Finite Group Databasebeta

← #10950815

Subgroups — #10950815

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#80743C211yesmaximal, characteristic, complemented
2#80699C211yesmaximal, characteristic
3C3×#2217C211yesmaximal, characteristic, complemented
4#71479C211yesmaximal, characteristic, complemented
5Dic3×SD32C211yesmaximal, characteristic, complemented
6#71414C211yesmaximal, characteristic, complemented
7#71399C211yesmaximal, characteristic, complemented
8#71084C211yesmaximal, characteristic, complemented
9#71074C211yesmaximal, characteristic, complemented
10#69343C211yesmaximal, characteristic, complemented
11#69337C211yesmaximal, characteristic, complemented
12#69218C211yesmaximal, characteristic, complemented
13#69208C211yesmaximal, characteristic, complemented
14#69186C211yesmaximal, characteristic, complemented
15#69176C211yesmaximal, characteristic, complemented
16#33743—31nomaximal, Sylow
17#5625—21no—
18#5619—21no—
19C2×C3×SD32C2211yescharacteristic
20C22×C3×C16C2211yescharacteristic
21C3×#467C2211yescharacteristic
22C3×#466C2211yescharacteristic
23#5183C2211yescharacteristic
24Dic3×Q16C2211yescharacteristic
25C2×#828C2211yescharacteristic
26#5151C2211yescharacteristic
27Dic3×D8C2211yescharacteristic
28C2×#827C2211yescharacteristic
29#5116C2211yescharacteristic
30#5115C2211yescharacteristic
31#5101C2211yescharacteristic
32C2×#812C2211yescharacteristic
33#5052—21no—
34#5039—21no—
35C2×#800C2211yescharacteristic
36#4878C2211yescharacteristic
37#4878C2211yescharacteristic
38#4873C2211yescharacteristic
39#4872C2211yescharacteristic
40#4797—21no—
41#4789—21no—
42#4785—21no—
43#4764—21no—
44C3×#367C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#357C2211yescharacteristic
47#4564C2211yescharacteristic
48#4562C2211yescharacteristic
49#4509C2211yescharacteristic
50#4505C2211yescharacteristic
51#4503C2211yescharacteristic
52#4502C2211yescharacteristic
53#4502C2211yescharacteristic
54Dic3×C16C2211yescharacteristic
55Dic3×C16C2211yescharacteristic
56#4492C2211yescharacteristic
57#4491C2211yescharacteristic
58#4489C2211yescharacteristic
59#4489C2211yescharacteristic
60#3177—31no—
61#3176—31no—
62#2253—31no—
63#2248—31no—
64#2245—31no—
65#2244—31no—
66#2220—31no—
67#2220—31no—
68#2217—31no—
69#2217D311yescharacteristic, complemented
70#2175—31no—
71#2175—31no—
72C4×SD32—31no—
73C4×SD32—31no—
74#2108—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×D8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C3×C22⋊Q8—21no—
79C3×#121—21no—
80Q8×Dic3—21no—
81#944—21no—
82C2×#216—21no—
83#940—21no—
84#939—41no—
85#938—41no—
86D4×Dic3—21no—
87C2×#213—21no—
88#930—21no—
89#926D411yescharacteristic
90#924—21no—
91C2×#208D411yescharacteristic
92C2×Dic12C2311yescharacteristic
93C2×D24C2311yescharacteristic
94#893—21no—
95#890—21no—
96#890—21no—
97#890—41no—
98#889—41no—
99#887—21no—
100#886—41no—
101#883—41no—
102#879—41no—
103#879—41no—
104C3×SD32—22no—
105C2×C3×C16—22no—
106C2×C3×C16C2311yescharacteristic
107C2×C3×C16C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#107C2311yescharacteristic
110C3×#103—21no—
111C3×#102—21no—
112#835—21no—
113#832—21no—
114#828—22no—
115#827—22no—
116#821—21no—
117#818C2311yescharacteristic
118#818C2311yescharacteristic
119#816—21no—
120#815D411yescharacteristic
121#815D411yescharacteristic
122C8×Dic3C2311yescharacteristic
123C8×Dic3C2311yescharacteristic
124#812—22no—
125C2×C3⋊C16C2311yescharacteristic
126C2×C3⋊C16C2311yescharacteristic
127#810—21no—
128#809—21no—
129#808—21no—
130#808—21no—
131#800—22no—
132#541—61no—
133#540—61no—
134C2×SD32—31no—
135C2×SD32—31no—
136C2×SD32—31no—
137C2×SD32C2×D311yescharacteristic
138C2×M32—31no—
139C22×C16C2×D311yescharacteristic
140#499—31no—
141#499—31no—
142#494—31no—
143#493—31no—
144#491—61no—
145#490—61no—
146#470—61no—
147#469—61no—
148#467—31no—
149#467C2×D311yescharacteristic
150#466—31no—
151#466C2×D311yescharacteristic
152#442—61no—
153#441—61no—
154C4×Q16—31no—
155C4×Q16—31no—
156C4×D8—31no—
157C4×D8—31no—
158#405—31no—
159#367—31no—
160#367C2×D311yescharacteristic
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#358—31no—
166#358C2×D311yescharacteristic
167#357—31no—
168#357—31no—
169#357—31no—
170#357C2×D311yescharacteristic
171#346—31no—
172#346—31no—
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×D4—21no—
177C2×C3×D4—41no—
178C22×C3×C4C2×D411yescharacteristic
179C2×#67—41no—
180C2×#67—41no—
181C2×#67—41no—
182C22×Dic3—41no—
183C2×D12—21no—
184C2×C4×D3—41no—
185C2×Dic6—21no—
186C3×Q16—21no—
187C3×Q16—21no—
188C3×D8—21no—
189C3×D8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8C2411yescharacteristic
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—41no—
197C3×C22⋊C4—41no—
198C3×C22⋊C4—41no—
199#217—41no—
200#217—41no—
201#217D811yescharacteristic
202#217D811yescharacteristic
203#216—42no—
204#213—42no—
205(C2×D3)⋊C4—41no—
206(C2×D3)⋊C4—41no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—41no—
210C4⋊Dic3—41no—
211C4⋊Dic3—41no—
212C4⋊Dic3D811yescharacteristic
213C4⋊Dic3D811yescharacteristic
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218Dic3⋊C4—41no—
219Dic3⋊C4—41no—
220C4×Dic3—41no—
221C4×Dic3—41no—
222C4×Dic3C2×D411yescharacteristic
223C4×Dic3C2×D411yescharacteristic
224#208—22no—
225C2×C3⋊C8C2×D411yescharacteristic
226C2×C3⋊C8C2×D411yescharacteristic
227Dic12—21no—
228Dic12—21no—
229D24—21no—
230D24—21no—
231C3×C16—21no—
232C3×C16#4112yes—
233C3⋊C16—21no—
234C3⋊C16#4112yes—
235C2×Q16—31no—
236C2×Q16—61no—
237C2×Q16C22×D311yescharacteristic
238C2×D8—31no—
239C2×D8—61no—
240C2×D8C22×D311yescharacteristic
241C2×M16—31no—
242C22×C8C22×D311yescharacteristic
243C4⋊Q8—61no—
244#127—61no—
245#126—121no—
246#126—121no—
247#125—61no—
248#125—61no—
249#125—121no—
250#125—121no—
251#123—121no—
252#123—121no—
253C22⋊Q8—21no—
254C22⋊Q8—61no—
255#121—21no—
256#121—61no—
257#121—121no—
258#121—121no—
259C4×Q8—61no—
260C4×Q8—61no—
261C4×D4—61no—
262C4×D4—61no—
263#117—31no—
264SD32—22no—
265SD32—62no—
266SD32—62no—
267SD32—62no—
268M32—62no—
269C2×C16—22no—
270C2×C16—31no—
271C2×C16—31no—
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—31no—
275#107—31no—
276#107C22×D311yescharacteristic
277#107C22×D311yescharacteristic
278#106—61no—
279#106—61no—
280#103—21no—
281#103—61no—
282#103—61no—
283#103—61no—
284#102—21no—
285#102—61no—
286#102—61no—
287#102—61no—
288#97—31no—
289#97—31no—
290C4×C8—31no—
291C4×C8—31no—
292C23×C3—41no—
293C23×C3C2×D811yescharacteristic
294C22×D3—41no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—161no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4C2×D811yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307#67—161no—
308#67—161no—
309#67—161no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—161no—
314C2×Dic3C2×D811yescharacteristic
315C2×Dic3C2×D811yescharacteristic
316C2×Dic3C2×D811yescharacteristic
317C2×Dic3C2×D811yescharacteristic
318D12—41no—
319D12—41no—
320C4×D3—161no—
321Dic6—41no—
322Dic6—41no—
323C3×C8—21no—
324C3×C8C2×#4111yescharacteristic
325C3×C8C2×#4111yescharacteristic
326C3⋊C8—21no—
327C3⋊C8#13512yes—
328C2×Q8—21no—
329C2×Q8—61no—
330C2×D4—21no—
331C2×D4—41no—
332C2×D4—61no—
333C2×D4—121no—
334C2×D4—121no—
335C2×D4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4D3×D411yescharacteristic
339Q16—21no—
340Q16—21no—
341Q16—61no—
342Q16—61no—
343Q16—122no—
344D8—21no—
345D8—21no—
346D8—61no—
347D8—61no—
348D8—122no—
349M16—62no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—31no—
353C2×C8—31no—
354C2×C8D3×D411yescharacteristic
355C2×C8C23×D311yescharacteristic
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—41no—
361C4⋊C4—61no—
362C4⋊C4—61no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C22⋊C4—31no—
373C22⋊C4—31no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C42—31no—
381C42—31no—
382C42—121no—
383C42—121no—
384C16—21no—
385C16—32no—
386C16—61no—
387C16#23712yes—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3C22×D811yescharacteristic
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C3×C4—21no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4—81no—
402C3×C4#53211yescharacteristic
403C3×C4C2×#13511yescharacteristic
404Dic3—21no—
405Dic3—21no—
406Dic3—21no—
407Dic3—21no—
408Dic3—81no—
409Dic3—81no—
410Dic3—81no—
411C23—41no—
412C23—121no—
413C23D3×D811yescharacteristic
414Q8—41no—
415Q8—41no—
416Q8—121no—
417Q8—121no—
418D4—41no—
419D4—41no—
420D4—121no—
421D4—121no—
422D4—161no—
423D4—481no—
424D4—481no—
425D4—481no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—481no—
439C2×C4—481no—
440C2×C4D3×D811yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C8—21no—
443C8—32no—
444C8—61no—
445C8C2×#23711yescharacteristic
446C8D3×#4111yescharacteristic
447C2×C3—41no—
448C2×C3—161no—
449C2×C3#316911yescharacteristic
450C2×C3#341611yescharacteristic
451C2×C3#341611yescharacteristic
452D3—161no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22C2×D3×D811yescharacteristic
463C4—21no—
464C4—61no—
465C4—61no—
466C4—61no—
467C4—61no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—241no—
472C4—241no—
473C4—241no—
474C4#559611yescharacteristic
475C4D3×#13511yescharacteristic
476C3#3374311yescharacteristic, complemented, Sylow
477C2—41no—
478C2—161no—
479C2—481no—
480C2#8069711yescharacteristic
481C2#8202411yescharacteristic
482C2#8203211yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2217C2
socleC22×C3C22×D8
prime radical#80699C2
hypercenter#2217D3
hypocenterC3#33743
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #80743C2semidirect
3. C3×#2217C2semidirect
4. #71479C2semidirect
5. Dic3×SD32C2semidirect
6. #71414C2semidirect
7. #71399C2semidirect
8. #71084C2semidirect
9. #71074C2semidirect
10. #69343C2semidirect
11. #69337C2semidirect
12. #69218C2semidirect
13. #69208C2semidirect
14. #69186C2semidirect
15. #69176C2semidirect
69. #2217D3semidirect
476. C3#33743semidirect

Maximal subgroups

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

subgroupindex
#807432
#806992
C3×#22172
#714792
Dic3×SD322
#714142
#713992
#710842
#710742
#693432
#693372
#692182
#692082
#691862
#691762
#337433