Finite Group Databasebeta

← #85818⋊C3 #11923189

Subgroups — #85818⋊C3 #11923189

Order 1152.

779 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 91 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×#1007⋊C3C212yesmaximal, complemented
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#5576⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
5C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
6#85818C311yesmaximal, characteristic, complemented
7#3497⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9#1007⋊C3—24no—
10C2×D3×SL(2,𝔽3)C2212yes—
11C2×C3⋊(#41⋊C3)C2212yes—
12C3×#930—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C2×#1007C2×C312yescomplemented
16C2×2−1+4⋊C3—32no—
17C2×C3×C4×Q8C2×C311yescharacteristic, complemented
18#5577—31no—
19#5577C2×C311yescharacteristic, complemented
20#5576—32no—
21#5576C2×C312yescomplemented
22Q8×D12—31no—
23Q8×D12C2×C311yescharacteristic, complemented
24#5510—32no—
25#5477—31no—
26#5477—31no—
27C4⋊(#41⋊C3)—31no—
28D4×SL(2,𝔽3)—31no—
29#548⋊C3—32no—
30C2×C4×SL(2,𝔽3)—21no—
31C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
32C22×C32×C4—41no—
33C2×C3×#67—42no—
34C2×C3×D12—41no—
35C2×C3×SL(2,𝔽3)D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37D3×SL(2,𝔽3)—22no—
38C3⋊(#41⋊C3)—22no—
39C3×(C2×D3)⋊C4—42no—
40C3×C4⋊Dic3—41no—
41#3497—31noSylow
42C22×C3×Q8C22×C311yescharacteristic
43#1007—24no—
44#1007—64no—
45C2×#239—32no—
46C2×#239C22×C312yes—
47C2×D3×Q8—32no—
48C2×D3×Q8C22×C312yes—
49C2×#235—32no—
50C2×#235—32no—
512−1+4⋊C3—64no—
52C2×#41⋊C3—32no—
53C22×SL(2,𝔽3)—21no—
54C22×SL(2,𝔽3)—32no—
55C22×SL(2,𝔽3)C2×D311yescharacteristic
56C3×#121—121no—
57C3×C4×Q8—31no—
58C3×C4×Q8—31no—
59C3×C4×Q8C22×C311yescharacteristic
60C3×C4×Q8C22×C311yescharacteristic
61C2×C3×C4⋊C4—31no—
62C2×C3×C42—31no—
63#930—32no—
64#930—31no—
65#930—32no—
66#930A411yescharacteristic
67#897—34no—
68#897—32no—
69#897—32no—
70#895—32no—
71#895—34no—
72#895—32no—
73#875—32no—
74#875—32no—
75#874—31no—
76#874—31no—
77C4×D12—32no—
78C4×D12—32no—
79C4×D12—31no—
80C4×D12—32no—
81C4×D12—31no—
82#869—31no—
83#869—31no—
84C4×SL(2,𝔽3)—21no—
85C4×SL(2,𝔽3)—21no—
86C4×SL(2,𝔽3)C2×D311yescharacteristic
87C4×SL(2,𝔽3)C2×D311yescharacteristic
88C23×C32—41no—
89C22×C3×D3—42no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4—82no—
93C3×#67—84no—
94C2×C3×Dic3—42no—
95C3×D12—82no—
96C3×SL(2,𝔽3)C2×D411yescharacteristic
97C2×2−1+4—32no—
98#550—31no—
99#550—91no—
100D4×Q8—31no—
101D4×Q8—91no—
102#548—32no—
103#548—92no—
104#537—92no—
105#532—91no—
106#532—91no—
107C2×C4×Q8C3×D311yescharacteristic, complemented
108C2×C3×Q8—31no—
109C2×C3×Q8—32no—
110C2×C3×Q8C3×D412yescomplemented
111C2×C3×Q8C23×C311yescharacteristic
112C2×C3×D4—121no—
113C2×C3×D4—122no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—41no—
117C22×C3×C4—81no—
118C22×C3×C4C2×A411yescharacteristic
119C2×#67—32no—
120C2×#67C2×A412yes—
121#239—22no—
122#239—64no—
123#239—64no—
124#239—62no—
125D3×Q8—22no—
126D3×Q8—64no—
127D3×Q8—64no—
128D3×Q8—62no—
129#235—64no—
130#235—64no—
131#235—64no—
132#235—64no—
133C2×D12—32no—
134C2×D12—32no—
135C2×D12—31no—
136C2×D12—32no—
137C2×D12—32no—
138C2×D12C2×A411yescharacteristic
139C2×C4×D3—34no—
140C2×C4×D3—32no—
141C2×C4×D3—32no—
142C2×Dic6—32no—
143C2×Dic6—32no—
144#41⋊C3—62no—
145C2×SL(2,𝔽3)—22no—
146C2×SL(2,𝔽3)—21no—
147C2×SL(2,𝔽3)—62no—
148C2×SL(2,𝔽3)D1212yescomplemented
149C2×SL(2,𝔽3)C22×D311yescharacteristic
150C3×C4⋊C4—31no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—121no—
154C3×C22⋊C4—122no—
155C3×C42—31no—
156C3×C42—31no—
157C3×C42—31no—
158C3×C42—31no—
159(C2×D3)⋊C4—34no—
160(C2×D3)⋊C4—32no—
161(C2×D3)⋊C4—32no—
162(C2×D3)⋊C4—32no—
163(C2×D3)⋊C4C2×A412yes—
164C4⋊Dic3—32no—
165C4⋊Dic3—31no—
166C4⋊Dic3—32no—
167C4⋊Dic3C2×A411yescharacteristic
168C22×C32—42no—
169C22×C32—41no—
170C22×C32—81no—
171C22×C32—81no—
172C2×C3×D3—82no—
173C2×C3×D3—82no—
174C2×C3×D3—82no—
175C32×C4—42no—
176C32×C4—81no—
177C3×Dic3—82no—
1782−1+4—64no—
1792−1+4—184no—
180C2×#41—32no—
181C2×#41—92no—
182C2×#41—92no—
183C2×#41—92no—
184C22×Q8—32no—
185C22×Q8—92no—
186C22×Q8C2×C3×D311yescharacteristic
187C4⋊Q8—91no—
188C4⋊Q8—91no—
189#127—91no—
190#127—91no—
191#124—92no—
192#124—92no—
193C22⋊Q8—94no—
194C22⋊Q8—92no—
195C22⋊Q8—92no—
196#121—31no—
197#121—92no—
198#121—94no—
199#121—92no—
200#121—92no—
201#121—91no—
202#121—92no—
203C4×Q8—31no—
204C4×Q8—31no—
205C4×Q8C2×C3×D311yescharacteristic
206C4×Q8C2×C3×D311yescharacteristic
207C4×D4—92no—
208C4×D4—91no—
209C4×D4—91no—
210C4×D4—92no—
211C4×D4—92no—
212C2×C4⋊C4—31no—
213C2×C42—31no—
214C23×C3—41no—
215C23×C3—81no—
216C23×C3—122no—
217C23×C3C22×A411yescharacteristic
218C22×D3—32no—
219C22×D3C22×A412yes—
220C3×Q8—32no—
221C3×Q8—31no—
222C3×Q8—32no—
223C3×Q8C2×C3×D411yescharacteristic
224C3×D4—242no—
225C3×D4—244no—
226C2×C3×C4—22no—
227C2×C3×C4—32no—
228C2×C3×C4—32no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—82no—
236C2×C3×C4—84no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4—122no—
240C2×C3×C4—121no—
241C2×C3×C4C22×A411yescharacteristic
242C2×C3×C4C22×A411yescharacteristic
243#67—24no—
244#67—64no—
245C2×Dic3—32no—
246C2×Dic3C22×A412yes—
247D12—22no—
248D12—64no—
249D12—62no—
250D12—62no—
251D12—62no—
252D12—122no—
253D12—122no—
254C4×D3—62no—
255C4×D3—62no—
256C4×D3—62no—
257C4×D3—62no—
258C4×D3—68no—
259Dic6—64no—
260Dic6—62no—
261Dic6—62no—
262SL(2,𝔽3)—21no—
263SL(2,𝔽3)C2×D1211yescharacteristic
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—82no—
268C3×D3—162no—
269#41—62no—
270#41—184no—
271#41—184no—
272#41—184no—
273#41—184no—
274#41—184no—
275#41—184no—
276#41—182no—
277C2×Q8—31no—
278C2×Q8—32no—
279C2×Q8—62no—
280C2×Q8—92no—
281C2×Q8—92no—
282C2×Q8—184no—
283C2×Q8—184no—
284C2×Q8—182no—
285C2×Q8C3×D1212yescomplemented
286C2×Q8C22×C3×D311yescharacteristic
287C2×D4—32no—
288C2×D4—31no—
289C2×D4—92no—
290C2×D4—92no—
291C2×D4—92no—
292C2×D4—92no—
293C2×D4—92no—
294C2×D4—91no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—92no—
298C22×C4—92no—
299C22×C4—94no—
300C22×C4D3×A411yescharacteristic
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—32no—
304C4⋊C4—31no—
305C4⋊C4—92no—
306C4⋊C4—92no—
307C4⋊C4—91no—
308C22⋊C4—32no—
309C22⋊C4—92no—
310C22⋊C4—92no—
311C22⋊C4—94no—
312C22⋊C4—92no—
313C42—31no—
314C42—31no—
315C42—31no—
316C42—31no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—42no—
320C22×C3—41no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—82no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3—81no—
327C22×C3—242no—
328C22×C3—242no—
329C22×C3—242no—
330C22×C3D4×A412yes—
331C22×C3C23×A411yescharacteristic
332C2×D3—22no—
333C2×D3—22no—
334C2×D3—22no—
335C2×D3—62no—
336C2×D3—62no—
337C2×D3—62no—
338C3×C4—21no—
339C3×C4—32no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—42no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—81no—
346C3×C4—82no—
347C3×C4—82no—
348C3×C4—242no—
349C3×C42−1+4⋊C312yescomplemented
350Dic3—22no—
351Dic3—62no—
352C32—41noSylow
353C23—32no—
354C23—92no—
355C23C2×D3×A411yescharacteristic
356Q8—32no—
357Q8—31no—
358Q8—32no—
359Q8—184no—
360Q8—182no—
361Q8—182no—
362Q8C2×C3×D1211yescharacteristic
363D4—64no—
364D4—62no—
365D4—184no—
366D4—182no—
367D4—182no—
368D4—184no—
369D4—182no—
370D4—362no—
371D4—362no—
372C2×C4—22no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—32no—
380C2×C4—92no—
381C2×C4—121no—
382C2×C4—182no—
383C2×C4—182no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—188no—
387C2×C4C2×D3×A411yescharacteristic
388C2×C4C2×D3×A411yescharacteristic
389C2×C3—22no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—82no—
394C2×C3—84no—
395C2×C3—81no—
396C2×C3—81no—
397C2×C3—81no—
398C2×C3—482no—
399C2×C3C2×D4×A411yescharacteristic
400C2×C3C2×2−1+4⋊C311yescharacteristic
401C2×C3#594511yescharacteristic
402D3—42no—
403D3—122no—
404C22—21no—
405C22—21no—
406C22—62no—
407C22—62no—
408C22—62no—
409C22—182no—
410C22—182no—
411C22—182no—
412C22A4×D1212yes—
413C22C22×D3×A411yescharacteristic
414C4—21no—
415C4—31no—
416C4—32no—
417C4—31no—
418C4—62no—
419C4—61no—
420C4—61no—
421C4—182no—
422C4#1007⋊C312yescomplemented
423C3—41no—
424C3—81no—
425C3#3497⋊C311yescharacteristic, complemented
426C2—22no—
427C2—122no—
428C2—362no—
429C2C2×A4×D1211yescharacteristic
430C2C2×#1007⋊C311yescharacteristic
431C2#5964⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×C4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#1007⋊C3C2semidirect
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. #5576⋊C3C2semidirect
4. SL(2,𝔽3)×D12C2semidirect
5. C4⋊(C3⋊(#41⋊C3))C2semidirect
6. #85818C3semidirect
15. C2×#1007C2×C3semidirect
17. C2×C3×C4×Q8C2×C3semidirect
19. #5577C2×C3semidirect
21. #5576C2×C3semidirect
23. Q8×D12C2×C3semidirect
31. C2×C4×SL(2,𝔽3)D3semidirect
35. C2×C3×SL(2,𝔽3)D4semidirect
107. C2×C4×Q8C3×D3semidirect
110. C2×C3×Q8C3×D4semidirect
148. C2×SL(2,𝔽3)D12semidirect
285. C2×Q8C3×D12semidirect
349. C3×C42−1+4⋊C3semidirect
422. C4#1007⋊C3semidirect
425. C3#3497⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1007⋊C32
C2×C3×C4×SL(2,𝔽3)2
#5576⋊C32
SL(2,𝔽3)×D122
C4⋊(C3⋊(#41⋊C3))2
#858183
#3497⋊C33
C3×#9304