Finite Group Databasebeta

← #10950830

Subgroups — #10950830

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81811C211yesmaximal, characteristic, complemented
2#81776C211yesmaximal, characteristic, complemented
3C3×#2250C211yesmaximal, characteristic, complemented
4#71488C211yesmaximal, characteristic, complemented
5#71487C211yesmaximal, characteristic, complemented
6#71474C211yesmaximal, characteristic, complemented
7Dic3×SD32C211yesmaximal, characteristic, complemented
8#71470C211yesmaximal, characteristic, complemented
9#71469C211yesmaximal, characteristic, complemented
10#71461C211yesmaximal, characteristic, complemented
11Dic3×D16C211yesmaximal, characteristic, complemented
12#71318C211yesmaximal, characteristic, complemented
13#71288C211yesmaximal, characteristic, complemented
14#69072C211yesmaximal, characteristic, complemented
15#69069C211yesmaximal, characteristic, complemented
16#33744—31nomaximal, Sylow
17#5719—21no—
18#5682—21no—
19C2×C3×SD32C2211yescharacteristic
20C2×C3×SD32C2211yescharacteristic
21C2×C3×D16C2211yescharacteristic
22C2×C3×D16C2211yescharacteristic
23C3×#495C2211yescharacteristic
24C3×#493C2211yescharacteristic
25#5188C2211yescharacteristic
26Dic3×Q16C2211yescharacteristic
27#5174—21no—
28#5164—21no—
29#5159—21no—
30#5157C2211yescharacteristic
31#5157C2211yescharacteristic
32#5157C2211yescharacteristic
33#5152—21no—
34Dic3×D8C2211yescharacteristic
35Dic3×D8C2211yescharacteristic
36Dic3×D8C2211yescharacteristic
37#5071—21no—
38#5055—21no—
39C2×#798C2211yescharacteristic
40C2×D3×C16C2211yescharacteristic
41#4689C2211yescharacteristic
42#4683C2211yescharacteristic
43#4682C2211yescharacteristic
44C3×#346C2211yescharacteristic
45#4564C2211yescharacteristic
46#4564C2211yescharacteristic
47#4562C2211yescharacteristic
48#4562C2211yescharacteristic
49#4562C2211yescharacteristic
50#4562C2211yescharacteristic
51#4562C2211yescharacteristic
52#4562C2211yescharacteristic
53#4505C2211yescharacteristic
54#4505C2211yescharacteristic
55#4504C2211yescharacteristic
56#4504C2211yescharacteristic
57#4502C2211yescharacteristic
58Dic3×C16C2211yescharacteristic
59C4×C3⋊C16C2211yescharacteristic
60#3176—31no—
61#3175—31no—
62#2255—31no—
63#2250D311yescharacteristic, complemented
64#2242—31no—
65#2240—31no—
66#2220—31no—
67#2218—31no—
68#2217—31no—
69#2215—31no—
70#2177—31no—
71#2175—31no—
72C4×SD32—31no—
73C4×D16—31no—
74#2108—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×SD16—21no—
77C2×C3×D8—21no—
78C2×C3×D8C2311yescharacteristic
79C2×C3×D8C2311yescharacteristic
80C2×C3×D8C2311yescharacteristic
81C3×#127—21no—
82C3×#124—21no—
83#946—21no—
84Q8×Dic3—21no—
85#939—41no—
86#939—41no—
87#938—21no—
88#938—21no—
89#938—21no—
90#935—41no—
91#935—41no—
92D4×Dic3—21no—
93D4×Dic3—21no—
94D4×Dic3—21no—
95C2×ΓL(1,𝔽25)D411yescharacteristic
96C2×D3×C8C2311yescharacteristic
97#886—41no—
98#885—41no—
99#879—41no—
100#878—41no—
101#872D411yescharacteristic
102#870—21no—
103#869—21no—
104C3×SD32—22no—
105C3×SD32—22no—
106C3×D16—22no—
107C3×D16—22no—
108C2×C3×C16C2311yescharacteristic
109C2×C3×C16C2311yescharacteristic
110C3×C4×C8C2311yescharacteristic
111#835—21no—
112#835—21no—
113#832—21no—
114#832—21no—
115#832—21no—
116#832—21no—
117#832—21no—
118#832—21no—
119#818—21no—
120#818C2311yescharacteristic
121#818C2311yescharacteristic
122#818C2311yescharacteristic
123#818C2311yescharacteristic
124#817—21no—
125#815D411yescharacteristic
126C8×Dic3C2311yescharacteristic
127C2×C3⋊C16C2311yescharacteristic
128C2×C3⋊C16C2311yescharacteristic
129#803D411yescharacteristic
130#798—22no—
131D3×C16—22no—
132#542—61no—
133#540—61no—
134C2×SD32C2×D311yescharacteristic
135C2×SD32C2×D311yescharacteristic
136C2×D16C2×D311yescharacteristic
137C2×D16C2×D311yescharacteristic
138C2×M32—31no—
139C22×C16—31no—
140#500—31no—
141#499—31no—
142#495C2×D311yescharacteristic
143#493C2×D311yescharacteristic
144#490—61no—
145#489—61no—
146#469—61no—
147#468—61no—
148#467—31no—
149#466—31no—
150#466—31no—
151#466—31no—
152#442—61no—
153#440—61no—
154C4×Q16—31no—
155C4×D8—31no—
156C4×D8—31no—
157C4×D8—31no—
158#405—31no—
159#367—31no—
160#367—31no—
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#357—31no—
166#357—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357—31no—
171#346—31no—
172#346C2×D311yescharacteristic
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—41no—
180C2×C3×D4—41no—
181C2×#67—41no—
182C2×#67—41no—
183C2×#67—41no—
184C22×Dic3—41no—
185C22×Dic3—41no—
186C22×Dic3—41no—
187C2×C4×D3C2×D411yescharacteristic
188C2×Dic6—41no—
189C3×Q16—21no—
190C3×Q16—21no—
191C3×SD16—42no—
192C3×D8—21no—
193C3×D8—21no—
194C3×D8—21no—
195C3×D8—21no—
196C3×D8—21no—
197C3×D8—21no—
198C3×D8—42no—
199C2×C3×C8C2×D411yescharacteristic
200C2×C3×C8C2411yescharacteristic
201C3×C22⋊C4—41no—
202C3×C22⋊C4—41no—
203C3×C42C2×D411yescharacteristic
204#217—41no—
205#217—41no—
206#217—41no—
207#217—41no—
208#217—41no—
209#217—41no—
210(C2×D3)⋊C4—41no—
211(C2×D3)⋊C4D811yescharacteristic
212(C2×D3)⋊C4D811yescharacteristic
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—41no—
218C4⋊Dic3—41no—
219Dic3⋊C4—41no—
220Dic3⋊C4—41no—
221Dic3⋊C4—41no—
222Dic3⋊C4D811yescharacteristic
223Dic3⋊C4D811yescharacteristic
224C4×Dic3—41no—
225C4×Dic3C2×D411yescharacteristic
226C2×C3⋊C8C2×D411yescharacteristic
227C2×C3⋊C8C2×D411yescharacteristic
228ΓL(1,𝔽25)—22no—
229D3×C8—21no—
230D3×C8—21no—
231C3×C16#4112yes—
232C3×C16#4112yes—
233C3⋊C16—21no—
234C3⋊C16—21no—
235C2×Q16C22×D311yescharacteristic
236C2×SD16—21no—
237C2×D8—21no—
238C2×D8C22×D311yescharacteristic
239C2×D8C22×D311yescharacteristic
240C2×D8C22×D311yescharacteristic
241C2×M16—31no—
242C22×C8—31no—
243C4⋊Q8—61no—
244#127—21no—
245#126—121no—
246#125—61no—
247#124—21no—
248#124—121no—
249#123—121no—
250#123—121no—
251#123—121no—
252C22⋊Q8—61no—
253C22⋊Q8—121no—
254#121—61no—
255#121—61no—
256#121—61no—
257#121—121no—
258#121—121no—
259C4×Q8—61no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—61no—
263#117—31no—
264SD32—22no—
265SD32—22no—
266D16—22no—
267D16—22no—
268M32—62no—
269C2×C16—31no—
270C2×C16—31no—
271C2×C16—62no—
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—31no—
275#107—31no—
276#107—31no—
277#107—31no—
278#107—61no—
279#106—61no—
280#103—61no—
281#103—61no—
282#102—61no—
283#102—61no—
284#102—61no—
285#102—61no—
286#102—61no—
287#102—61no—
288#97—31no—
289#97—31no—
290C4×C8—31no—
291C4×C8C22×D311yescharacteristic
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C22×D3C2×D811yescharacteristic
296C3×Q8—41no—
297C3×Q8—41no—
298C3×D4—41no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—41no—
302C3×D4—41no—
303C3×D4—41no—
304C3×D4—161no—
305C3×D4—161no—
306C2×C3×C4—41no—
307C2×C3×C4C2×D811yescharacteristic
308C2×C3×C4C2×D811yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310#67—161no—
311#67—161no—
312#67—161no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—161no—
318C2×Dic3—161no—
319C2×Dic3—161no—
320C2×Dic3C2×D811yescharacteristic
321C2×Dic3C2×D811yescharacteristic
322C2×Dic3C2×D811yescharacteristic
323C4×D3—21no—
324C4×D3—21no—
325Dic6—161no—
326C3×C8#13512yes—
327C3×C8C2×#4111yescharacteristic
328C3×C8C2×#4111yescharacteristic
329C3⋊C8—21no—
330C3⋊C8—21no—
331C2×Q8—21no—
332C2×Q8—121no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—121no—
339C2×D4—121no—
340C2×D4—121no—
341C22×C4—31no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4—121no—
345Q16—21no—
346Q16—21no—
347SD16—42no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—21no—
352D8—21no—
353D8—21no—
354D8—42no—
355M16—62no—
356C2×C8—31no—
357C2×C8—31no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8D3×D411yescharacteristic
361C2×C8C23×D311yescharacteristic
362C4⋊C4—31no—
363C4⋊C4—31no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C4⋊C4—121no—
373C22⋊C4—31no—
374C22⋊C4—31no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—121no—
383C22⋊C4—121no—
384C42—31no—
385C42—121no—
386C42D3×D411yescharacteristic
387C16—61no—
388C16—61no—
389C16#23712yes—
390C16#23712yes—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3C22×D811yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—81no—
407C3×C4#53211yescharacteristic
408C3×C4C2×#13511yescharacteristic
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—81no—
413Dic3—81no—
414Dic3—81no—
415Dic3—81no—
416C23—31no—
417C23—41no—
418C23—41no—
419C23—41no—
420Q8—41no—
421Q8—41no—
422Q8—481no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—161no—
430D4—161no—
431D4—481no—
432D4—481no—
433D4—481no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—41no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—481no—
445C2×C4—481no—
446C2×C4—481no—
447C2×C4D3×D811yescharacteristic
448C2×C4D3×D811yescharacteristic
449C2×C4C2×D3×D411yescharacteristic
450C8—61no—
451C8—61no—
452C8#91212yes—
453C8C2×#23711yescharacteristic
454C8C2×#23711yescharacteristic
455C2×C3—161no—
456C2×C3—161no—
457C2×C3—161no—
458C2×C3#316911yescharacteristic
459C2×C3#341511yescharacteristic
460C2×C3#341611yescharacteristic
461D3—41no—
462C22—61no—
463C22—61no—
464C22—61no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22C2×D3×D811yescharacteristic
475C4—21no—
476C4—21no—
477C4—61no—
478C4—61no—
479C4—61no—
480C4—81no—
481C4—241no—
482C4—241no—
483C4—241no—
484C4—241no—
485C4#571611yescharacteristic
486C4C2×#91211yescharacteristic
487C3#3374411yescharacteristic, complemented, Sylow
488C2—121no—
489C2—161no—
490C2—161no—
491C2—161no—
492C2#8177511yescharacteristic
493C2#8203611yescharacteristic
494C2#8203711yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2250C2
socleC22×C3C22×D8
hypercenter#2250D3
hypocenterC3#33744
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2250

Complements

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

NH ≅ G/Ntype
1. #81811C2semidirect
2. #81776C2semidirect
3. C3×#2250C2semidirect
4. #71488C2semidirect
5. #71487C2semidirect
6. #71474C2semidirect
7. Dic3×SD32C2semidirect
8. #71470C2semidirect
9. #71469C2semidirect
10. #71461C2semidirect
11. Dic3×D16C2semidirect
12. #71318C2semidirect
13. #71288C2semidirect
14. #69072C2semidirect
15. #69069C2semidirect
63. #2250D3semidirect
487. C3#33744semidirect

Maximal subgroups

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

subgroupindex
#818112
#817762
C3×#22502
#714882
#714872
#714742
Dic3×SD322
#714702
#714692
#714612
Dic3×D162
#713182
#712882
#690722
#690692
#337443