Finite Group Databasebeta

← #10950848

Subgroups — #10950848

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#78577C211yesmaximal, characteristic, complemented
2#78568C211yesmaximal, characteristic
3C3×#2108C211yesmaximal, characteristic, complemented
4#71447C211yesmaximal, characteristic, complemented
5#71444C211yesmaximal, characteristic
6#71416C211yesmaximal, characteristic, complemented
7#71414C211yesmaximal, characteristic
8#69084C211yesmaximal, characteristic, complemented
9#69082C211yesmaximal, characteristic, complemented
10#69077C211yesmaximal, characteristic, complemented
11#69069C211yesmaximal, characteristic, complemented
12#69065C211yesmaximal, characteristic, complemented
13C4×Dic24C211yesmaximal, characteristic, complemented
14C4×#800C211yesmaximal, characteristic, complemented
15#69059C211yesmaximal, characteristic, complemented
16#33745—31nomaximal, Sylow
17#5534—21no—
18#5518—21no—
19C2×C3×M32C2211yescharacteristic
20C22×C3×C16C2211yescharacteristic
21C3×#405C2211yescharacteristic
22#5137—21no—
23#5134—21no—
24#5116C2211yescharacteristic
25#5116C2211yescharacteristic
26#5116C2211yescharacteristic
27#5115C2211yescharacteristic
28C2×Dic24C2211yescharacteristic
29C2×Dic24C2211yescharacteristic
30C2×#800C2211yescharacteristic
31C2×#800C2211yescharacteristic
32#4716—21no—
33#4714—21no—
34#4710—21no—
35#4703—21no—
36#4699C2211yescharacteristic
37C4×Dic12C2211yescharacteristic
38C4×Dic12C2211yescharacteristic
39C4×Dic12C2211yescharacteristic
40#4696C2211yescharacteristic
41C4×D24C2211yescharacteristic
42#4683C2211yescharacteristic
43#4682C2211yescharacteristic
44C3×#346C2211yescharacteristic
45C3×#346C2211yescharacteristic
46C3×C4×C16C2211yescharacteristic
47C3×C4×C16C2211yescharacteristic
48#4509C2211yescharacteristic
49#4509C2211yescharacteristic
50#4505C2211yescharacteristic
51#4505C2211yescharacteristic
52#4504C2211yescharacteristic
53#4504C2211yescharacteristic
54#4503C2211yescharacteristic
55#4503C2211yescharacteristic
56#4503C2211yescharacteristic
57#4503C2211yescharacteristic
58#4503C2211yescharacteristic
59#4503C2211yescharacteristic
60#3177—31no—
61#3175—31no—
62#2255—31no—
63#2251—31no—
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#2108D311yescharacteristic, complemented
75C2×C3×M16D411yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C3×#117D411yescharacteristic
78#930—21no—
79#924—21no—
80#924—21no—
81#924—21no—
82C2×Dic12—21no—
83C2×Dic12C2311yescharacteristic
84C2×Dic12C2311yescharacteristic
85C2×Dic12C2311yescharacteristic
86C2×D24C2311yescharacteristic
87C2×#204—21no—
88#898—41no—
89#896—41no—
90#889—41no—
91#889—41no—
92#885—41no—
93#883—41no—
94#879—41no—
95#879—41no—
96#875—21no—
97C4×D12—21no—
98#870—21no—
99#869—21no—
100#869—21no—
101C4×Dic6—21no—
102C4×Dic6—21no—
103C4×Dic6—21no—
104C3×M32—22no—
105C2×C3×C16—22no—
106C2×C3×C16C2311yescharacteristic
107C2×C3×C16C2311yescharacteristic
108C2×C3×C16C2311yescharacteristic
109C2×C3×C16C2311yescharacteristic
110C3×#97D411yescharacteristic
111C3×#97D411yescharacteristic
112C3×C4×C8C2311yescharacteristic
113C3×C4×C8C2311yescharacteristic
114#821—21no—
115#821—21no—
116#818—21no—
117#818C2311yescharacteristic
118#818C2311yescharacteristic
119#818C2311yescharacteristic
120#818C2311yescharacteristic
121#817—21no—
122#816—21no—
123#816—21no—
124#816—21no—
125#816—21no—
126#816—21no—
127#816—21no—
128Dic24—22no—
129Dic24—22no—
130#800—22no—
131#800—22no—
132#542—61no—
133#541—61no—
134C2×Q32—31no—
135C2×Q32—31no—
136C2×SD32—31no—
137C2×SD32—31no—
138C2×M32C2×D311yescharacteristic
139C22×C16C2×D311yescharacteristic
140#500—31no—
141#499—31no—
142#495—31no—
143#494—31no—
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#405C2×D311yescharacteristic
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#346C2×D311yescharacteristic
172#346C2×D311yescharacteristic
173C4×C16C2×D311yescharacteristic
174C4×C16C2×D311yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C2×#67—41no—
177C2×D12—21no—
178C2×C4×D3—41no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—41no—
183C3×M16—22no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8C2×D411yescharacteristic
187C2×C3×C8C2×D411yescharacteristic
188C2×C3×C8C2×D411yescharacteristic
189C2×C3×C8C2411yescharacteristic
190C3×C4⋊C4D811yescharacteristic
191C3×C4⋊C4D811yescharacteristic
192C3×C22⋊C4D811yescharacteristic
193C3×C22⋊C4D811yescharacteristic
194C3×C42C2×D411yescharacteristic
195C3×C42C2×D411yescharacteristic
196#217—41no—
197#217—41no—
198#217—41no—
199(C2×D3)⋊C4—41no—
200(C2×D3)⋊C4—41no—
201(C2×D3)⋊C4—41no—
202(C2×D3)⋊C4—41no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—41no—
208C4⋊Dic3—41no—
209C4⋊Dic3—41no—
210Dic3⋊C4—41no—
211Dic3⋊C4—41no—
212Dic3⋊C4—41no—
213Dic3⋊C4—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218C4×Dic3—41no—
219C4×Dic3—41no—
220C4×Dic3—41no—
221Dic12—21no—
222Dic12—21no—
223Dic12—21no—
224Dic12—21no—
225Dic12—21no—
226Dic12—21no—
227Dic12—42no—
228D24—21no—
229D24—21no—
230#204—42no—
231C3×C16—21no—
232C3×C16—21no—
233C3×C16#4112yes—
234C3×C16#4112yes—
235C2×Q16—31no—
236C2×Q16—31no—
237C2×Q16—31no—
238C2×Q16—61no—
239C2×SD16—61no—
240C2×D8—31no—
241C2×M16D1211yescharacteristic
242C22×C8C22×D311yescharacteristic
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245#126—121no—
246#126—121no—
247#126—121no—
248#125—61no—
249#125—121no—
250#125—121no—
251#124—61no—
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#117D1211yescharacteristic
264Q32—62no—
265Q32—62no—
266SD32—62no—
267SD32—62no—
268M32—22no—
269C2×C16—22no—
270C2×C16C22×D311yescharacteristic
271C2×C16C22×D311yescharacteristic
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#97D1211yescharacteristic
289#97D1211yescharacteristic
290C4×C8C22×D311yescharacteristic
291C4×C8C22×D311yescharacteristic
292C23×C3C2×D811yescharacteristic
293C22×D3—41no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4C2×D811yescharacteristic
297C2×C3×C4C2×D811yescharacteristic
298C2×C3×C4C2×D811yescharacteristic
299C2×C3×C4C2×D811yescharacteristic
300C2×C3×C4C2×D811yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302#67—161no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310D12—41no—
311D12—41no—
312C4×D3—161no—
313Dic6—41no—
314Dic6—41no—
315Dic6—41no—
316Dic6—41no—
317Dic6—41no—
318Dic6—41no—
319Dic6—161no—
320C3×C8—21no—
321C3×C8—21no—
322C3×C8#13512yes—
323C3×C8C2×#4111yescharacteristic
324C3×C8C2×#4111yescharacteristic
325C2×Q8—61no—
326C2×Q8—61no—
327C2×Q8—61no—
328C2×Q8—121no—
329C2×D4—61no—
330C2×D4—121no—
331C22×C4—121no—
332C22×C4C2×D1211yescharacteristic
333Q16—61no—
334Q16—61no—
335Q16—61no—
336Q16—61no—
337Q16—61no—
338Q16—61no—
339Q16—122no—
340SD16—122no—
341D8—61no—
342D8—61no—
343M16—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8C2×D1211yescharacteristic
347C2×C8C2×D1211yescharacteristic
348C2×C8C2×D1211yescharacteristic
349C2×C8C23×D311yescharacteristic
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—121no—
355C4⋊C4—121no—
356C4⋊C4—121no—
357C4⋊C4—121no—
358C4⋊C4—121no—
359C4⋊C4—121no—
360C4⋊C4—121no—
361C4⋊C4—121no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4D2411yescharacteristic
366C4⋊C4D2411yescharacteristic
367C22⋊C4—121no—
368C22⋊C4—121no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4D2411yescharacteristic
375C22⋊C4D2411yescharacteristic
376C42—121no—
377C42—121no—
378C42—121no—
379C42C2×D1211yescharacteristic
380C42C2×D1211yescharacteristic
381C16—21no—
382C16—21no—
383C16#23512yes—
384C16#23512yes—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3C22×D811yescharacteristic
389C2×D3—81no—
390C2×D3—81no—
391C2×D3—81no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4#53211yescharacteristic
398C3×C4C2×#13511yescharacteristic
399Dic3—81no—
400Dic3—81no—
401Dic3—81no—
402Dic3—81no—
403Dic3—81no—
404Dic3—81no—
405Dic3—81no—
406C23—121no—
407C23C2×D2411yescharacteristic
408Q8—121no—
409Q8—121no—
410Q8—121no—
411Q8—121no—
412Q8—121no—
413Q8—121no—
414Q8—481no—
415D4—121no—
416D4—121no—
417D4—481no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—481no—
428C2×C4C2×D2411yescharacteristic
429C2×C4C2×D2411yescharacteristic
430C2×C4C2×D2411yescharacteristic
431C2×C4C2×D2411yescharacteristic
432C2×C4C2×D2411yescharacteristic
433C2×C4C22×D1211yescharacteristic
434C8—21no—
435C8—21no—
436C8#90412yes—
437C8C2×#23511yescharacteristic
438C8C2×#23511yescharacteristic
439C2×C3—41no—
440C2×C3#316911yescharacteristic
441C2×C3#341611yescharacteristic
442C2×C3#341711yescharacteristic
443D3—161no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22C22×D2411yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—241no—
457C4—241no—
458C4—241no—
459C4—241no—
460C4—241no—
461C4—241no—
462C4—241no—
463C4#547711yescharacteristic
464C4C2×#90411yescharacteristic
465C3#3374511yescharacteristic, complemented, Sylow
466C2—41no—
467C2—481no—
468C2#7847011yescharacteristic
469C2#8201411yescharacteristic
470C2#8201511yescharacteristic
subgroupG/H
centerC22C22×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2108C2
socleC22×C3C22×D8
prime radical#5115C22
hypercenter#2108D3
hypocenterC3#33745
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
1. #78577C2semidirect
3. C3×#2108C2semidirect
4. #71447C2semidirect
6. #71416C2semidirect
8. #69084C2semidirect
9. #69082C2semidirect
10. #69077C2semidirect
11. #69069C2semidirect
12. #69065C2semidirect
13. C4×Dic24C2semidirect
14. C4×#800C2semidirect
15. #69059C2semidirect
74. #2108D3semidirect
465. C3#33745semidirect

Maximal subgroups

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

subgroupindex
#785772
#785682
C3×#21082
#714472
#714442
#714162
#714142
#690842
#690822
#690772
#690692
#690652
C4×Dic242
C4×#8002
#690592
#337453