Finite Group Databasebeta

← #85834⋊C3 #11923199

Subgroups — #85834⋊C3 #11923199

Order 1152.

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

normal subgroups only

This group has 753 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×C4×#41⋊C3C211yesmaximal, characteristic, complemented
3Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
4#5572⋊C3C212yesmaximal, complemented
5#5574⋊C3C212yesmaximal, complemented
6#85834C311yesmaximal, 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×#872—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14Dic3×SL(2,𝔽3)C2212yescomplemented
15C2×D3×#41C2×C311yescharacteristic
16C22×#41⋊C3—31no—
17C3×C4×#41C2×C311yescharacteristic, complemented
18Dic3×#41C2×C311yescharacteristic, complemented
19#5574C2×C312yescomplemented
20#5572C2×C312yes—
21#5546—32no—
22#5545—32no—
23#5522—31no—
24D3×#117—31no—
25C4×#235—31no—
26C2×#872—31no—
27#520⋊C3—32no—
28#519⋊C3—32no—
29C4×#41⋊C3—21no—
30C4×#41⋊C3—31no—
31C4×#41⋊C3D311yescharacteristic, complemented
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×C42—41no—
38C3×(C2×D3)⋊C4—42no—
39C3×Dic3⋊C4—42no—
40C3×C4×Dic3—41no—
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×C42—31no—
61Q8×Dic3C22×C312yes—
62D4×Dic3—32no—
63C4×#67—34no—
64C2×(C2×D3)⋊C4—32no—
65#926—31no—
66C2×Dic3⋊C4—32no—
67C2×C4×Dic3—31no—
68#893—32no—
69#892—32no—
70D3×C4⋊C4—32no—
71#887—32no—
72#67⋊C4—34no—
73D3×C22⋊C4—32no—
74#877—32no—
75C4×D12—32no—
76#872—31no—
77#872—31no—
78#872—32no—
79#872—32no—
80#872—31no—
81#872A411yescharacteristic
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—32no—
96#520—92no—
97#519—32no—
98#518—92no—
99C4×#41—31no—
100C4×#41—91no—
101C4×#41—91no—
102C4×#41C3×D311yescharacteristic, complemented
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—34no—
133C2×C4×D3C2×A411yescharacteristic
134C2×Dic6—31no—
135#41⋊C3—22no—
136#41⋊C3—32no—
137#41⋊C3C4×D312yescomplemented
138C2×SL(2,𝔽3)—21no—
139C2×SL(2,𝔽3)—32no—
140C2×SL(2,𝔽3)C22×D311yescharacteristic
141C3×C4⋊C4—32no—
142C3×C4⋊C4—122no—
143C3×C22⋊C4—32no—
144C3×C22⋊C4—122no—
145C3×C42—32no—
146C3×C42—31no—
147C3×C42—41no—
148C3×C42—81no—
149C3×C42—121no—
150C3×C42C2×A411yescharacteristic
151#217—32no—
152(C2×D3)⋊C4—32no—
153(C2×D3)⋊C4—32no—
154(C2×D3)⋊C4—32no—
155(C2×D3)⋊C4C2×A412yes—
156C4⋊Dic3—32no—
157Dic3⋊C4—32no—
158Dic3⋊C4—32no—
159Dic3⋊C4—32no—
160Dic3⋊C4C2×A412yes—
161C4×Dic3—34no—
162C4×Dic3—31no—
163C4×Dic3—32no—
164C4×Dic3C2×A411yescharacteristic
165C22×C32—41no—
166C2×C3×D3—42no—
167C2×C3×D3—82no—
168C32×C4—42no—
169C32×C4—82no—
170C3×Dic3—42no—
171C3×Dic3—82no—
172C2×#41—34no—
173C2×#41—31no—
174C2×#41—91no—
175C2×#41—91no—
176C2×#41—94no—
177C2×#41C2×C3×D311yescharacteristic
178C22×Q8—31no—
179C22×D4—91no—
180C23×C4—91no—
181C4×Q8—32no—
182C4×Q8—92no—
183C4×Q8—92no—
184C4×Q8C2×C3×D312yes—
185C4×D4—32no—
186C4×D4—92no—
187C4×D4—94no—
188C4×D4—94no—
189C4×D4—92no—
190C4×D4—92no—
191#117—31no—
192#117—31no—
193#117—91no—
194#117—92no—
195#117—92no—
196#117—91no—
197#117—92no—
198#117—92no—
199#117—91no—
200#117—91no—
201C2×C4⋊C4—92no—
202C2×C4⋊C4—92no—
203C2×C22⋊C4—92no—
204C2×C22⋊C4—92no—
205C2×C42—31no—
206C2×C42—91no—
207C2×C42—92no—
208C23×C3—31no—
209C23×C3—121no—
210C22×D3—31no—
211C22×D3—34no—
212C22×D3—122no—
213C22×D3C22×A411yescharacteristic
214C3×Q8—31no—
215C3×Q8C22×C3×C411yescharacteristic
216C3×D4—32no—
217C3×D4—32no—
218C2×C3×C4—31no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—32no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—82no—
225C2×C3×C4—81no—
226C2×C3×C4—124no—
227C2×C3×C4—121no—
228C2×C3×C4—122no—
229C2×C3×C4—122no—
230C2×C3×C4C22×A412yes—
231C2×C3×C4C22×A411yescharacteristic
232#67—38no—
233C2×Dic3—31no—
234C2×Dic3—34no—
235C2×Dic3—32no—
236C2×Dic3—122no—
237C2×Dic3C22×A412yes—
238C2×Dic3C22×A411yescharacteristic
239D12—32no—
240D12—32no—
241C4×D3—32no—
242C4×D3—32no—
243C4×D3—34no—
244C4×D3—32no—
245C4×D3—32no—
246C4×D3—124no—
247C4×D3C4×A414yes—
248Dic6—32no—
249Dic6—32no—
250SL(2,𝔽3)—21no—
251SL(2,𝔽3)C2×C4×D311yescharacteristic
252C2×C32—41no—
253C2×C32—42no—
254C3×D3—82no—
255C24—91no—
256#41—32no—
257#41—32no—
258#41—94no—
259#41—94no—
260#41—94no—
261#41—94no—
262#41—92no—
263#41C3×C4×D312yes—
264C2×Q8—32no—
265C2×Q8—91no—
266C2×Q8—92no—
267C2×Q8C22×C3×D311yescharacteristic
268C2×D4—31no—
269C2×D4—91no—
270C2×D4—94no—
271C2×D4—94no—
272C2×D4—92no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—32no—
276C22×C4—91no—
277C22×C4—94no—
278C22×C4—94no—
279C22×C4—91no—
280C22×C4—91no—
281C22×C4—92no—
282C22×C4—94no—
283C22×C4—91no—
284C4⋊C4—32no—
285C4⋊C4—32no—
286C4⋊C4—92no—
287C4⋊C4—92no—
288C4⋊C4—92no—
289C4⋊C4—92no—
290C22⋊C4—32no—
291C22⋊C4—32no—
292C22⋊C4—92no—
293C22⋊C4—92no—
294C22⋊C4—92no—
295C22⋊C4—92no—
296C42—32no—
297C42—31no—
298C42—31no—
299C42—92no—
300C42—94no—
301C42—91no—
302C42D3×A411yescharacteristic
303C22×C3—32no—
304C22×C3—41no—
305C22×C3—62no—
306C22×C3—81no—
307C22×C3—122no—
308C22×C3—242no—
309C22×C3C23×A411yescharacteristic
310C2×D3—22no—
311C2×D3—32no—
312C2×D3—62no—
313C2×D3—124no—
314C2×D3C2×C4×A412yes—
315C3×C4—22no—
316C3×C4—31no—
317C3×C4—31no—
318C3×C4—42no—
319C3×C4—62no—
320C3×C4—82no—
321C3×C4—84no—
322C3×C4—82no—
323C3×C4—122no—
324C3×C4—242no—
325C3×C4C2×C4×A412yes—
326Dic3—22no—
327Dic3—32no—
328Dic3—62no—
329Dic3C2×C4×A412yes—
330C32—41noSylow
331C23—31no—
332C23—31no—
333C23—94no—
334C23—91no—
335C23—362no—
336Q8—31no—
337Q8—92no—
338Q8—92no—
339Q8C2×C3×C4×D311yescharacteristic
340D4—32no—
341D4—32no—
342D4—92no—
343D4—92no—
344D4—98no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—34no—
350C2×C4—32no—
351C2×C4—32no—
352C2×C4—92no—
353C2×C4—92no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—94no—
357C2×C4—91no—
358C2×C4—94no—
359C2×C4—92no—
360C2×C4—122no—
361C2×C4—362no—
362C2×C4—364no—
363C2×C4C2×D3×A412yes—
364C2×C4C2×D3×A411yescharacteristic
365C2×C3—41no—
366C2×C3—42no—
367C2×C3—62no—
368C2×C3—81no—
369C2×C3—82no—
370C2×C3—242no—
371C2×C3C22×C4×A411yescharacteristic
372C2×C3#594512yes—
373D3—22no—
374D3—62no—
375C22—32no—
376C22—32no—
377C22—62no—
378C22—62no—
379C22—92no—
380C22—182no—
381C22—364no—
382C22C22×D3×A411yescharacteristic
383C4—22no—
384C4—31no—
385C4—31no—
386C4—32no—
387C4—62no—
388C4—62no—
389C4—92no—
390C4—182no—
391C4C4×D3×A412yes—
392C3—41no—
393C3—81no—
394C3#3428⋊C311yescharacteristic, complemented
395C2—62no—
396C2—62no—
397C2—182no—
398C2C2×C4×D3×A411yescharacteristic
399C2#5964⋊C312yes—
subgroupG/H
centerC2×C4C2×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FrattiniC22C22×D3×A4
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
prime radicalC2×D3×#41⋊C3C2
cosocleC2×C3×Q8C23×C3
hypercenterC42D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×#41⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. #5572⋊C3C2semidirect
5. #5574⋊C3C2semidirect
6. #85834C3semidirect
9. D3×#41⋊C3C4semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. Dic3×SL(2,𝔽3)C22semidirect
17. C3×C4×#41C2×C3semidirect
18. Dic3×#41C2×C3semidirect
19. #5574C2×C3semidirect
31. C4×#41⋊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
102. C4×#41C3×D3semidirect
137. #41⋊C3C4×D3semidirect
394. 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×C4×#41⋊C32
Dic3×#41⋊C32
#5572⋊C32
#5574⋊C32
#858343
#3428⋊C33
C3×#8724