Finite Group Databasebeta

← #86760⋊C3 #11923262

Subgroups — #86760⋊C3 #11923262

Order 1152.

779 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 85 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
2C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#5817⋊C3C212yesmaximal, complemented
4C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
6#86760C311yesmaximal, characteristic, complemented
7#3497⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2212yes—
9#1007⋊C3—24no—
10C2×D3×SL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊(#41⋊C3)C2212yes—
12C3×#938—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14Dic3×SL(2,𝔽3)C2211yescharacteristic
15C2×#1007C2×C312yescomplemented
16C2×2−1+4⋊C3—32no—
17C3×D4×Q8C2×C311yescharacteristic, complemented
18#5817C2×C312yescomplemented
19#5727—31no—
20#5725—31no—
21#5633—32no—
22#5629—32no—
23#5577C2×C311yescharacteristic, complemented
24C4×D3×Q8C2×C311yescharacteristic, complemented
25#5553—31no—
26#5548—31no—
27C4⋊(#41⋊C3)—31no—
28D4×SL(2,𝔽3)—21no—
29D4×SL(2,𝔽3)D311yescharacteristic, complemented
30#548⋊C3—32no—
31C2×C4×SL(2,𝔽3)—31no—
32C2×C32×D4—41no—
33C2×C3×#67—42no—
34C2×C3×C4×D3—41no—
35C2×C3×SL(2,𝔽3)—22no—
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37D3×SL(2,𝔽3)D412yescomplemented
38C3⋊(#41⋊C3)—22no—
39C3×#217—42no—
40C3×C4⋊Dic3—41no—
41#3497—31noSylow
42C22×C3×Q8C22×C312yes—
43#1007—24no—
44#1007—64no—
45C2×#239—32no—
46C2×#239C22×C312yes—
47C2×D3×Q8—32no—
48C2×D3×Q8C22×C311yescharacteristic
49C2×#235—32no—
50C2×#235—32no—
512−1+4⋊C3—64no—
52C2×#41⋊C3—32no—
53C22×SL(2,𝔽3)—22no—
54C22×SL(2,𝔽3)—31no—
55C22×SL(2,𝔽3)C2×D312yes—
56C3×C4⋊Q8—31no—
57C3×C22⋊Q8—32no—
58C3×#121—121no—
59C3×C4×Q8C22×C311yescharacteristic
60C3×C4×D4—31no—
61#947—32no—
62#946—32no—
63Q8×Dic3C22×C311yescharacteristic
64#940—31no—
65#938A411yescharacteristic
66#930—32no—
67C4×#67—32no—
68#896—34no—
69#895—32no—
70#895—31no—
71#895—32no—
72#893—32no—
73#893—32no—
74D3×C4⋊C4—31no—
75#888—31no—
76#887—31no—
77#885—32no—
78#884—34no—
79#884—32no—
80#874—31no—
81C4×D12—31no—
82C42×D3—31no—
83C4×Dic6—31no—
84C4×SL(2,𝔽3)—21no—
85C4×SL(2,𝔽3)—31no—
86C4×SL(2,𝔽3)C2×D311yescharacteristic
87C23×C32—42no—
88C22×C3×D3—41no—
89C32×D4—82no—
90C2×C32×C4—41no—
91C3×#67—84no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—42no—
94C3×C4×D3—82no—
95C3×SL(2,𝔽3)C2×D411yescharacteristic
96C2×2−1+4—32no—
97#550—31no—
98#550—91no—
99D4×Q8—91no—
100D4×Q8C3×D311yescharacteristic, complemented
101#548—32no—
102#548—92no—
103#537—92no—
104#532—91no—
105#532—91no—
106C2×C4×Q8—31no—
107C2×C3×Q8—22no—
108C2×C3×Q8—32no—
109C2×C3×Q8—62no—
110C2×C3×Q8C23×C311yescharacteristic
111C2×C3×D4—41no—
112C2×C3×D4—81no—
113C2×C3×D4—122no—
114C2×C3×D4C2×A411yescharacteristic
115C22×C3×C4—32no—
116C22×C3×C4—121no—
117C2×#67—32no—
118C2×#67—32no—
119C2×#67C2×A412yes—
120#239—22no—
121#239—64no—
122#239—64no—
123#239—62no—
124D3×Q8—32no—
125D3×Q8—64no—
126D3×Q8—64no—
127D3×Q8C3×D412yescomplemented
128#235—64no—
129#235—64no—
130#235—64no—
131#235—64no—
132C2×D12—32no—
133C2×D12—32no—
134C2×D12—31no—
135C2×D12—32no—
136C2×C4×D3—31no—
137C2×C4×D3—34no—
138C2×C4×D3—32no—
139C2×C4×D3—31no—
140C2×C4×D3C2×A411yescharacteristic
141C2×Dic6—31no—
142C2×Dic6—32no—
143#41⋊C3—62no—
144C2×SL(2,𝔽3)—22no—
145C2×SL(2,𝔽3)—21no—
146C2×SL(2,𝔽3)—32no—
147C2×SL(2,𝔽3)—42no—
148C2×SL(2,𝔽3)C22×D311yescharacteristic
149C3×C4⋊C4—31no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—121no—
153C3×C22⋊C4—32no—
154C3×C22⋊C4—122no—
155C3×C42—31no—
156#217C2×A412yes—
157(C2×D3)⋊C4—34no—
158(C2×D3)⋊C4—32no—
159(C2×D3)⋊C4—32no—
160C4⋊Dic3—31no—
161C4⋊Dic3C2×A411yescharacteristic
162Dic3⋊C4—32no—
163Dic3⋊C4—32no—
164C4×Dic3—31no—
165C4×Dic3—31no—
166C4×Dic3—31no—
167C22×C32—41no—
168C22×C32—82no—
169C22×C32—82no—
170C22×C32—82no—
171C2×C3×D3—42no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C32×C4—42no—
175C3×Dic3—81no—
176C3×Dic3—82no—
1772−1+4—64no—
1782−1+4—184no—
179C2×#41—32no—
180C2×#41—92no—
181C2×#41—92no—
182C2×#41—92no—
183C22×Q8—31no—
184C22×Q8—92no—
185C22×Q8C2×C3×D312yes—
186C4⋊Q8—31no—
187C4⋊Q8—91no—
188#127—91no—
189#127—91no—
190#124—92no—
191#124—92no—
192C22⋊Q8—32no—
193C22⋊Q8—94no—
194C22⋊Q8—92no—
195#121—31no—
196#121—92no—
197#121—94no—
198#121—91no—
199#121—92no—
200#121—92no—
201#121—92no—
202C4×Q8—31no—
203C4×Q8—91no—
204C4×Q8—91no—
205C4×Q8C2×C3×D311yescharacteristic
206C4×D4—31no—
207C4×D4—91no—
208C4×D4—92no—
209C4×D4—92no—
210C4×D4—92no—
211C2×C4⋊C4—91no—
212C2×C42—91no—
213C23×C3—42no—
214C23×C3—82no—
215C23×C3—121no—
216C23×C3C22×A412yes—
217C22×D3—32no—
218C22×D3C22×A411yescharacteristic
219C3×Q8—31no—
220C3×Q8—62no—
221C3×Q8—62no—
222C3×Q8C2×C3×D411yescharacteristic
223C3×D4—22no—
224C3×D4—82no—
225C3×D4—84no—
226C3×D4—244no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—32no—
230C2×C3×C4—41no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—81no—
234C2×C3×C4—121no—
235C2×C3×C4—122no—
236C2×C3×C4—242no—
237C2×C3×C4C22×A411yescharacteristic
238#67—24no—
239#67—64no—
240#67—122no—
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3C22×A411yescharacteristic
244C2×Dic3C22×A412yes—
245D12—62no—
246D12—62no—
247D12—64no—
248D12—62no—
249D12—122no—
250C4×D3—22no—
251C4×D3—32no—
252C4×D3—32no—
253C4×D3—62no—
254C4×D3—62no—
255C4×D3—68no—
256C4×D3—121no—
257Dic6—32no—
258Dic6—32no—
259Dic6—64no—
260SL(2,𝔽3)—21no—
261SL(2,𝔽3)D3×D411yescharacteristic
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—162no—
266C3×D3—82no—
267#41—62no—
268#41—184no—
269#41—184no—
270#41—184no—
271#41—184no—
272#41—184no—
273#41—184no—
274#41—182no—
275C2×Q8—22no—
276C2×Q8—32no—
277C2×Q8—32no—
278C2×Q8—62no—
279C2×Q8—92no—
280C2×Q8—91no—
281C2×Q8—92no—
282C2×Q8—184no—
283C2×Q8—184no—
284C2×Q8C22×C3×D311yescharacteristic
285C2×D4—32no—
286C2×D4—92no—
287C2×D4—92no—
288C2×D4—92no—
289C2×D4—92no—
290C2×D4—92no—
291C2×D4—91no—
292C2×D4D3×A411yescharacteristic
293C22×C4—32no—
294C22×C4—31no—
295C22×C4—92no—
296C22×C4—91no—
297C22×C4—94no—
298C22×C4—91no—
299C4⋊C4—31no—
300C4⋊C4—32no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—92no—
304C4⋊C4—91no—
305C4⋊C4—92no—
306C22⋊C4—32no—
307C22⋊C4—32no—
308C22⋊C4—92no—
309C22⋊C4—92no—
310C22⋊C4—94no—
311C42—31no—
312C42—91no—
313C42—91no—
314C42—91no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—41no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—82no—
322C22×C3—84no—
323C22×C3—81no—
324C22×C3—122no—
325C22×C3—162no—
326C22×C3—162no—
327C22×C3—241no—
328C22×C3—241no—
329C22×C3C23×A411yescharacteristic
330C2×D3—21no—
331C2×D3—21no—
332C2×D3—62no—
333C2×D3—62no—
334C2×D3—62no—
335C2×D3D4×A412yes—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—42no—
339C3×C4—62no—
340C3×C4—61no—
341C3×C4—82no—
342C3×C4—241no—
343C3×C4—242no—
344C3×C42−1+4⋊C312yescomplemented
345Dic3—22no—
346Dic3—21no—
347Dic3—32no—
348Dic3—61no—
349C32—41noSylow
350C23—31no—
351C23—92no—
352C23C2×D3×A412yes—
353Q8—31no—
354Q8—62no—
355Q8—62no—
356Q8—92no—
357Q8—92no—
358Q8—184no—
359Q8C3×D3×D411yescharacteristic
360D4—22no—
361D4—64no—
362D4—182no—
363D4—182no—
364D4—184no—
365D4—184no—
366D4—182no—
367D4—362no—
368D4—362no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—92no—
378C2×C4—92no—
379C2×C4—91no—
380C2×C4—91no—
381C2×C4—182no—
382C2×C4—182no—
383C2×C4—188no—
384C2×C4—361no—
385C2×C4C2×D3×A411yescharacteristic
386C2×C3—42no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—81no—
391C2×C3—81no—
392C2×C3—81no—
393C2×C3—162no—
394C2×C3—164no—
395C2×C3—242no—
396C2×C3C2×D4×A411yescharacteristic
397C2×C3C2×2−1+4⋊C311yescharacteristic
398C2×C3#594511yescharacteristic
399D3—22no—
400D3—122no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—32no—
405C22—61no—
406C22—61no—
407C22—182no—
408C22—182no—
409C22—182no—
410C22C22×D3×A411yescharacteristic
411C4—31no—
412C4—31no—
413C4—62no—
414C4—62no—
415C4—61no—
416C4—61no—
417C4—92no—
418C4—181no—
419C4#1007⋊C312yescomplemented
420C3—41no—
421C3—81no—
422C3#3497⋊C311yescharacteristic, complemented
423C2—42no—
424C2—62no—
425C2—362no—
426C2D3×D4×A411yescharacteristic
427C2C2×#1007⋊C311yescharacteristic
428C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×D4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#1007⋊C3C2semidirect
2. C3×D4×SL(2,𝔽3)C2semidirect
3. #5817⋊C3C2semidirect
4. C4×D3×SL(2,𝔽3)C2semidirect
5. C4⋊(C3⋊(#41⋊C3))C2semidirect
6. #86760C3semidirect
15. C2×#1007C2×C3semidirect
17. C3×D4×Q8C2×C3semidirect
18. #5817C2×C3semidirect
23. #5577C2×C3semidirect
24. C4×D3×Q8C2×C3semidirect
29. D4×SL(2,𝔽3)D3semidirect
37. D3×SL(2,𝔽3)D4semidirect
100. D4×Q8C3×D3semidirect
127. D3×Q8C3×D4semidirect
344. C3×C42−1+4⋊C3semidirect
419. C4#1007⋊C3semidirect
422. 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
C3×D4×SL(2,𝔽3)2
#5817⋊C32
C4×D3×SL(2,𝔽3)2
C4⋊(C3⋊(#41⋊C3))2
#867603
#3497⋊C33
C3×#9384