Finite Group Databasebeta

← #87290⋊C3 #11923398

Subgroups — #87290⋊C3 #11923398

Order 1152.

775 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 87 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C22×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#5834⋊C3C212yesmaximal, complemented
4#5812⋊C3C211yesmaximal, characteristic, complemented
5#5817⋊C3C212yesmaximal, complemented
6#87290C311yesmaximal, characteristic, complemented
7#3487⋊C3—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic, complemented
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C2×C3⋊(#41⋊C3)—22no—
13C2×C3⋊(#41⋊C3)C2212yes—
14C3×#935—41nomaximal
15Dic3×SL(2,𝔽3)C2212yescomplemented
16C2×C3×2−1+4C2×C311yescharacteristic
17C22×#239C2×C311yescharacteristic, complemented
18C2×2−1+4⋊C3—21no—
19C2×2−1+4⋊C3D311yescharacteristic, complemented
20C22×#41⋊C3—31no—
21#5834—32no—
22#5834C2×C312yescomplemented
23#5830—31no—
24#5830—31no—
25#5828—31no—
26#5828—31no—
27#5817—32no—
28#5817C2×C312yescomplemented
29#5812C2×C311yescharacteristic, complemented
30C4⋊(#41⋊C3)—32no—
31#548⋊C3—32no—
32#519⋊C3—31no—
33C2×C32×D4—41no—
34C22×C3×Dic3—41no—
35C3×#41⋊C3—21no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)D412yescomplemented
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C3⋊(#41⋊C3)—22no—
40C3×#217—42no—
41C3×#217—41no—
42C3×Dic3⋊C4—42no—
43#3487—31noSylow
44C3×2−1+4—22no—
45C3×2−1+4—62no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41—31no—
49C2×C3×#41C22×C311yescharacteristic
50C22×C3×Q8—31no—
51C22×C3×Q8C22×C311yescharacteristic
52C22×C3×Q8C22×C311yescharacteristic
53C2×#239—22no—
54C2×#239—32no—
55C2×#239—62no—
56C2×#239C22×C312yes—
57C22×D12—31no—
58C22×C4×D3—31no—
592−1+4⋊C3—24no—
602−1+4⋊C3—22no—
61C2×#41⋊C3—21no—
62C2×#41⋊C3—32no—
63C2×#41⋊C3—62no—
64C2×#41⋊C3C2×D311yescharacteristic, complemented
65C22×SL(2,𝔽3)—21no—
66C22×SL(2,𝔽3)—21no—
67C22×SL(2,𝔽3)C2×D311yescharacteristic
68C22×SL(2,𝔽3)C2×D311yescharacteristic, complemented
69C3×#123—121no—
70#947—32no—
71#947—32no—
72#946—32no—
73#946—32no—
74Q8×Dic3—32no—
75Q8×Dic3C22×C312yes—
76#940—32no—
77#940—32no—
78#938—32no—
79#938—32no—
80#937—32no—
81#937—31no—
82#937—31no—
83#935—31no—
84#935A411yescharacteristic
85#930—34no—
86#930—32no—
87#930—32no—
88C4×#67—34no—
89C4×#67—32no—
90C4×#67—32no—
91#926—31no—
92#926—31no—
93C4×SL(2,𝔽3)—32no—
94C23×C32—41no—
95C23×C32—41no—
96C32×D4—82no—
97C2×C32×C4—41no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—42no—
100C2×C3×Dic3—82no—
101C3×SL(2,𝔽3)C2×D411yescharacteristic
102C2×2−1+4C3×D311yescharacteristic
103C22×#41—31no—
104#550—32no—
105#550—92no—
106#548—32no—
107#548—92no—
108#535—91no—
109#535—91no—
110#525—91no—
111#525—91no—
112#519—31no—
113C23×D3—31no—
114C3×#41—21no—
115C3×#41—62no—
116C3×#41—62no—
117C3×#41—62no—
118C3×#41—62no—
119C3×#41—62no—
120C3×#41—62no—
121C3×#41—61no—
122C2×C3×Q8—21no—
123C2×C3×Q8—31no—
124C2×C3×Q8—31no—
125C2×C3×Q8—32no—
126C2×C3×Q8—31no—
127C2×C3×Q8—62no—
128C2×C3×Q8—62no—
129C2×C3×Q8—61no—
130C2×C3×Q8C3×D412yes—
131C2×C3×Q8C23×C311yescharacteristic
132C2×C3×D4—31no—
133C2×C3×D4—31no—
134C2×C3×D4—31no—
135C2×C3×D4—41no—
136C2×C3×D4—81no—
137C2×C3×D4C2×A411yescharacteristic
138C22×C3×C4—31no—
139C22×C3×C4—32no—
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—121no—
143C2×#67—34no—
144C2×#67—32no—
145C2×#67—32no—
146C22×Dic3C2×A411yescharacteristic
147#239—22no—
148#239—64no—
149#239—64no—
150#239—62no—
151C2×D12—32no—
152C2×D12—32no—
153C2×D12—62no—
154C2×D12—62no—
155C2×C4×D3—32no—
156C2×C4×D3—32no—
157C2×C4×D3—62no—
158C2×C4×D3—62no—
159#41⋊C3—22no—
160#41⋊C3—21no—
161#41⋊C3—62no—
162C2×SL(2,𝔽3)—22no—
163C2×SL(2,𝔽3)—21no—
164C2×SL(2,𝔽3)—22no—
165C2×SL(2,𝔽3)—21no—
166C2×SL(2,𝔽3)#6712yescomplemented
167C2×SL(2,𝔽3)C22×D311yescharacteristic
168C3×C4⋊C4—122no—
169C3×C22⋊C4—121no—
170C3×C22⋊C4—122no—
171#217—31no—
172#217—32no—
173#217C2×A411yescharacteristic
174#217C2×A412yes—
175(C2×D3)⋊C4—34no—
176(C2×D3)⋊C4—32no—
177(C2×D3)⋊C4—32no—
178C4⋊Dic3—32no—
179C4⋊Dic3—32no—
180Dic3⋊C4—32no—
181Dic3⋊C4C2×A412yes—
182C4×Dic3—32no—
183C4×Dic3—32no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—81no—
187C22×C32—82no—
188C22×C32—82no—
189C32×C4—81no—
190C3×Dic3—82no—
191C3×Dic3—82no—
1922−1+4—22no—
1932−1+4—62no—
194C2×#41—31no—
195C2×#41—31no—
196C2×#41—31no—
197C2×#41—32no—
198C2×#41—62no—
199C2×#41—92no—
200C2×#41—182no—
201C2×#41C2×C3×D311yescharacteristic
202C22×Q8—31no—
203C22×Q8C2×C3×D311yescharacteristic
204C22×Q8C2×C3×D311yescharacteristic
205C22×D4—91no—
206C23×C4—91no—
207#127—92no—
208#127—92no—
209#124—92no—
210#124—92no—
211#123—31no—
212#123—91no—
213C22⋊Q8—92no—
214C22⋊Q8—92no—
215#121—92no—
216#121—92no—
217#121—94no—
218#121—92no—
219#121—92no—
220#120—91no—
221#120—92no—
222#120—91no—
223C4×Q8—32no—
224C4×Q8—92no—
225C4×D4—92no—
226C4×D4—92no—
227C4×D4—94no—
228#117—91no—
229#117—91no—
230C23×C3—31no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—81no—
234C23×C3—81no—
235C23×C3C22×A411yescharacteristic
236C23×C3C22×A411yescharacteristic
237C22×D3—32no—
238C22×D3—62no—
239C22×D3—122no—
240C3×Q8—32no—
241C3×Q8—31no—
242C3×Q8—32no—
243C3×Q8—62no—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×Q8C2×C3×D411yescharacteristic
247C3×D4—22no—
248C3×D4—62no—
249C3×D4—61no—
250C3×D4—61no—
251C3×D4—62no—
252C3×D4—82no—
253C3×D4—84no—
254C2×C3×C4—32no—
255C2×C3×C4—32no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—41no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—64no—
265C2×C3×C4—81no—
266C2×C3×C4—122no—
267C2×C3×C4—122no—
268C2×C3×C4—242no—
269C2×C3×C4C22×A411yescharacteristic
270#67—122no—
271#67—122no—
272#67—124no—
273C2×Dic3—22no—
274C2×Dic3—32no—
275C2×Dic3C22×A412yes—
276C2×Dic3C22×A412yes—
277D12—62no—
278D12—62no—
279D12—64no—
280C4×D3—62no—
281C4×D3—64no—
282C4×D3—62no—
283SL(2,𝔽3)—21no—
284SL(2,𝔽3)C2×#6711yescharacteristic
285C2×C32—41no—
286C2×C32—42no—
287C2×C32—82no—
288C2×C32—161no—
289C24—91no—
290#41—21no—
291#41—62no—
292#41—62no—
293#41—62no—
294#41—62no—
295#41—62no—
296#41—62no—
297#41—61no—
298#41—62no—
299#41—184no—
300#41—184no—
301#41—182no—
302C2×Q8—21no—
303C2×Q8—31no—
304C2×Q8—31no—
305C2×Q8—32no—
306C2×Q8—31no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—61no—
310C2×Q8C3×#6712yes—
311C2×Q8C22×C3×D311yescharacteristic
312C2×D4—31no—
313C2×D4—31no—
314C2×D4—31no—
315C2×D4—92no—
316C2×D4—92no—
317C2×D4—92no—
318C2×D4—94no—
319C2×D4—92no—
320C2×D4—182no—
321C2×D4—182no—
322C2×D4D3×A411yescharacteristic
323C22×C4—31no—
324C22×C4—32no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—92no—
329C22×C4—92no—
330C22×C4—182no—
331C22×C4—182no—
332C4⋊C4—32no—
333C4⋊C4—92no—
334C4⋊C4—92no—
335C4⋊C4—92no—
336C22⋊C4—31no—
337C22⋊C4—32no—
338C22⋊C4—92no—
339C22⋊C4—92no—
340C22⋊C4—94no—
341C22⋊C4—91no—
342C22⋊C4—92no—
343C42—92no—
344C42—92no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—42no—
349C22×C3—41no—
350C22×C3—61no—
351C22×C3—62no—
352C22×C3—82no—
353C22×C3—81no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—81no—
358C22×C3—162no—
359C22×C3—162no—
360C22×C3D4×A412yes—
361C22×C3C23×A411yescharacteristic
362C2×D3—62no—
363C2×D3—62no—
364C2×D3—62no—
365C2×D3—124no—
366C3×C4—21no—
367C3×C4—31no—
368C3×C4—32no—
369C3×C4—31no—
370C3×C4—61no—
371C3×C4—81no—
372C3×C4—82no—
373C3×C4—242no—
374C3×C4—242no—
375Dic3—22no—
376Dic3—22no—
377Dic3—62no—
378C32—41noSylow
379C23—31no—
380C23—92no—
381C23—182no—
382C23—362no—
383C23C2×D3×A411yescharacteristic
384C23C2×D3×A411yescharacteristic
385Q8—32no—
386Q8—31no—
387Q8—32no—
388Q8—62no—
389Q8—61no—
390Q8—61no—
391Q8C2×C3×#6711yescharacteristic
392D4—22no—
393D4—62no—
394D4—61no—
395D4—61no—
396D4—62no—
397D4—182no—
398D4—182no—
399D4—184no—
400D4—362no—
401D4—362no—
402D4—364no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—31no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—64no—
415C2×C4—62no—
416C2×C4—92no—
417C2×C4—182no—
418C2×C4—182no—
419C2×C4—184no—
420C2×C4C2×D3×A411yescharacteristic
421C2×C3—22no—
422C2×C3—41no—
423C2×C3—42no—
424C2×C3—41no—
425C2×C3—82no—
426C2×C3—82no—
427C2×C3—81no—
428C2×C3—121no—
429C2×C3—162no—
430C2×C3—161no—
431C2×C3—162no—
432C2×C3C2×D4×A411yescharacteristic
433C2×C3#594512yes—
434D3—122no—
435C22—22no—
436C22—22no—
437C22—21no—
438C22—62no—
439C22—61no—
440C22—182no—
441C22—182no—
442C22—182no—
443C22—364no—
444C22A4×#6712yes—
445C22C22×D3×A411yescharacteristic
446C4—21no—
447C4—31no—
448C4—32no—
449C4—31no—
450C4—62no—
451C4—61no—
452C4—62no—
453C4—182no—
454C3—41no—
455C3—81no—
456C3#3487⋊C311yescharacteristic, complemented
457C2—22no—
458C2—41no—
459C2—121no—
460C2—362no—
461C2#5968⋊C312yes—
462C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×2−1+4C2×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×C3×2−1+4⋊C3C2semidirect
2. C22×C3⋊(#41⋊C3)C2semidirect
3. #5834⋊C3C2semidirect
4. #5812⋊C3C2semidirect
5. #5817⋊C3C2semidirect
6. #87290C3semidirect
9. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
15. Dic3×SL(2,𝔽3)C22semidirect
17. C22×#239C2×C3semidirect
19. C2×2−1+4⋊C3D3semidirect
22. #5834C2×C3semidirect
28. #5817C2×C3semidirect
29. #5812C2×C3semidirect
37. C2×C3×SL(2,𝔽3)D4semidirect
64. C2×#41⋊C3C2×D3semidirect
68. C22×SL(2,𝔽3)C2×D3semidirect
166. C2×SL(2,𝔽3)#67semidirect
456. C3#3487⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C22×C3⋊(#41⋊C3)2
#5834⋊C32
#5812⋊C32
#5817⋊C32
#872903
#3487⋊C33
C3×#9354