Finite Group Databasebeta

← #85886⋊C3 #11923218

Subgroups — #85886⋊C3 #11923218

Order 1152.

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

normal subgroups only

This group has 781 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×C3⋊#994C211yesmaximal, characteristic
2C3×#520⋊C3C211yesmaximal, characteristic, complemented
3Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
4#5574⋊C3C212yesmaximal, complemented
5C4×C3⋊(#41⋊C3)C212yesmaximal, complemented
6#85886C311yesmaximal, characteristic, complemented
7C4×#994—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3⋊#994C414yescomplemented
10C2×C3⋊(#41⋊C3)C2212yes—
11C3×#887—41nomaximal
12C3×C4×SL(2,𝔽3)C2212yescomplemented
13Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
14Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
15C2×#1009C2×C311yescharacteristic
16C2×#994—31no—
17C3×#520C2×C311yescharacteristic
18Dic3×#41C2×C311yescharacteristic
19#5574C2×C312yes—
20C4×#239C2×C312yes—
21#5545—32no—
22C4×D3×D4—32no—
23#5522—31no—
24#5521—31no—
25#5504—31no—
26C2×#893—31no—
27#520⋊C3—21no—
28#520⋊C3—32no—
29#520⋊C3D311yescharacteristic, complemented
30C4×#41⋊C3—31no—
31C4×#41⋊C3—32no—
32C2×C3×Dic6—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊(#41⋊C3)C2×C414yescomplemented
36C32×C4⋊C4—41no—
37C3×Dic3⋊C4—42no—
38C3×C4×Dic3—42no—
39C3×C4×Dic3—41no—
40C4×H(5,𝔽2)—31noSylow
41C2×C3×#41C22×C311yescharacteristic
42#1009—34no—
43#1009C3×C414yescomplemented
44C2×#239C22×C312yes—
45C2×D3×D4—32no—
46C2×#235—31no—
47C22×D12—31no—
48C22×C4×D3—32no—
49#994—34no—
50C2×#41⋊C3—21no—
51C2×#41⋊C3—32no—
52C2×#41⋊C3C2×D311yescharacteristic
53C3×C4×Q8—121no—
54C3×C4×Q8C22×C312yes—
55C3×C4×D4—32no—
56C3×#117—31no—
57C2×C3×C4⋊C4—31no—
58Q8×Dic3C22×C311yescharacteristic
59Q8×Dic3C22×C311yescharacteristic
60D4×Dic3—31no—
61D4×Dic3—31no—
62C4×#67—34no—
63C2×(C2×D3)⋊C4—32no—
64#926—31no—
65C2×C4×Dic3—31no—
66#893—32no—
67#893—32no—
68#893—31no—
69#893—32no—
70#893—31no—
71#892—32no—
72D3×C4⋊C4—32no—
73#887—31no—
74#887A411yescharacteristic
75#67⋊C4—32no—
76#67⋊C4—32no—
77D3×C22⋊C4—34no—
78C4×D12—32no—
79C4×D12—32no—
80#872—32no—
81C42×D3—32no—
82C4×SL(2,𝔽3)—22no—
83C4×SL(2,𝔽3)—31no—
84C4×SL(2,𝔽3)—31no—
85C4×SL(2,𝔽3)C2×D312yescomplemented
86C2×C32×C4—41no—
87C2×C32×C4—42no—
88C2×C3×Dic3—41no—
89C2×C3×Dic3—42no—
90C2×C3×Dic3—41no—
91C3×Dic6—44no—
92C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
93C2×H(5,𝔽2)—31no—
94#520—32no—
95#520—91no—
96#520C3×D311yescharacteristic
97#518—91no—
98#518—92no—
99C4×#41—31no—
100C4×#41—32no—
101C4×#41—91no—
102C2×C4×D4—91no—
103C2×C4×D4—92no—
104C23×D3—32no—
105C3×#41—32no—
106C3×#41C2×C3×C412yes—
107C2×C3×Q8—121no—
108C2×C3×Q8C23×C311yescharacteristic
109C2×C3×D4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C2×#67—32no—
113C22×Dic3—31no—
114C22×Dic3—31no—
115#239—34no—
116#239C2×C3×C414yes—
117D3×D4—38no—
118D3×D4—38no—
119#235—34no—
120#235—34no—
121C2×D12—32no—
122C2×D12—34no—
123C2×D12—31no—
124C2×D12—34no—
125C2×C4×D3—32no—
126C2×C4×D3—32no—
127C2×C4×D3—34no—
128C2×C4×D3—32no—
129C2×C4×D3—124no—
130C2×Dic6C2×A411yescharacteristic
131#41⋊C3—22no—
132#41⋊C3—34no—
133#41⋊C3C4×D312yescomplemented
134C2×SL(2,𝔽3)—21no—
135C2×SL(2,𝔽3)C22×D311yescharacteristic
136C3×C4⋊C4—32no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—81no—
140C3×C4⋊C4—122no—
141C3×C4⋊C4C2×A411yescharacteristic
142C3×C22⋊C4—32no—
143C3×C42—32no—
144C3×C42—121no—
145C3×C42—122no—
146#217—32no—
147(C2×D3)⋊C4—34no—
148(C2×D3)⋊C4—32no—
149(C2×D3)⋊C4—32no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152Dic3⋊C4—32no—
153Dic3⋊C4C2×A412yes—
154C4×Dic3—31no—
155C4×Dic3—31no—
156C4×Dic3—32no—
157C4×Dic3—31no—
158C4×Dic3C2×A412yes—
159C4×Dic3C2×A411yescharacteristic
160C22×C32—41no—
161C32×C4—42no—
162C32×C4—82no—
163C3×Dic3—44no—
164C3×Dic3—42no—
165C3×Dic3—81no—
166H(5,𝔽2)—34no—
167H(5,𝔽2)—94no—
168C2×#41—32no—
169C2×#41—91no—
170C2×#41C2×C3×D311yescharacteristic
171C22×D4—91no—
172C22×D4—92no—
173C23×C4—92no—
174C4×Q8—31no—
175C4×Q8—31no—
176C4×Q8—31no—
177C4×Q8—91no—
178C4×Q8C2×C3×D312yes—
179C4×D4—32no—
180C4×D4—92no—
181C4×D4—91no—
182C4×D4—92no—
183C4×D4—92no—
184C4×D4—92no—
185C4×D4—92no—
186C4×D4—92no—
187C4×D4—94no—
188C4×D4—91no—
189C4×D4—91no—
190C4×D4—91no—
191C4×D4—92no—
192#117—31no—
193#117—92no—
194#117—92no—
195#117—91no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—92no—
198C2×C22⋊C4—94no—
199C2×C22⋊C4—92no—
200C2×C42—92no—
201C2×C42—91no—
202C23×C3—31no—
203C22×D3—32no—
204C22×D3—38no—
205C22×D3—122no—
206C22×D3—122no—
207C3×Q8—31no—
208C3×Q8—124no—
209C3×Q8C22×C3×C411yescharacteristic
210C3×D4—32no—
211C3×D4—32no—
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—122no—
222C2×C3×C4—121no—
223C2×C3×C4—122no—
224C2×C3×C4C22×A412yes—
225C2×C3×C4C22×A411yescharacteristic
226#67—38no—
227C2×Dic3—31no—
228C2×Dic3—31no—
229C2×Dic3—62no—
230C2×Dic3—121no—
231C2×Dic3C22×A411yescharacteristic
232C2×Dic3C22×A412yes—
233C2×Dic3C22×A411yescharacteristic
234D12—34no—
235D12—34no—
236D12—34no—
237C4×D3—34no—
238C4×D3—34no—
239C4×D3—64no—
240C4×D3—122no—
241C4×D3—124no—
242Dic6C4×A414yes—
243SL(2,𝔽3)—21no—
244SL(2,𝔽3)C2×C4×D311yescharacteristic, complemented
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C24—92no—
249#41—32no—
250#41—34no—
251#41—94no—
252#41—94no—
253#41—94no—
254#41C3×C4×D312yes—
255C2×Q8—31no—
256C2×Q8C22×C3×D311yescharacteristic
257C2×D4—31no—
258C2×D4—92no—
259C2×D4—92no—
260C2×D4—94no—
261C2×D4—94no—
262C2×D4—98no—
263C2×D4—98no—
264C2×D4—91no—
265C22×C4—31no—
266C22×C4—32no—
267C22×C4—92no—
268C22×C4—92no—
269C22×C4—91no—
270C22×C4—91no—
271C22×C4—94no—
272C22×C4—92no—
273C22×C4—364no—
274C4⋊C4—32no—
275C4⋊C4—31no—
276C4⋊C4—32no—
277C4⋊C4—92no—
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4D3×A411yescharacteristic
281C22⋊C4—32no—
282C22⋊C4—92no—
283C22⋊C4—94no—
284C22⋊C4—92no—
285C22⋊C4—92no—
286C42—32no—
287C42—32no—
288C42—31no—
289C42—91no—
290C42—91no—
291C42—92no—
292C42—91no—
293C22×C3—32no—
294C22×C3—41no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—81no—
298C22×C3C23×A411yescharacteristic
299C2×D3—34no—
300C2×D3—62no—
301C2×D3—62no—
302C2×D3—128no—
303C3×C4—22no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—42no—
307C3×C4—62no—
308C3×C4—82no—
309C3×C4—82no—
310C3×C4—124no—
311C3×C4—122no—
312C3×C4—162no—
313C3×C4—241no—
314C3×C4C2×C4×A412yes—
315Dic3—21no—
316Dic3—61no—
317Dic3—61no—
318Dic3C2×C4×A414yes—
319Dic3#99412yescomplemented
320C32—41noSylow
321C23—31no—
322C23—98no—
323C23—92no—
324C23—362no—
325C23—362no—
326Q8—31no—
327Q8—34no—
328Q8C2×C3×C4×D311yescharacteristic
329D4—32no—
330D4—32no—
331D4—98no—
332D4—94no—
333D4—94no—
334D4—94no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—31no—
342C2×C4—94no—
343C2×C4—94no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—122no—
347C2×C4—184no—
348C2×C4—182no—
349C2×C4—362no—
350C2×C4—361no—
351C2×C4—364no—
352C2×C4C2×D3×A412yes—
353C2×C4C2×D3×A411yescharacteristic
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—62no—
358C2×C3—81no—
359C2×C3—81no—
360C2×C3—81no—
361C2×C3C22×C4×A411yescharacteristic
362C2×C3C2×#99411yescharacteristic
363C2×C3#594511yescharacteristic
364D3—64no—
365C22—32no—
366C22—61no—
367C22—61no—
368C22—94no—
369C22—182no—
370C22—182no—
371C22—368no—
372C22C22×D3×A411yescharacteristic
373C4—22no—
374C4—31no—
375C4—31no—
376C4—34no—
377C4—32no—
378C4—62no—
379C4—61no—
380C4—181no—
381C4—181no—
382C4C4×D3×A412yes—
383C3—41no—
384C3—81no—
385C3C4×#99411yescharacteristic, complemented
386C2—62no—
387C2—184no—
388C2C2×C4×D3×A411yescharacteristic
389C2D3×#99411yescharacteristic
390C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FittingC3×#520C2×C3
socleC22×C3C23×A4
prime radicalC2×C3⋊#994C2
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. #5574⋊C3C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #85886C3semidirect
9. C3⋊#994C4semidirect
12. C3×C4×SL(2,𝔽3)C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
14. Dic3×SL(2,𝔽3)C22semidirect
29. #520⋊C3D3semidirect
33. C3×#41⋊C3C2×C4semidirect
35. C3⋊(#41⋊C3)C2×C4semidirect
43. #1009C3×C4semidirect
85. C4×SL(2,𝔽3)C2×D3semidirect
92. C3×SL(2,𝔽3)C22×C4semidirect
133. #41⋊C3C4×D3semidirect
244. SL(2,𝔽3)C2×C4×D3semidirect
319. Dic3#994semidirect
385. C3C4×#994semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#9942
C3×#520⋊C32
Dic3×#41⋊C32
#5574⋊C32
C4×C3⋊(#41⋊C3)2
#858863
C4×#9943
C3×#8874