Finite Group Databasebeta

← #10950845

Subgroups — #10950845

Order 768.

496 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81811C211yesmaximal, characteristic
2#81793C211yesmaximal, characteristic, complemented
3C3×#2251C211yesmaximal, characteristic, complemented
4#71505C211yesmaximal, characteristic, complemented
5#71504C211yesmaximal, characteristic, complemented
6#71496C211yesmaximal, characteristic, complemented
7Dic3×Q32C211yesmaximal, characteristic, complemented
8#71488C211yesmaximal, characteristic
9#71487C211yesmaximal, characteristic
10#71474C211yesmaximal, characteristic, complemented
11Dic3×SD32C211yesmaximal, characteristic, complemented
12#71318C211yesmaximal, characteristic, complemented
13#71310C211yesmaximal, characteristic, complemented
14#69072C211yesmaximal, characteristic, complemented
15#69069C211yesmaximal, characteristic, complemented
16#33745—31nomaximal, Sylow
17#5731—21no—
18#5682—21no—
19C2×C3×Q32C2211yescharacteristic
20C2×C3×Q32C2211yescharacteristic
21C2×C3×SD32C2211yescharacteristic
22C2×C3×SD32C2211yescharacteristic
23C3×#495C2211yescharacteristic
24C3×#494C2211yescharacteristic
25#5189—21no—
26#5188C2211yescharacteristic
27#5188C2211yescharacteristic
28#5188C2211yescharacteristic
29#5184—21no—
30Dic3×Q16C2211yescharacteristic
31Dic3×Q16C2211yescharacteristic
32Dic3×Q16C2211yescharacteristic
33#5174—21no—
34#5164—21no—
35#5157C2211yescharacteristic
36Dic3×D8C2211yescharacteristic
37#5082—21no—
38#5055—21no—
39C2×#798C2211yescharacteristic
40C2×D3×C16C2211yescharacteristic
41#4689C2211yescharacteristic
42#4683C2211yescharacteristic
43#4682C2211yescharacteristic
44C3×#346C2211yescharacteristic
45#4564C2211yescharacteristic
46#4564C2211yescharacteristic
47#4564C2211yescharacteristic
48#4564C2211yescharacteristic
49#4564C2211yescharacteristic
50#4564C2211yescharacteristic
51#4562C2211yescharacteristic
52#4562C2211yescharacteristic
53#4505C2211yescharacteristic
54#4505C2211yescharacteristic
55#4504C2211yescharacteristic
56#4504C2211yescharacteristic
57#4502C2211yescharacteristic
58Dic3×C16C2211yescharacteristic
59C4×C3⋊C16C2211yescharacteristic
60#3177—31no—
61#3175—31no—
62#2255—31no—
63#2251D311yescharacteristic, complemented
64#2242—31no—
65#2241—31no—
66#2220—31no—
67#2219—31no—
68#2217—31no—
69#2216—31no—
70#2176—31no—
71#2175—31no—
72C4×Q32—31no—
73C4×SD32—31no—
74#2108—31no—
75C2×C3×Q16—21no—
76C2×C3×Q16C2311yescharacteristic
77C2×C3×Q16C2311yescharacteristic
78C2×C3×Q16C2311yescharacteristic
79C2×C3×SD16—21no—
80C2×C3×D8C2311yescharacteristic
81C3×C4⋊Q8—21no—
82C3×#124—21no—
83#946—21no—
84#946—21no—
85#946—21no—
86Q8×Dic3—21no—
87Q8×Dic3—21no—
88Q8×Dic3—21no—
89#938—21no—
90D4×Dic3—21no—
91C2×ΓL(1,𝔽25)D411yescharacteristic
92C2×D3×C8C2311yescharacteristic
93#898—41no—
94#898—41no—
95#889—41no—
96#889—41no—
97#886—41no—
98#885—41no—
99#879—41no—
100#878—41no—
101#872D411yescharacteristic
102#870—21no—
103#869—21no—
104C3×Q32—22no—
105C3×Q32—22no—
106C3×SD32—22no—
107C3×SD32—22no—
108C2×C3×C16C2311yescharacteristic
109C2×C3×C16C2311yescharacteristic
110C3×C4×C8C2311yescharacteristic
111#835—21no—
112#835—21no—
113#835—21no—
114#835—21no—
115#835—21no—
116#835—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#541—61no—
134C2×Q32C2×D311yescharacteristic
135C2×Q32C2×D311yescharacteristic
136C2×SD32C2×D311yescharacteristic
137C2×SD32C2×D311yescharacteristic
138C2×M32—31no—
139C22×C16—31no—
140#500—31no—
141#499—31no—
142#495C2×D311yescharacteristic
143#494C2×D311yescharacteristic
144#491—61no—
145#489—61no—
146#470—61no—
147#468—61no—
148#467—31no—
149#467—31no—
150#467—31no—
151#466—31no—
152#441—61no—
153#440—61no—
154C4×Q16—31no—
155C4×Q16—31no—
156C4×Q16—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#358—31no—
166#358—31no—
167#358—31no—
168#358—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×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×D4—21no—
179C2×#67—41no—
180C22×Dic3—41no—
181C2×C4×D3C2×D411yescharacteristic
182C2×Dic6—41no—
183C3×Q16—21no—
184C3×Q16—21no—
185C3×Q16—21no—
186C3×Q16—21no—
187C3×Q16—21no—
188C3×Q16—21no—
189C3×Q16—42no—
190C3×SD16—42no—
191C3×D8—21no—
192C3×D8—21no—
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2411yescharacteristic
195C3×C4⋊C4—41no—
196C3×C4⋊C4—41no—
197C3×C22⋊C4—41no—
198C3×C22⋊C4—41no—
199C3×C42C2×D411yescharacteristic
200#217—41no—
201#217—41no—
202(C2×D3)⋊C4—41no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4D811yescharacteristic
206(C2×D3)⋊C4D811yescharacteristic
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—41no—
212C4⋊Dic3—41no—
213C4⋊Dic3—41no—
214C4⋊Dic3—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218Dic3⋊C4—41no—
219Dic3⋊C4—41no—
220Dic3⋊C4D811yescharacteristic
221Dic3⋊C4D811yescharacteristic
222C4×Dic3—41no—
223C4×Dic3—41no—
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×Q16—21no—
236C2×Q16C22×D311yescharacteristic
237C2×Q16C22×D311yescharacteristic
238C2×Q16C22×D311yescharacteristic
239C2×SD16—21no—
240C2×D8C22×D311yescharacteristic
241C2×M16—31no—
242C22×C8—31no—
243C4⋊Q8—21no—
244C4⋊Q8—61no—
245#126—121no—
246#126—121no—
247#126—121no—
248#125—61no—
249#125—121no—
250#125—121no—
251#124—21no—
252#124—121no—
253#123—121no—
254C22⋊Q8—61no—
255C22⋊Q8—61no—
256C22⋊Q8—61no—
257C22⋊Q8—121no—
258#121—61no—
259C4×Q8—61no—
260C4×Q8—61no—
261C4×Q8—61no—
262C4×D4—61no—
263#117—31no—
264Q32—22no—
265Q32—22no—
266SD32—22no—
267SD32—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#103—61no—
283#103—61no—
284#103—61no—
285#103—61no—
286#102—61no—
287#102—61no—
288#97—31no—
289#97—31no—
290C4×C8—31no—
291C4×C8C22×D311yescharacteristic
292C23×C3—41no—
293C22×D3C2×D811yescharacteristic
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×Q8—41no—
300C3×D4—41no—
301C3×D4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4C2×D811yescharacteristic
306C2×C3×C4C2×D811yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308#67—161no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—161no—
314C2×Dic3C2×D811yescharacteristic
315C2×Dic3C2×D811yescharacteristic
316C2×Dic3C2×D811yescharacteristic
317C4×D3—21no—
318C4×D3—21no—
319Dic6—161no—
320C3×C8#13512yes—
321C3×C8C2×#4111yescharacteristic
322C3×C8C2×#4111yescharacteristic
323C3⋊C8—21no—
324C3⋊C8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—121no—
329C2×D4—21no—
330C2×D4—121no—
331C22×C4—31no—
332C22×C4—121no—
333Q16—21no—
334Q16—21no—
335Q16—21no—
336Q16—21no—
337Q16—21no—
338Q16—21no—
339Q16—42no—
340SD16—42no—
341D8—21no—
342D8—21no—
343M16—62no—
344C2×C8—31no—
345C2×C8—31no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8D3×D411yescharacteristic
349C2×C8C23×D311yescharacteristic
350C4⋊C4—31no—
351C4⋊C4—31no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—61no—
357C4⋊C4—61no—
358C4⋊C4—121no—
359C4⋊C4—121no—
360C4⋊C4—121no—
361C4⋊C4—121no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C22⋊C4—31no—
368C22⋊C4—31no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C42—31no—
377C42—121no—
378C42—121no—
379C42—121no—
380C42D3×D411yescharacteristic
381C16—61no—
382C16—61no—
383C16#23712yes—
384C16#23712yes—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3C22×D811yescharacteristic
389C2×D3—21no—
390C2×D3—21no—
391C2×D3—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4—81no—
397C3×C4#53211yescharacteristic
398C3×C4C2×#13511yescharacteristic
399Dic3—21no—
400Dic3—21no—
401Dic3—21no—
402Dic3—81no—
403Dic3—81no—
404Dic3—81no—
405Dic3—81no—
406C23—31no—
407C23—41no—
408Q8—41no—
409Q8—41no—
410Q8—41no—
411Q8—41no—
412Q8—41no—
413Q8—41no—
414Q8—481no—
415D4—41no—
416D4—41no—
417D4—481no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—481no—
431C2×C4D3×D811yescharacteristic
432C2×C4D3×D811yescharacteristic
433C2×C4C2×D3×D411yescharacteristic
434C8—61no—
435C8—61no—
436C8#91212yes—
437C8C2×#23711yescharacteristic
438C8C2×#23711yescharacteristic
439C2×C3—161no—
440C2×C3#316911yescharacteristic
441C2×C3#341611yescharacteristic
442C2×C3#341711yescharacteristic
443D3—41no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22C2×D3×D811yescharacteristic
451C4—21no—
452C4—21no—
453C4—61no—
454C4—61no—
455C4—61no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—241no—
460C4—241no—
461C4—241no—
462C4—241no—
463C4#571611yescharacteristic
464C4C2×#91211yescharacteristic
465C3#3374511yescharacteristic, complemented, Sylow
466C2—121no—
467C2—161no—
468C2#8177511yescharacteristic
469C2#8204011yescharacteristic
470C2#8204111yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2251C2
socleC22×C3C22×D8
prime radical#5157C22
hypercenter#2251D3
hypocenterC3#33745
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2251

Complements

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

NH ≅ G/Ntype
2. #81793C2semidirect
3. C3×#2251C2semidirect
4. #71505C2semidirect
5. #71504C2semidirect
6. #71496C2semidirect
7. Dic3×Q32C2semidirect
10. #71474C2semidirect
11. Dic3×SD32C2semidirect
12. #71318C2semidirect
13. #71310C2semidirect
14. #69072C2semidirect
15. #69069C2semidirect
63. #2251D3semidirect
465. C3#33745semidirect

Maximal subgroups

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

subgroupindex
#818112
#817932
C3×#22512
#715052
#715042
#714962
Dic3×Q322
#714882
#714872
#714742
Dic3×SD322
#713182
#713102
#690722
#690692
#337453