Finite Group Databasebeta

← #85845⋊C3 #11923193

Subgroups — #85845⋊C3 #11923193

Order 1152.

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

normal subgroups only

This group has 607 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
1C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
2#5834⋊C3C212yesmaximal, complemented
3C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
4C4×C3⋊(#41⋊C3)C212yesmaximal, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#85845C311yesmaximal, characteristic, complemented
7#85604—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3⋊(#41⋊C3)C2212yes—
10C3×C4×Dic6—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×C4×#41C2×C311yescharacteristic
15#5834C2×C312yes—
16#5577C2×C311yescharacteristic
17C4×#239C2×C312yes—
18Q8×Dic6C2×C311yescharacteristic
19#5563—31no—
20#5553—31no—
21#5548—32no—
22#5532—32no—
23#5477—31no—
24C4×#235—31no—
25Q8×SL(2,𝔽3)—31no—
26C4⋊(#41⋊C3)—32no—
27C4⋊(#41⋊C3)—31no—
28C4×#41⋊C3—21no—
29C4×#41⋊C3—32no—
30C4×#41⋊C3D311yescharacteristic, complemented
31C2×C3×Dic6—41no—
32C3×#41⋊C3—21no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊(#41⋊C3)—22no—
35C32×C42—41no—
36C3×C4⋊Dic3—41no—
37C3×Dic3⋊C4—42no—
38C3×C4×Dic3—42no—
39#3483—31noSylow
40C2×C3×#41C22×C311yescharacteristic
41C2×#239C22×C312yes—
42C2×#235—32no—
43C2×#235—31no—
44C2×#41⋊C3—21no—
45C2×#41⋊C3—32no—
46C2×#41⋊C3C2×D311yescharacteristic
47C3×C4×Q8—121no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×Q8C22×C311yescharacteristic
50C3×C4×D4—31no—
51C3×C4×D4—31no—
52C3×#117—31no—
53C2×C3×C42—31no—
54Q8×Dic3C22×C312yes—
55#940—32no—
56#938—32no—
57#930—32no—
58#930—32no—
59C4×#67—32no—
60C4×#67—32no—
61#926—31no—
62#895—32no—
63#893—32no—
64#892—32no—
65#888—32no—
66#885—34no—
67#884—34no—
68#875—32no—
69#874—31no—
70C4×D12—32no—
71C4×D12—31no—
72C4×D12—31no—
73#872—32no—
74C42×D3—32no—
75#869—31no—
76C4×Dic6—31no—
77C4×Dic6A411yescharacteristic
78C4×SL(2,𝔽3)—21no—
79C4×SL(2,𝔽3)—21no—
80C4×SL(2,𝔽3)—32no—
81C4×SL(2,𝔽3)C2×D311yescharacteristic
82C4×SL(2,𝔽3)C2×D311yescharacteristic
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—42no—
88C3×Dic6—44no—
89C3×SL(2,𝔽3)#4111yescharacteristic, complemented
90Q82—31no—
91#555—92no—
92#555—91no—
93#550—32no—
94#550—31no—
95#550—91no—
96#532—91no—
97#532—92no—
98C4×#41—32no—
99C4×#41—91no—
100C4×#41C3×D311yescharacteristic
101C3×#41—21no—
102C3×#41—61no—
103C2×C3×Q8—121no—
104C2×C3×Q8C23×C311yescharacteristic
105C2×C3×D4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C2×#67—34no—
110C2×#67—32no—
111#239—22no—
112#239—62no—
113#235—64no—
114#235—124no—
115C2×D12—32no—
116C2×D12—31no—
117C2×D12—32no—
118C2×C4×D3—32no—
119C2×C4×D3—32no—
120C2×C4×D3—32no—
121C2×Dic6—32no—
122C2×Dic6C2×A411yescharacteristic
123#41⋊C3—22no—
124#41⋊C3—21no—
125#41⋊C3—62no—
126C2×SL(2,𝔽3)—21no—
127C2×SL(2,𝔽3)C22×D311yescharacteristic
128C3×C4⋊C4—31no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—122no—
131C3×C4⋊C4—121no—
132C3×C22⋊C4—32no—
133C3×C42—31no—
134C3×C42—31no—
135C3×C42—31no—
136C3×C42—41no—
137C3×C42—81no—
138C3×C42—122no—
139C3×C42C2×A411yescharacteristic
140#217—32no—
141(C2×D3)⋊C4—34no—
142(C2×D3)⋊C4—32no—
143(C2×D3)⋊C4—32no—
144C4⋊Dic3—31no—
145C4⋊Dic3—32no—
146C4⋊Dic3C2×A411yescharacteristic
147Dic3⋊C4—32no—
148Dic3⋊C4C2×A412yes—
149C4×Dic3—32no—
150C4×Dic3—32no—
151C4×Dic3C2×A412yes—
152C22×C32—41no—
153C32×C4—42no—
154C32×C4—42no—
155C32×C4—81no—
156C3×Dic3—44no—
157C3×Dic3—82no—
158C2×#41—32no—
159C2×#41—91no—
160C2×#41—92no—
161C2×#41C2×C3×D311yescharacteristic
162C4⋊Q8—91no—
163C4⋊Q8—92no—
164#127—91no—
165#127—92no—
166#124—92no—
167#124—94no—
168#121—94no—
169#121—92no—
170#121—92no—
171#121—92no—
172#121—92no—
173C4×Q8—31no—
174C4×Q8—32no—
175C4×Q8—91no—
176C4×Q8C2×C3×D311yescharacteristic
177C4×Q8C2×C3×D311yescharacteristic
178C4×D4—31no—
179C4×D4—31no—
180C4×D4—91no—
181C4×D4—92no—
182C4×D4—92no—
183C4×D4—92no—
184C4×D4—92no—
185C4×D4—91no—
186#117—31no—
187#117—92no—
188#117—91no—
189#117—92no—
190C2×C42—31no—
191C2×C42—92no—
192C23×C3—31no—
193C22×D3—32no—
194C3×Q8—31no—
195C3×Q8—124no—
196C3×Q8C3×#4111yescharacteristic
197C3×D4—61no—
198C3×D4—61no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—122no—
213C2×C3×C4—122no—
214C2×C3×C4C22×A411yescharacteristic
215C2×C3×C4C22×A411yescharacteristic
216C2×C3×C4C22×A411yescharacteristic
217#67—64no—
218#67—124no—
219C2×Dic3—32no—
220C2×Dic3C22×A412yes—
221C2×Dic3C22×A412yes—
222D12—62no—
223D12—62no—
224D12—62no—
225D12—122no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—64no—
229C4×D3—64no—
230Dic6—122no—
231Dic6#41⋊C314yescomplemented
232SL(2,𝔽3)—21no—
233SL(2,𝔽3)#23511yescharacteristic, complemented
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237#41—21no—
238#41—61no—
239#41—62no—
240#41—184no—
241#41—182no—
242#41—364no—
243C2×Q8—31no—
244C2×Q8—92no—
245C2×Q8C22×C3×D311yescharacteristic
246C2×D4—31no—
247C2×D4—92no—
248C2×D4—92no—
249C2×D4—92no—
250C2×D4—94no—
251C2×D4—91no—
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—92no—
256C22×C4—92no—
257C22×C4—92no—
258C4⋊C4—31no—
259C4⋊C4—31no—
260C4⋊C4—32no—
261C4⋊C4—31no—
262C4⋊C4—92no—
263C4⋊C4—92no—
264C4⋊C4—91no—
265C22⋊C4—32no—
266C22⋊C4—92no—
267C22⋊C4—92no—
268C22⋊C4—94no—
269C22⋊C4—92no—
270C42—31no—
271C42—31no—
272C42—31no—
273C42—32no—
274C42—92no—
275C42—92no—
276C42D3×A411yescharacteristic
277C22×C3—41no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—81no—
282C22×C3C23×A411yescharacteristic
283C2×D3—62no—
284C2×D3—62no—
285C2×D3—62no—
286C3×C4—21no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—42no—
290C3×C4—42no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—81no—
294C3×C4—82no—
295C3×C4—82no—
296C3×C4—82no—
297C3×C4—124no—
298C3×C4—242no—
299C3×C4C2×#41⋊C312yes—
300C3×C4#99412yes—
301Dic3—22no—
302Dic3—62no—
303Dic3C2×#41⋊C314yes—
304C32—41noSylow
305C23—31no—
306C23—92no—
307Q8—31no—
308Q8—34no—
309Q8—362no—
310Q8C3×#23511yescharacteristic
311D4—61no—
312D4—61no—
313D4—184no—
314D4—182no—
315D4—182no—
316D4—182no—
317D4—362no—
318D4—364no—
319C2×C4—31no—
320C2×C4—32no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—32no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—92no—
329C2×C4—182no—
330C2×C4—184no—
331C2×C4—182no—
332C2×C4—184no—
333C2×C4C2×D3×A411yescharacteristic
334C2×C4C2×D3×A411yescharacteristic
335C2×C4C2×D3×A411yescharacteristic
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—81no—
340C2×C3—81no—
341C2×C3—81no—
342C2×C3—121no—
343C2×C3C22×#41⋊C311yescharacteristic
344C2×C3A4×#4111yescharacteristic
345C2×C3C2×#99411yescharacteristic
346D3—122no—
347C22—61no—
348C22—61no—
349C22—61no—
350C22—182no—
351C22—182no—
352C22—182no—
353C22C22×D3×A411yescharacteristic
354C4—21no—
355C4—31no—
356C4—31no—
357C4—34no—
358C4—62no—
359C4—61no—
360C4—61no—
361C4—182no—
362C4C3⋊#99412yes—
363C4D3×#41⋊C312yes—
364C3—41no—
365C3—81no—
366C3#8560411yescharacteristic, complemented
367C2—121no—
368C2—362no—
369C2A4×#23511yescharacteristic
370C2C2×C3⋊#99411yescharacteristic
371C2C2×D3×#41⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
hypercenterC42D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#41⋊C3C2semidirect
2. #5834⋊C3C2semidirect
3. C4⋊(C3⋊(#41⋊C3))C2semidirect
4. C4×C3⋊(#41⋊C3)C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #85845C3semidirect
30. C4×#41⋊C3D3semidirect
89. C3×SL(2,𝔽3)#41semidirect
231. Dic6#41⋊C3semidirect
233. SL(2,𝔽3)#235semidirect
366. C3#85604semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
#5834⋊C32
C4⋊(C3⋊(#41⋊C3))2
C4×C3⋊(#41⋊C3)2
SL(2,𝔽3)×Dic62
#858453
#856043
C3×C4×Dic64