#220211yescharacteristic, complemented
378C2—21no—
379C2—81no—
380C2—81no—
381C2—242no—
382C2C3×#503211yescharacteristic
subgroupG/H
centerC2×C3#5032
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC2×Q8C22×C3×D3
FittingC32×#454C2
socleC2×C32B(4,𝔽2)
hypercenterC3×#454D3
hypocenterC3C3×#2202
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#454

Complements

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

NH ≅ G/Ntype
1. C3×#5683C2semidirect
2. C32×#454C2semidirect
3. C3×#5245C2semidirect
4. C3×#4542C2semidirect
5. C3×#4477C2semidirect
7. #70987C3direct (N × H)
8. C32×2−1+4C22semidirect
18. C32×#101C22semidirect
20. C3×#836C22semidirect
22. #5683C2×C3semidirect
25. C3×#454D3semidirect
26. C3×#454C2×C3semidirect
27. #5245C2×C3semidirect
30. #4542C2×C3semidirect
31. #4477C2×C3semidirect
58. C3×2−1+4C2×D3semidirect
59. C3×2−1+4C22×C3semidirect
84. C3×#101C2×D3semidirect
85. C3×#101C22×C3semidirect
87. #836C22×C3semidirect
113. #454C3×D3semidirect
188. 2−1+4C2×C3×D3semidirect
200. #101C2×C3×D3semidirect
324. C32#2202semidirect
376. C3#70987direct (N × H)
377. C3C3×#2202semidirect

Maximal subgroups

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

subgroupindex
C3×#56832
C32×#4542
C3×#52452
C3×#45422
C3×#44772
C3×#22023
#709873