Finite Group Databasebeta

← #85887⋊C3 #11923225

Subgroups — #85887⋊C3 #11923225

Order 1152.

765 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 765 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#1010⋊C3C211yesmaximal, characteristic
2C3×#520⋊C3C211yesmaximal, characteristic, complemented
3Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
4#5572⋊C3C212yesmaximal, complemented
5C4×D3×SL(2,𝔽3)C212yesmaximal, complemented
6#85887C311yesmaximal, characteristic, complemented
7C4×2−1+4⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#1010⋊C3C414yescomplemented
10C2×D3×SL(2,𝔽3)C2212yes—
11C3×#893—41nomaximal
12C3×C4×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14Dic3×SL(2,𝔽3)C2211yescharacteristic
15C2×#1010C2×C311yescharacteristic
16C2×2−1+4⋊C3—31no—
17C3×#520C2×C311yescharacteristic, complemented
18Dic3×#41C2×C311yescharacteristic, complemented
19#5572C2×C312yescomplemented
20C4×D3×Q8C2×C312yescomplemented
21#5546—32no—
22C4×#237—32no—
23#5522—31no—
24#5516—31no—
25#5504—31no—
26C2×#887—31no—
27#520⋊C3—21no—
28#520⋊C3D311yescharacteristic, complemented
29#519⋊C3—32no—
30C4×#41⋊C3—31no—
31C2×C4×SL(2,𝔽3)—32no—
32C2×C3×D12—41no—
33C2×C3×C4×D3—42no—
34C3×#41⋊C3C2×C412yescomplemented
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36D3×SL(2,𝔽3)C2×C414yescomplemented
37C32×C4⋊C4—41no—
38C3×(C2×D3)⋊C4—42no—
39C3×C4×Dic3—41no—
40C4×2−1+4—31noSylow
41C2×C3×#41C22×C311yescharacteristic
42#1010—34no—
43#1010C3×C414yescomplemented
44C2×D3×Q8C22×C312yes—
45C2×#237—32no—
46C2×#235—31no—
47C22×Dic6—31no—
482−1+4⋊C3—34no—
49C2×#41⋊C3—21no—
50C2×#41⋊C3C2×D311yescharacteristic
51C22×SL(2,𝔽3)—32no—
52C3×C4×Q8C22×C312yes—
53C3×C4×D4—32no—
54C3×C4×D4—121no—
55C3×#117—31no—
56C2×C3×C4⋊C4—31no—
57Q8×Dic3C22×C311yescharacteristic
58Q8×Dic3C22×C311yescharacteristic
59D4×Dic3—31no—
60D4×Dic3—31no—
61C4×#67—34no—
62#926—31no—
63C2×Dic3⋊C4—32no—
64C2×C4×Dic3—32no—
65C2×C4×Dic3—31no—
66#893—31no—
67#893A411yescharacteristic
68#892—32no—
69D3×C4⋊C4—32no—
70#887—32no—
71#887—31no—
72#887—32no—
73#887—31no—
74#887—32no—
75#67⋊C4—32no—
76#67⋊C4—32no—
77#877—34no—
78#872—32no—
79C42×D3—32no—
80C4×Dic6—32no—
81C4×Dic6—32no—
82C4×SL(2,𝔽3)—22no—
83C4×SL(2,𝔽3)—31no—
84C4×SL(2,𝔽3)—31no—
85C4×SL(2,𝔽3)C2×D312yes—
86C22×C3×D3—42no—
87C2×C32×C4—41no—
88C2×C32×C4—42no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—41no—
91C3×D12—44no—
92C3×C4×D3—84no—
93C3×SL(2,𝔽3)C22×C411yescharacteristic
94C2×2−1+4—31no—
95#520—91no—
96#520—92no—
97#520C3×D311yescharacteristic, complemented
98#519—32no—
99#519—91no—
100C4×#41—31no—
101C4×#41—91no—
102C4×#41—92no—
103C2×C4×Q8—32no—
104C2×C4×Q8—91no—
105C3×#41—32no—
106C3×#41C2×C3×C412yescomplemented
107C2×C3×Q8C23×C311yescharacteristic
108C2×C3×D4—31no—
109C2×C3×D4—121no—
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—122no—
113C2×#67—32no—
114C22×Dic3—32no—
115C22×Dic3—31no—
116C22×Dic3—31no—
117D3×Q8—34no—
118D3×Q8C2×C3×C414yescomplemented
119#237—38no—
120#237—38no—
121#235—34no—
122#235—34no—
123C2×D12C2×A411yescharacteristic
124C2×C4×D3—32no—
125C2×C4×D3—32no—
126C2×C4×D3C2×A412yes—
127C2×Dic6—34no—
128C2×Dic6—31no—
129C2×Dic6—32no—
130C2×Dic6—34no—
131#41⋊C3—22no—
132#41⋊C3C4×D312yescomplemented
133C2×SL(2,𝔽3)—21no—
134C2×SL(2,𝔽3)—34no—
135C2×SL(2,𝔽3)C22×D311yescharacteristic
136C3×C4⋊C4—32no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—81no—
140C3×C4⋊C4C2×A411yescharacteristic
141C3×C22⋊C4—32no—
142C3×C22⋊C4—122no—
143C3×C42—32no—
144C3×C42—121no—
145#217—32no—
146(C2×D3)⋊C4—32no—
147(C2×D3)⋊C4C2×A412yes—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150Dic3⋊C4—34no—
151Dic3⋊C4—32no—
152Dic3⋊C4—32no—
153C4×Dic3—34no—
154C4×Dic3—32no—
155C4×Dic3—31no—
156C4×Dic3—31no—
157C4×Dic3—32no—
158C4×Dic3—31no—
159C4×Dic3C2×A411yescharacteristic
160C22×C32—41no—
161C2×C3×D3—44no—
162C2×C3×D3—82no—
163C2×C3×D3—82no—
164C32×C4—42no—
165C32×C4—82no—
166C3×Dic3—42no—
167C3×Dic3—81no—
1682−1+4—34no—
1692−1+4—94no—
170C2×#41—91no—
171C2×#41—92no—
172C2×#41C2×C3×D311yescharacteristic
173C22×Q8—32no—
174C22×Q8—91no—
175C4×Q8—31no—
176C4×Q8—31no—
177C4×Q8—91no—
178C4×Q8—92no—
179C4×Q8—92no—
180C4×Q8—92no—
181C4×Q8—92no—
182C4×Q8—91no—
183C4×Q8—92no—
184C4×Q8C2×C3×D312yes—
185C4×D4—31no—
186C4×D4—32no—
187C4×D4—92no—
188C4×D4—92no—
189C4×D4—91no—
190C4×D4—94no—
191C4×D4—91no—
192C4×D4—91no—
193#117—31no—
194#117—92no—
195#117—94no—
196#117—92no—
197#117—91no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—92no—
200C2×C4⋊C4—92no—
201C2×C42—92no—
202C2×C42—91no—
203C2×C42—92no—
204C23×C3—31no—
205C23×C3—122no—
206C22×D3C22×A412yes—
207C3×Q8—31no—
208C3×Q8C22×C3×C411yescharacteristic
209C3×D4—32no—
210C3×D4—32no—
211C3×D4—124no—
212C2×C3×C4—31no—
213C2×C3×C4—32no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—81no—
219C2×C3×C4—82no—
220C2×C3×C4—121no—
221C2×C3×C4—121no—
222C2×C3×C4—122no—
223C2×C3×C4—244no—
224C2×C3×C4C22×A412yes—
225C2×C3×C4C22×A411yescharacteristic
226#67—38no—
227C2×Dic3—38no—
228C2×Dic3—32no—
229C2×Dic3—31no—
230C2×Dic3—31no—
231C2×Dic3—62no—
232C2×Dic3—121no—
233C2×Dic3C22×A411yescharacteristic
234C2×Dic3C22×A411yescharacteristic
235D12C4×A414yes—
236C4×D3—24no—
237C4×D3—34no—
238C4×D3—34no—
239C4×D3—122no—
240Dic6—34no—
241Dic6—34no—
242Dic6—34no—
243SL(2,𝔽3)—21no—
244SL(2,𝔽3)C2×C4×D311yescharacteristic
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C3×D3—84no—
249#41—32no—
250#41—94no—
251#41—94no—
252#41—98no—
253#41—98no—
254#41C3×C4×D312yescomplemented
255C2×Q8—34no—
256C2×Q8—94no—
257C2×Q8—94no—
258C2×Q8—92no—
259C2×Q8—91no—
260C2×Q8—94no—
261C2×Q8C22×C3×D311yescharacteristic
262C2×D4—31no—
263C2×D4—31no—
264C2×D4—92no—
265C22×C4—31no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—92no—
269C22×C4—91no—
270C22×C4—91no—
271C22×C4—92no—
272C22×C4—92no—
273C4⋊C4—32no—
274C4⋊C4—31no—
275C4⋊C4—92no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—94no—
279C4⋊C4—92no—
280C4⋊C4D3×A411yescharacteristic
281C22⋊C4—32no—
282C22⋊C4—32no—
283C22⋊C4—92no—
284C22⋊C4—92no—
285C42—32no—
286C42—31no—
287C42—91no—
288C42—92no—
289C42—91no—
290C42—94no—
291C42—92no—
292C42—91no—
293C22×C3—32no—
294C22×C3—41no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—81no—
298C22×C3—124no—
299C22×C3—242no—
300C22×C3—242no—
301C22×C3C23×A411yescharacteristic
302C2×D3—22no—
303C2×D3—22no—
304C2×D3C2×C4×A414yes—
305C3×C4—22no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—42no—
309C3×C4—62no—
310C3×C4—82no—
311C3×C4—82no—
312C3×C4—122no—
313C3×C4—162no—
314C3×C4—241no—
315C3×C4C2×C4×A412yes—
316Dic3—21no—
317Dic3—34no—
318Dic3—61no—
319Dic3—61no—
320Dic32−1+4⋊C312yescomplemented
321C32—41noSylow
322C23—31no—
323C23—32no—
324Q8—31no—
325Q8—94no—
326Q8—94no—
327Q8—94no—
328Q8C2×C3×C4×D311yescharacteristic
329D4—32no—
330D4—32no—
331D4—34no—
332D4—98no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—31no—
339C2×C4—64no—
340C2×C4—94no—
341C2×C4—94no—
342C2×C4—98no—
343C2×C4—92no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—122no—
347C2×C4—182no—
348C2×C4—362no—
349C2×C4—361no—
350C2×C4C2×D3×A412yes—
351C2×C4C2×D3×A411yescharacteristic
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—62no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3—81no—
359C2×C3—244no—
360C2×C3C22×C4×A411yescharacteristic
361C2×C3C2×2−1+4⋊C311yescharacteristic
362C2×C3#594511yescharacteristic
363D3—24no—
364C22—32no—
365C22—34no—
366C22—61no—
367C22—62no—
368C22—62no—
369C22—61no—
370C22C22×D3×A411yescharacteristic
371C4—22no—
372C4—31no—
373C4—31no—
374C4—32no—
375C4—62no—
376C4—61no—
377C4—94no—
378C4—181no—
379C4—181no—
380C4C4×D3×A412yes—
381C3—41no—
382C3—81no—
383C3C4×2−1+4⋊C311yescharacteristic, complemented
384C2—64no—
385C2—62no—
386C2C2×C4×D3×A411yescharacteristic
387C2#1059662311yescharacteristic
388C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FittingC3×#520C2×C3
socleC22×C3C23×A4
prime radicalC2×#1010⋊C3C2
cosocleC2×C3×Q8C23×C3
hypercenterC4⋊C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. #5572⋊C3C2semidirect
5. C4×D3×SL(2,𝔽3)C2semidirect
6. #85887C3semidirect
9. #1010⋊C3C4semidirect
17. C3×#520C2×C3semidirect
18. Dic3×#41C2×C3semidirect
19. #5572C2×C3semidirect
20. C4×D3×Q8C2×C3semidirect
28. #520⋊C3D3semidirect
34. C3×#41⋊C3C2×C4semidirect
36. D3×SL(2,𝔽3)C2×C4semidirect
43. #1010C3×C4semidirect
97. #520C3×D3semidirect
106. C3×#41C2×C3×C4semidirect
118. D3×Q8C2×C3×C4semidirect
132. #41⋊C3C4×D3semidirect
254. #41C3×C4×D3semidirect
320. Dic32−1+4⋊C3semidirect
383. C3C4×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1010⋊C32
C3×#520⋊C32
Dic3×#41⋊C32
#5572⋊C32
C4×D3×SL(2,𝔽3)2
#858873
C4×2−1+4⋊C33
C3×#8934