Finite Group Databasebeta

← #86766⋊C3 #11923271

Subgroups — #86766⋊C3 #11923271

Order 1152.

817 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 93 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⋊C3C212yesmaximal, complemented
2C3×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#5831⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
5C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
6#86766C311yesmaximal, characteristic, complemented
7D4×#41⋊C3—31nomaximal
8C2×C3×#41⋊C3C2212yescomplemented
9D3×#41⋊C3—24no—
10C2×D3×SL(2,𝔽3)C2212yescomplemented
11C2×C3⋊(#41⋊C3)C2211yescharacteristic
12C3×#893—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
14Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
15C2×D3×#41C2×C312yescomplemented
16C22×#41⋊C3—32no—
17C3×#550C2×C311yescharacteristic, complemented
18#5831C2×C312yescomplemented
19#5717—31no—
20#5716—31no—
21#5612—32no—
22#5606—32no—
23Q8×D12C2×C311yescharacteristic
24C4×#239C2×C311yescharacteristic, complemented
25#5551—31no—
26C4×D3×D4—31no—
27C4⋊(#41⋊C3)—21no—
28C4⋊(#41⋊C3)D311yescharacteristic, complemented
29D4×SL(2,𝔽3)—31no—
30#548⋊C3—32no—
31C4×#41⋊C3—31no—
32C2×C3×D12—41no—
33C2×C3×C4×D3—42no—
34C3×#41⋊C3—22no—
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36D3×SL(2,𝔽3)—22no—
37C3⋊(#41⋊C3)D412yescomplemented
38C32×C4⋊C4—41no—
39C3×(C2×D3)⋊C4—42no—
40C3×C4×Dic3—41no—
41D4×#41—31noSylow
42C2×C3×#41C22×C312yes—
43D3×#41—24no—
44D3×#41—64no—
45C2×#239C22×C311yescharacteristic
46C2×D3×Q8C22×C312yes—
47C2×#237—32no—
48C2×D3×D4—32no—
49C2×D3×D4—31no—
50C2×#235—32no—
51C22×C4×D3—32no—
52C2×#41⋊C3—22no—
53C2×#41⋊C3—31no—
54C2×#41⋊C3—64no—
55C2×#41⋊C3C2×D312yescomplemented
56C22×SL(2,𝔽3)—32no—
57C3×#127—31no—
58C3×#121—32no—
59C3×C4×Q8C22×C311yescharacteristic
60C3×C4×D4—31no—
61C3×C4×D4—121no—
62Q8×Dic3C22×C311yescharacteristic
63#938—32no—
64#938—32no—
65#938—32no—
66#937—34no—
67#936—32no—
68D4×Dic3—32no—
69D4×Dic3—31no—
70C4×#67—32no—
71#924—32no—
72#897—32no—
73#893—31no—
74#893A411yescharacteristic
75#892—31no—
76D3×C4⋊C4—31no—
77#883—34no—
78#67⋊C4—32no—
79D3×C22⋊C4—32no—
80C4×D12—31no—
81C4×D12—31no—
82C42×D3—31no—
83#869—31no—
84C4×SL(2,𝔽3)—21no—
85C4×SL(2,𝔽3)—31no—
86C4×SL(2,𝔽3)C2×D311yescharacteristic, complemented
87C22×C3×D3—42no—
88C2×C32×C4—41no—
89C2×C32×C4—42no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—41no—
92C3×D12—44no—
93C3×C4×D3—84no—
94C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
95C22×#41—32no—
96#550C3×D311yescharacteristic, complemented
97D4×Q8—31no—
98#548—32no—
99#547—91no—
100#546—92no—
101D42—91no—
102#532—91no—
103#525—92no—
104C4×#41—31no—
105C2×C4×D4—91no—
106C23×D3—32no—
107C3×#41—22no—
108C3×#41—62no—
109C2×C3×Q8C23×C311yescharacteristic
110C2×C3×D4—32no—
111C2×C3×D4—32no—
112C2×C3×D4—31no—
113C2×C3×D4—121no—
114C22×C3×C4—32no—
115C22×C3×C4—122no—
116C2×#67—34no—
117C2×#67—32no—
118C22×Dic3—32no—
119#239—32no—
120#239C3×D412yescomplemented
121D3×Q8—22no—
122D3×Q8—62no—
123#237—64no—
124#237—64no—
125D3×D4—64no—
126D3×D4—64no—
127D3×D4—64no—
128#235—64no—
129#235—64no—
130C2×D12—31no—
131C2×D12C2×A411yescharacteristic
132C2×C4×D3—31no—
133C2×C4×D3—32no—
134C2×C4×D3—31no—
135C2×C4×D3—32no—
136C2×C4×D3—31no—
137C2×C4×D3—64no—
138C2×C4×D3—64no—
139C2×C4×D3C2×A412yes—
140C2×Dic6—32no—
141#41⋊C3—22no—
142#41⋊C3—32no—
143#41⋊C3—42no—
144C2×SL(2,𝔽3)—21no—
145C2×SL(2,𝔽3)—62no—
146C2×SL(2,𝔽3)C22×D311yescharacteristic
147C3×C4⋊C4—31no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—81no—
150C3×C4⋊C4C2×A411yescharacteristic
151C3×C22⋊C4—32no—
152C3×C22⋊C4—122no—
153C3×C42—31no—
154C3×C42—121no—
155#217—34no—
156#217—32no—
157(C2×D3)⋊C4—32no—
158(C2×D3)⋊C4—32no—
159(C2×D3)⋊C4C2×A412yes—
160C4⋊Dic3—32no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163Dic3⋊C4—32no—
164C4×Dic3—31no—
165C4×Dic3—31no—
166C4×Dic3C2×A411yescharacteristic
167C22×C32—41no—
168C2×C3×D3—44no—
169C2×C3×D3—82no—
170C2×C3×D3—82no—
171C32×C4—42no—
172C32×C4—82no—
173C3×Dic3—42no—
174C3×Dic3—81no—
175C2×#41—31no—
176C2×#41—64no—
177C2×#41—92no—
178C2×#41—92no—
179C2×#41—184no—
180C2×#41C2×C3×D312yes—
181C22×Q8—32no—
182C22×D4—91no—
183C22×D4—92no—
184C23×C4—92no—
185C4⋊Q8—91no—
186#127—31no—
187#124—92no—
188#123—94no—
189C22⋊Q8—92no—
190C22⋊Q8—92no—
191#121—32no—
192#121—92no—
193#121—92no—
194#121—92no—
195#120—94no—
196C4×Q8—31no—
197C4×Q8C2×C3×D311yescharacteristic
198C4×D4—31no—
199C4×D4—31no—
200C4×D4—91no—
201C4×D4—91no—
202C4×D4—91no—
203C4×D4—92no—
204C4×D4—92no—
205C4×D4—91no—
206C4×D4—92no—
207#117—91no—
208C2×C4⋊C4—91no—
209C2×C22⋊C4—92no—
210C2×C42—91no—
211C23×C3—32no—
212C23×C3—122no—
213C22×D3—31no—
214C22×D3—64no—
215C22×D3—64no—
216C22×D3—122no—
217C22×D3C22×A412yes—
218C3×Q8—31no—
219C3×Q8C2×C3×D411yescharacteristic
220C3×D4—62no—
221C3×D4—62no—
222C3×D4—62no—
223C3×D4—124no—
224C3×D4—122no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—81no—
232C2×C3×C4—82no—
233C2×C3×C4—121no—
234C2×C3×C4—121no—
235C2×C3×C4—244no—
236C2×C3×C4C22×A412yes—
237C2×C3×C4C22×A411yescharacteristic
238#67—68no—
239#67—64no—
240C2×Dic3—31no—
241C2×Dic3—32no—
242C2×Dic3—64no—
243C2×Dic3C22×A411yescharacteristic
244C2×Dic3C22×A411yescharacteristic
245D12—32no—
246D12—32no—
247D12#41⋊C314yescomplemented
248C4×D3—24no—
249C4×D3—32no—
250C4×D3—32no—
251C4×D3—64no—
252C4×D3—62no—
253C4×D3—62no—
254C4×D3—62no—
255C4×D3—121no—
256Dic6—62no—
257Dic6—62no—
258SL(2,𝔽3)—21no—
259SL(2,𝔽3)D3×D411yescharacteristic, complemented
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C3×D3—84no—
264C24—92no—
265#41—22no—
266#41—32no—
267#41—62no—
268#41—92no—
269#41—184no—
270#41—184no—
271#41—184no—
272#41—184no—
273C2×Q8—62no—
274C2×Q8—92no—
275C2×Q8—182no—
276C2×Q8C22×C3×D311yescharacteristic
277C2×D4—32no—
278C2×D4—32no—
279C2×D4—31no—
280C2×D4—31no—
281C2×D4—91no—
282C2×D4—92no—
283C2×D4—94no—
284C2×D4—184no—
285C2×D4—184no—
286C2×D4—184no—
287C22×C4—32no—
288C22×C4—32no—
289C22×C4—91no—
290C22×C4—91no—
291C22×C4—92no—
292C22×C4—92no—
293C22×C4—91no—
294C22×C4—92no—
295C22×C4—184no—
296C22×C4—184no—
297C4⋊C4—31no—
298C4⋊C4—91no—
299C4⋊C4—92no—
300C4⋊C4—91no—
301C4⋊C4—92no—
302C4⋊C4D3×A411yescharacteristic
303C22⋊C4—32no—
304C22⋊C4—32no—
305C22⋊C4—92no—
306C22⋊C4—92no—
307C22⋊C4—92no—
308C22⋊C4—94no—
309C42—31no—
310C42—31no—
311C42—91no—
312C42—91no—
313C22×C3—41no—
314C22×C3—62no—
315C22×C3—62no—
316C22×C3—62no—
317C22×C3—81no—
318C22×C3—124no—
319C22×C3—242no—
320C22×C3—242no—
321C22×C3C23×A411yescharacteristic
322C2×D3—22no—
323C2×D3—22no—
324C2×D3—32no—
325C2×D3—61no—
326C2×D3—61no—
327C2×D3—128no—
328C2×D3C2×#41⋊C314yescomplemented
329C3×C4—22no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—42no—
333C3×C4—61no—
334C3×C4—82no—
335C3×C4—82no—
336C3×C4—122no—
337C3×C4—162no—
338C3×C4—241no—
339C3×C4C2×#41⋊C312yescomplemented
340Dic3—21no—
341Dic3—62no—
342Dic3—61no—
343Dic3D4×A412yes—
344C32—41noSylow
345C23—32no—
346C23—32no—
347C23—91no—
348C23—184no—
349C23—184no—
350C23—362no—
351Q8—31no—
352Q8—182no—
353Q8—182no—
354Q8C3×D3×D411yescharacteristic
355D4—34no—
356D4—62no—
357D4—62no—
358D4—62no—
359D4—92no—
360D4—92no—
361D4—122no—
362D4—184no—
363D4—188no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—64no—
371C2×C4—92no—
372C2×C4—92no—
373C2×C4—91no—
374C2×C4—92no—
375C2×C4—184no—
376C2×C4—182no—
377C2×C4—182no—
378C2×C4—184no—
379C2×C4—182no—
380C2×C4—361no—
381C2×C4C2×D3×A412yes—
382C2×C4C2×D3×A411yescharacteristic
383C2×C3—41no—
384C2×C3—41no—
385C2×C3—41no—
386C2×C3—81no—
387C2×C3—81no—
388C2×C3—81no—
389C2×C3—122no—
390C2×C3—244no—
391C2×C3C2×D4×A411yescharacteristic
392C2×C3C22×#41⋊C311yescharacteristic
393C2×C3#594511yescharacteristic
394D3—24no—
395D3—62no—
396C22—34no—
397C22—62no—
398C22—62no—
399C22—62no—
400C22—62no—
401C22—62no—
402C22—92no—
403C22—181no—
404C22—181no—
405C22—368no—
406C22C22×D3×A411yescharacteristic
407C4—22no—
408C4—31no—
409C4—31no—
410C4—32no—
411C4—61no—
412C4—61no—
413C4—182no—
414C4—181no—
415C4D3×#41⋊C312yescomplemented
416C3—41no—
417C3—81no—
418C3D4×#41⋊C311yescharacteristic, complemented
419C2—64no—
420C2—122no—
421C2—182no—
422C2D3×D4×A411yescharacteristic
423C2C2×D3×#41⋊C311yescharacteristic
424C2#1059662311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#550C2×C3
socleC22×C3C23×A4
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×#41⋊C3C2semidirect
2. C3×C4⋊(#41⋊C3)C2semidirect
3. #5831⋊C3C2semidirect
4. SL(2,𝔽3)×D12C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #86766C3semidirect
8. C2×C3×#41⋊C3C22semidirect
10. C2×D3×SL(2,𝔽3)C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. Dic3×SL(2,𝔽3)C22semidirect
15. C2×D3×#41C2×C3semidirect
17. C3×#550C2×C3semidirect
18. #5831C2×C3semidirect
24. C4×#239C2×C3semidirect
28. C4⋊(#41⋊C3)D3semidirect
37. C3⋊(#41⋊C3)D4semidirect
55. C2×#41⋊C3C2×D3semidirect
86. C4×SL(2,𝔽3)C2×D3semidirect
94. C3×SL(2,𝔽3)C2×D4semidirect
96. #550C3×D3semidirect
120. #239C3×D4semidirect
247. D12#41⋊C3semidirect
259. SL(2,𝔽3)D3×D4semidirect
328. C2×D3C2×#41⋊C3semidirect
339. C3×C4C2×#41⋊C3semidirect
415. C4D3×#41⋊C3semidirect
418. C3D4×#41⋊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⋊C3)2
#5831⋊C32
SL(2,𝔽3)×D122
C4×C3⋊(#41⋊C3)2
#867663
D4×#41⋊C33
C3×#8934