Finite Group Databasebeta

← #87251⋊C3 #11923371

Subgroups — #87251⋊C3 #11923371

Order 1152.

887 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 887 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#1007⋊C3C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#67C212yesmaximal, complemented
4#5812⋊C3C211yesmaximal, characteristic, complemented
5#5817⋊C3C212yesmaximal, complemented
6#87251C311yesmaximal, characteristic, complemented
7#3486⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)—22no—
9C22×C3×SL(2,𝔽3)C2212yes—
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#1007⋊C3—22no—
12C2×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
14C3×#953—41nomaximal
15Dic3×SL(2,𝔽3)C2212yescomplemented
16C23×C3×Q8C2×C311yescharacteristic, complemented
17C2×#1007C2×C311yescharacteristic, complemented
18C2×2−1+4⋊C3—31no—
19C23×SL(2,𝔽3)—21no—
20C23×SL(2,𝔽3)D311yescharacteristic, complemented
21#5817—32no—
22#5817C2×C312yescomplemented
23#5816—31no—
24#5816—31no—
25Q8×#67—32no—
26Q8×#67C2×C312yescomplemented
27#5812C2×C311yescharacteristic, complemented
28#5791—31no—
29#5791—31no—
30D4×SL(2,𝔽3)—32no—
31#548⋊C3—32no—
32#519⋊C3—31no—
33C24×C32—41no—
34C2×C3×#67—41no—
35C2×C3×#67—42no—
36C2×C3×SL(2,𝔽3)—22no—
37C2×C3×SL(2,𝔽3)D412yescomplemented
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39D3×SL(2,𝔽3)—21no—
40C3⋊(#41⋊C3)—21no—
41C3×#217—42no—
42C3×#217—41no—
43#3486—31noSylow
44C22×C3×Q8—22no—
45C22×C3×Q8—32no—
46C22×C3×Q8—31no—
47C22×C3×Q8—62no—
48C22×C3×Q8C22×C312yes—
49C22×C3×Q8C22×C311yescharacteristic
50C23×C3×C4—31no—
51#1007—22no—
52#1007—62no—
53C2×#239—31no—
54C2×#239C22×C311yescharacteristic, complemented
55C2×D3×Q8—31no—
56C2×D3×Q8C22×C311yescharacteristic, complemented
57C2×#235—31no—
58C2×#235—31no—
592−1+4⋊C3—62no—
60C2×#41⋊C3—31no—
61C22×SL(2,𝔽3)—24no—
62C22×SL(2,𝔽3)—22no—
63C22×SL(2,𝔽3)—22no—
64C22×SL(2,𝔽3)—21no—
65C22×SL(2,𝔽3)—31no—
66C22×SL(2,𝔽3)C2×D312yes—
67C22×SL(2,𝔽3)C2×D311yescharacteristic
68C3×#120—121no—
69#953—31no—
70#953A411yescharacteristic
71#947—32no—
72#947—32no—
73#946—34no—
74#946—32no—
75#946—32no—
76Q8×Dic3—32no—
77Q8×Dic3C22×C312yescomplemented
78#944—32no—
79#944—32no—
80#930—32no—
81#930—32no—
82#929—32no—
83#929—31no—
84#929—31no—
85C4×#67—34no—
86C4×#67—32no—
87C4×#67—32no—
88#926—31no—
89#926—31no—
90#924—32no—
91#924—32no—
92C4×SL(2,𝔽3)—32no—
93C23×C32—41no—
94C23×C32—42no—
95C23×C32—84no—
96C23×C32—82no—
97C22×C3×D3—41no—
98C3×#67—82no—
99C3×#67—84no—
100C2×C3×Dic3—42no—
101C2×C3×Dic3—41no—
102C3×SL(2,𝔽3)C2×D411yescharacteristic
103C2×2−1+4—31no—
104C23×Q8C3×D311yescharacteristic, complemented
105D4×Q8—32no—
106D4×Q8—92no—
107#548—32no—
108#548—92no—
109#536—91no—
110#536—91no—
111#525—91no—
112#525—91no—
113#519—31no—
114C24×C3—41no—
115C24×C3—81no—
116C24×C3C2×A411yescharacteristic
117C2×C3×Q8—22no—
118C2×C3×Q8—32no—
119C2×C3×Q8—31no—
120C2×C3×Q8—32no—
121C2×C3×Q8—32no—
122C2×C3×Q8—62no—
123C2×C3×Q8—62no—
124C2×C3×Q8—64no—
125C2×C3×Q8—62no—
126C2×C3×Q8—64no—
127C2×C3×Q8C3×D412yescomplemented
128C2×C3×Q8C23×C311yescharacteristic
129C2×C3×D4—121no—
130C2×C3×D4—122no—
131C22×C3×C4—31no—
132C22×C3×C4—32no—
133C22×C3×C4—32no—
134C22×C3×C4—31no—
135C22×C3×C4—62no—
136C22×C3×C4—62no—
137C2×#67—32no—
138C2×#67—31no—
139C2×#67C2×A412yes—
140C2×#67C2×A411yescharacteristic
141#239—21no—
142#239—62no—
143#239—62no—
144#239—61no—
145D3×Q8—21no—
146D3×Q8—62no—
147D3×Q8—62no—
148D3×Q8—61no—
149#235—62no—
150#235—62no—
151#235—62no—
152#235—62no—
153C2×D12—31no—
154C2×D12—31no—
155C2×C4×D3—32no—
156C2×C4×D3—31no—
157C2×C4×D3—31no—
158C2×Dic6—31no—
159C2×Dic6—31no—
160#41⋊C3—61no—
161C2×SL(2,𝔽3)—24no—
162C2×SL(2,𝔽3)—22no—
163C2×SL(2,𝔽3)—22no—
164C2×SL(2,𝔽3)—21no—
165C2×SL(2,𝔽3)—61no—
166C2×SL(2,𝔽3)#6712yescomplemented
167C2×SL(2,𝔽3)C22×D311yescharacteristic
168C3×C22⋊C4—121no—
169C3×C22⋊C4—122no—
170#217—32no—
171#217—31no—
172#217C2×A412yes—
173#217C2×A411yescharacteristic
174(C2×D3)⋊C4—34no—
175(C2×D3)⋊C4—32no—
176(C2×D3)⋊C4—32no—
177C4⋊Dic3—32no—
178C4⋊Dic3—32no—
179Dic3⋊C4—34no—
180Dic3⋊C4—32no—
181Dic3⋊C4—32no—
182C4×Dic3—32no—
183C4×Dic3—32no—
184C22×C32—42no—
185C22×C32—44no—
186C22×C32—41no—
187C22×C32—82no—
188C22×C32—82no—
189C22×C32—82no—
190C22×C32—88no—
191C2×C3×D3—82no—
192C2×C3×D3—81no—
193C3×Dic3—82no—
194C3×Dic3—81no—
1952−1+4—62no—
1962−1+4—182no—
197C2×#41—31no—
198C2×#41—91no—
199C2×#41—91no—
200C2×#41—91no—
201C22×Q8—22no—
202C22×Q8—31no—
203C22×Q8—32no—
204C22×Q8—31no—
205C22×Q8—62no—
206C22×Q8—91no—
207C22×Q8C2×C3×D312yes—
208C22×Q8C2×C3×D311yescharacteristic
209C23×C4—31no—
210C4⋊Q8—92no—
211C4⋊Q8—92no—
212#124—92no—
213#124—92no—
214#123—91no—
215#123—91no—
216#123—92no—
217C22⋊Q8—92no—
218C22⋊Q8—94no—
219C22⋊Q8—92no—
220C22⋊Q8—92no—
221C22⋊Q8—92no—
222#121—92no—
223#121—92no—
224#120—31no—
225#120—91no—
226C4×Q8—32no—
227C4×Q8—92no—
228C4×D4—92no—
229C4×D4—92no—
230C4×D4—94no—
231#117—91no—
232#117—91no—
233C23×C3—22no—
234C23×C3—24no—
235C23×C3—42no—
236C23×C3—41no—
237C23×C3—82no—
238C23×C3—84no—
239C23×C3—84no—
240C23×C3—88no—
241C23×C3—82no—
242C23×C3—81no—
243C23×C3—121no—
244C23×C3C22×A411yescharacteristic
245C23×C3C22×A412yes—
246C22×D3—31no—
247C22×D3C22×A411yescharacteristic
248C3×Q8—32no—
249C3×Q8—31no—
250C3×Q8—32no—
251C3×Q8—62no—
252C3×Q8—62no—
253C3×Q8—64no—
254C3×Q8C2×C3×D411yescharacteristic
255C3×D4—242no—
256C3×D4—244no—
257C2×C3×C4—32no—
258C2×C3×C4—32no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—32no—
262C2×C3×C4—62no—
263C2×C3×C4—64no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—122no—
267C2×C3×C4—121no—
268#67—22no—
269#67—24no—
270#67—62no—
271#67—64no—
272C2×Dic3—32no—
273C2×Dic3—31no—
274C2×Dic3C22×A412yes—
275C2×Dic3C22×A411yescharacteristic
276D12—62no—
277D12—61no—
278D12—61no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—64no—
284Dic6—62no—
285Dic6—61no—
286Dic6—61no—
287SL(2,𝔽3)—21no—
288SL(2,𝔽3)C2×#6711yescharacteristic
289C2×C32—41no—
290C2×C32—42no—
291C2×C32—82no—
292C2×C32—84no—
293C3×D3—161no—
294C24D3×A411yescharacteristic
295#41—61no—
296#41—182no—
297#41—182no—
298#41—182no—
299#41—182no—
300#41—182no—
301#41—182no—
302#41—181no—
303C2×Q8—22no—
304C2×Q8—32no—
305C2×Q8—31no—
306C2×Q8—32no—
307C2×Q8—32no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—64no—
311C2×Q8—62no—
312C2×Q8—64no—
313C2×Q8—61no—
314C2×Q8—91no—
315C2×Q8—91no—
316C2×Q8—182no—
317C2×Q8—182no—
318C2×Q8—181no—
319C2×Q8C3×#6712yescomplemented
320C2×Q8C22×C3×D311yescharacteristic
321C2×D4—32no—
322C2×D4—31no—
323C2×D4—91no—
324C2×D4—91no—
325C2×D4—92no—
326C2×D4—91no—
327C22×C4—31no—
328C22×C4—32no—
329C22×C4—32no—
330C22×C4—31no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—91no—
334C22×C4—91no—
335C22×C4—92no—
336C4⋊C4—92no—
337C4⋊C4—92no—
338C4⋊C4—92no—
339C4⋊C4—92no—
340C4⋊C4—94no—
341C22⋊C4—31no—
342C22⋊C4—32no—
343C22⋊C4—92no—
344C22⋊C4—92no—
345C22⋊C4—94no—
346C22⋊C4—91no—
347C22⋊C4—92no—
348C42—92no—
349C42—92no—
350C22×C3—28no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—44no—
355C22×C3—42no—
356C22×C3—41no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—88no—
361C22×C3—816no—
362C22×C3—84no—
363C22×C3—84no—
364C22×C3—84no—
365C22×C3—84no—
366C22×C3—82no—
367C22×C3—81no—
368C22×C3—242no—
369C22×C3—241no—
370C22×C3D4×A412yes—
371C22×C32−1+4⋊C314yescomplemented
372C22×C3C23×A411yescharacteristic
373C2×D3—21no—
374C2×D3—22no—
375C2×D3—61no—
376C2×D3—62no—
377C3×C4—32no—
378C3×C4—31no—
379C3×C4—31no—
380C3×C4—62no—
381C3×C4—242no—
382C3×C4—241no—
383Dic3—21no—
384Dic3—22no—
385Dic3—61no—
386Dic3—62no—
387C32—41noSylow
388C23—24no—
389C23—22no—
390C23—31no—
391C23—91no—
392C23C2×D3×A412yes—
393C23C2×D3×A411yescharacteristic
394Q8—32no—
395Q8—31no—
396Q8—32no—
397Q8—62no—
398Q8—62no—
399Q8—64no—
400Q8—182no—
401Q8—181no—
402Q8—181no—
403Q8C2×C3×#6711yescharacteristic
404D4—64no—
405D4—62no—
406D4—182no—
407D4—181no—
408D4—181no—
409D4—184no—
410D4—182no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—32no—
416C2×C4—31no—
417C2×C4—32no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—64no—
421C2×C4—62no—
422C2×C4—92no—
423C2×C4—91no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4—184no—
429C2×C3—22no—
430C2×C3—24no—
431C2×C3—42no—
432C2×C3—41no—
433C2×C3—84no—
434C2×C3—82no—
435C2×C3—88no—
436C2×C3—84no—
437C2×C3—82no—
438C2×C3—81no—
439C2×C3—481no—
440C2×C3C2×D4×A411yescharacteristic
441C2×C3C2×2−1+4⋊C312yes—
442D3—41no—
443D3—121no—
444C22—28no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—62no—
449C22—61no—
450C22—182no—
451C22—181no—
452C22#1007⋊C314yescomplemented
453C22A4×#6712yes—
454C22C22×D3×A411yescharacteristic
455C4—31no—
456C4—32no—
457C4—31no—
458C4—62no—
459C4—62no—
460C4—61no—
461C4—182no—
462C4—181no—
463C3—41no—
464C3—81no—
465C3#3486⋊C311yescharacteristic, complemented
466C2—24no—
467C2—22no—
468C2—121no—
469C2—361no—
470C2C2×#1007⋊C312yes—
471C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC23×C3×Q8C2×C3
socleC22×C3C23×A4
hypercenterC24D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C2×#1007⋊C3C2semidirect
3. SL(2,𝔽3)×#67C2semidirect
4. #5812⋊C3C2semidirect
5. #5817⋊C3C2semidirect
6. #87251C3semidirect
12. C2×D3×SL(2,𝔽3)C22semidirect
13. C2×C3⋊(#41⋊C3)C22semidirect
15. Dic3×SL(2,𝔽3)C22semidirect
16. C23×C3×Q8C2×C3semidirect
17. C2×#1007C2×C3semidirect
20. C23×SL(2,𝔽3)D3semidirect
22. #5817C2×C3semidirect
26. Q8×#67C2×C3semidirect
27. #5812C2×C3semidirect
37. C2×C3×SL(2,𝔽3)D4semidirect
54. C2×#239C22×C3semidirect
56. C2×D3×Q8C22×C3semidirect
77. Q8×Dic3C22×C3semidirect
104. C23×Q8C3×D3semidirect
127. C2×C3×Q8C3×D4semidirect
166. C2×SL(2,𝔽3)#67semidirect
319. C2×Q8C3×#67semidirect
371. C22×C32−1+4⋊C3semidirect
452. C22#1007⋊C3semidirect
465. C3#3486⋊C3semidirect

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×#1007⋊C32
SL(2,𝔽3)×#672
#5812⋊C32
#5817⋊C32
#872513
#3486⋊C33
C3×#9534