Finite Group Databasebeta

← #85885⋊C3 #11923224

Subgroups — #85885⋊C3 #11923224

Order 1152.

751 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 111 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×D3×#41⋊C3C211yesmaximal, characteristic
2C3×#520⋊C3C211yesmaximal, characteristic, complemented
3#5827⋊C3C211yesmaximal, characteristic, complemented
4#5572⋊C3C212yesmaximal, complemented
5C4×C3⋊(#41⋊C3)C212yesmaximal, complemented
6#85885C311yesmaximal, characteristic, complemented
7#3428⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9D3×#41⋊C3C414yescomplemented
10C2×D3×SL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊(#41⋊C3)C2211yescharacteristic
12C3×#892—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14Dic3×SL(2,𝔽3)C2212yescomplemented
15C2×D3×#41C2×C311yescharacteristic
16C22×#41⋊C3—31no—
17C3×#520C2×C311yescharacteristic, complemented
18#5827C2×C311yescharacteristic, complemented
19C4×#239C2×C312yescomplemented
20#5572C2×C312yes—
21#5545—32no—
22C4×#237—32no—
23#5523—31no—
24D3×#117—31no—
25#5504—31no—
26C2×#892—31no—
27#520⋊C3—21no—
28#520⋊C3—31no—
29#520⋊C3D311yescharacteristic, complemented
30#519⋊C3—32no—
31C4×#41⋊C3—32no—
32C2×C3×C4×D3—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35D3×SL(2,𝔽3)C2×C412yescomplemented
36C3⋊(#41⋊C3)C2×C412yescomplemented
37C32×C4⋊C4—41no—
38C3×(C2×D3)⋊C4—42no—
39C3×C4⋊Dic3—41no—
40C3×C4×Dic3—42no—
41#3428—31noSylow
42C2×C3×#41C22×C311yescharacteristic
43D3×#41—34no—
44D3×#41C3×C414yescomplemented
45C2×#239C22×C311yescharacteristic
46C2×D3×Q8C22×C311yescharacteristic
47C2×#237—31no—
48C2×D3×D4—31no—
49C2×#235—31no—
50C22×C4×D3—31no—
51C2×#41⋊C3—21no—
52C2×#41⋊C3—34no—
53C2×#41⋊C3—31no—
54C2×#41⋊C3C2×D311yescharacteristic
55C22×SL(2,𝔽3)—31no—
56C3×C4×Q8C22×C312yes—
57C3×C4×D4—32no—
58C3×#117—31no—
59C3×#117—121no—
60C2×C3×C4⋊C4—31no—
61Q8×Dic3C22×C312yes—
62D4×Dic3—32no—
63C4×#67—34no—
64C2×(C2×D3)⋊C4—32no—
65#926—31no—
66C2×C4⋊Dic3—31no—
67C2×C4×Dic3—32no—
68#893—32no—
69#892—32no—
70#892—31no—
71#892—32no—
72#892—32no—
73#892A411yescharacteristic
74D3×C4⋊C4—31no—
75D3×C4⋊C4—31no—
76#887—32no—
77#67⋊C4—34no—
78D3×C22⋊C4—32no—
79#877—32no—
80C4×D12—32no—
81#872—32no—
82C42×D3—32no—
83C4×Dic6—32no—
84C4×SL(2,𝔽3)—22no—
85C4×SL(2,𝔽3)—32no—
86C4×SL(2,𝔽3)C2×D312yescomplemented
87C22×C3×D3—41no—
88C2×C32×C4—41no—
89C2×C32×C4—42no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—42no—
92C3×C4×D3—44no—
93C3×SL(2,𝔽3)C22×C411yescharacteristic
94C22×#41—31no—
95#520—31no—
96#520—91no—
97#520—91no—
98#520C3×D311yescharacteristic, complemented
99#519—32no—
100#518—92no—
101C4×#41—32no—
102C4×#41—92no—
103C2×#117—91no—
104C2×#117—91no—
105C23×D3—31no—
106C3×#41—32no—
107C3×#41C2×C3×C412yes—
108C2×C3×Q8C23×C311yescharacteristic
109C2×C3×D4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—121no—
113C2×#67—32no—
114C22×Dic3—31no—
115C22×Dic3—32no—
116#239—32no—
117#239C2×C3×C412yes—
118D3×Q8—32no—
119D3×Q8C2×C3×C412yes—
120#237—34no—
121#237—34no—
122D3×D4—34no—
123D3×D4—34no—
124#235—34no—
125#235—34no—
126C2×D12—31no—
127C2×C4×D3—31no—
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×C4×D3—34no—
131C2×C4×D3—34no—
132C2×C4×D3—32no—
133C2×C4×D3—32no—
134C2×C4×D3C2×A411yescharacteristic
135C2×Dic6—31no—
136#41⋊C3—22no—
137#41⋊C3—32no—
138#41⋊C3C4×D312yescomplemented
139C2×SL(2,𝔽3)—21no—
140C2×SL(2,𝔽3)—32no—
141C2×SL(2,𝔽3)C22×D311yescharacteristic
142C3×C4⋊C4—32no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—81no—
146C3×C4⋊C4—121no—
147C3×C4⋊C4C2×A411yescharacteristic
148C3×C22⋊C4—32no—
149C3×C22⋊C4—122no—
150C3×C42—32no—
151C3×C42—122no—
152#217—32no—
153(C2×D3)⋊C4—34no—
154(C2×D3)⋊C4—32no—
155(C2×D3)⋊C4C2×A412yes—
156C4⋊Dic3—31no—
157C4⋊Dic3—32no—
158C4⋊Dic3C2×A411yescharacteristic
159Dic3⋊C4—32no—
160Dic3⋊C4—32no—
161C4×Dic3—34no—
162C4×Dic3—32no—
163C4×Dic3—32no—
164C4×Dic3C2×A412yes—
165C22×C32—41no—
166C2×C3×D3—42no—
167C2×C3×D3—81no—
168C2×C3×D3—81no—
169C32×C4—42no—
170C32×C4—82no—
171C3×Dic3—42no—
172C3×Dic3—82no—
173C2×#41—34no—
174C2×#41—31no—
175C2×#41—91no—
176C2×#41—91no—
177C2×#41—94no—
178C2×#41C2×C3×D311yescharacteristic
179C22×Q8—31no—
180C22×D4—91no—
181C23×C4—91no—
182C4×Q8—32no—
183C4×Q8—92no—
184C4×Q8—92no—
185C4×Q8C2×C3×D312yes—
186C4×D4—32no—
187C4×D4—92no—
188C4×D4—94no—
189C4×D4—92no—
190C4×D4—94no—
191C4×D4—92no—
192#117—31no—
193#117—31no—
194#117—92no—
195#117—92no—
196#117—91no—
197#117—91no—
198#117—92no—
199#117—92no—
200#117—92no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—91no—
203C2×C4⋊C4—91no—
204C2×C4⋊C4—91no—
205C2×C22⋊C4—92no—
206C2×C22⋊C4—92no—
207C2×C42—92no—
208C2×C42—92no—
209C23×C3—31no—
210C23×C3—121no—
211C22×D3—31no—
212C22×D3—34no—
213C22×D3—121no—
214C22×D3—121no—
215C22×D3C22×A411yescharacteristic
216C3×Q8—31no—
217C3×Q8C22×C3×C411yescharacteristic
218C3×D4—32no—
219C3×D4—32no—
220C2×C3×C4—31no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—32no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—81no—
227C2×C3×C4—82no—
228C2×C3×C4—124no—
229C2×C3×C4—121no—
230C2×C3×C4—122no—
231C2×C3×C4—122no—
232C2×C3×C4C22×A412yes—
233C2×C3×C4C22×A411yescharacteristic
234#67—38no—
235C2×Dic3—31no—
236C2×Dic3—34no—
237C2×Dic3—32no—
238C2×Dic3—122no—
239C2×Dic3C22×A412yes—
240C2×Dic3C22×A411yescharacteristic
241D12—32no—
242D12—32no—
243C4×D3—32no—
244C4×D3—32no—
245C4×D3—34no—
246C4×D3—32no—
247C4×D3—32no—
248C4×D3—122no—
249C4×D3—122no—
250C4×D3C4×A414yes—
251Dic6—32no—
252Dic6—32no—
253SL(2,𝔽3)—21no—
254SL(2,𝔽3)C2×C4×D311yescharacteristic
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C3×D3—82no—
259C24—91no—
260#41—32no—
261#41—32no—
262#41—94no—
263#41—94no—
264#41—94no—
265#41—94no—
266#41—92no—
267#41C3×C4×D312yes—
268C2×Q8—32no—
269C2×Q8—91no—
270C2×Q8—92no—
271C2×Q8C22×C3×D311yescharacteristic
272C2×D4—31no—
273C2×D4—91no—
274C2×D4—94no—
275C2×D4—94no—
276C2×D4—92no—
277C22×C4—31no—
278C22×C4—31no—
279C22×C4—32no—
280C22×C4—91no—
281C22×C4—94no—
282C22×C4—94no—
283C22×C4—92no—
284C22×C4—91no—
285C22×C4—91no—
286C22×C4—91no—
287C22×C4—92no—
288C22×C4—92no—
289C4⋊C4—32no—
290C4⋊C4—31no—
291C4⋊C4—31no—
292C4⋊C4—92no—
293C4⋊C4—92no—
294C4⋊C4—91no—
295C4⋊C4—92no—
296C4⋊C4D3×A411yescharacteristic
297C22⋊C4—32no—
298C22⋊C4—32no—
299C22⋊C4—92no—
300C22⋊C4—92no—
301C22⋊C4—94no—
302C42—32no—
303C42—32no—
304C42—92no—
305C42—92no—
306C42—94no—
307C22×C3—32no—
308C22×C3—41no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—81no—
312C22×C3—122no—
313C22×C3—241no—
314C22×C3—241no—
315C22×C3C23×A411yescharacteristic
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—32no—
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—124no—
322C2×D3C2×C4×A412yes—
323C3×C4—22no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—42no—
327C3×C4—62no—
328C3×C4—82no—
329C3×C4—82no—
330C3×C4—122no—
331C3×C4—162no—
332C3×C4—242no—
333C3×C4C2×C4×A412yes—
334Dic3—22no—
335Dic3—32no—
336Dic3—62no—
337Dic3C2×C4×A412yes—
338C32—41noSylow
339C23—31no—
340C23—31no—
341C23—94no—
342C23—91no—
343C23—361no—
344C23—361no—
345Q8—31no—
346Q8—92no—
347Q8—92no—
348Q8C2×C3×C4×D311yescharacteristic
349D4—32no—
350D4—32no—
351D4—92no—
352D4—92no—
353D4—98no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—32no—
357C2×C4—32no—
358C2×C4—34no—
359C2×C4—32no—
360C2×C4—32no—
361C2×C4—92no—
362C2×C4—92no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—94no—
366C2×C4—91no—
367C2×C4—94no—
368C2×C4—92no—
369C2×C4—122no—
370C2×C4—362no—
371C2×C4—362no—
372C2×C4—362no—
373C2×C4C2×D3×A412yes—
374C2×C4C2×D3×A411yescharacteristic
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—62no—
379C2×C3—81no—
380C2×C3—81no—
381C2×C3—81no—
382C2×C3—242no—
383C2×C3C22×C4×A411yescharacteristic
384C2×C3#594511yescharacteristic
385C2×C3#594511yescharacteristic
386D3—22no—
387D3—62no—
388C22—32no—
389C22—32no—
390C22—61no—
391C22—61no—
392C22—61no—
393C22—61no—
394C22—92no—
395C22—181no—
396C22—181no—
397C22—364no—
398C22C22×D3×A411yescharacteristic
399C4—22no—
400C4—31no—
401C4—31no—
402C4—32no—
403C4—62no—
404C4—62no—
405C4—92no—
406C4—182no—
407C4C4×D3×A412yes—
408C3—41no—
409C3—81no—
410C3#3428⋊C311yescharacteristic, complemented
411C2—62no—
412C2—62no—
413C2—182no—
414C2C2×C4×D3×A411yescharacteristic
415C2#1059662311yescharacteristic
416C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FittingC3×#520C2×C3
socleC22×C3C23×A4
prime radicalC2×D3×#41⋊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. #5827⋊C3C2semidirect
4. #5572⋊C3C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #85885C3semidirect
9. D3×#41⋊C3C4semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. Dic3×SL(2,𝔽3)C22semidirect
17. C3×#520C2×C3semidirect
18. #5827C2×C3semidirect
19. C4×#239C2×C3semidirect
29. #520⋊C3D3semidirect
33. C3×#41⋊C3C2×C4semidirect
35. D3×SL(2,𝔽3)C2×C4semidirect
36. C3⋊(#41⋊C3)C2×C4semidirect
44. D3×#41C3×C4semidirect
86. C4×SL(2,𝔽3)C2×D3semidirect
98. #520C3×D3semidirect
138. #41⋊C3C4×D3semidirect
410. C3#3428⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#41⋊C32
C3×#520⋊C32
#5827⋊C32
#5572⋊C32
C4×C3⋊(#41⋊C3)2
#858853
#3428⋊C33
C3×#8924