Finite Group Databasebeta

← #10690035

Subgroups — #10690035

Order 768.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83351C211yesmaximal, characteristic
2M16×D12C211yesmaximal, characteristic
3C3×#1986C211yesmaximal, characteristic, complemented
4#71826C211yesmaximal, characteristic, complemented
5C4×#809C211yesmaximal, characteristic
6#67667C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#8989—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×D3×M16—21no—
11C2×C4×D12C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5126—21no—
15#5084C411yescharacteristic, complemented
16#5084C411yescharacteristic, complemented
17#5083C411yescharacteristic, complemented
18#5083C411yescharacteristic, complemented
19#5046—21no—
20#5043—21no—
21C4×#215—21no—
22C4×#215—21no—
23C2×#809—21no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4706—22no—
28#4688—21no—
29C8×D12—21no—
30C3×#339—21no—
31C3×#339—21no—
32#4535—22no—
33#4529—21no—
34#3101—31no—
35D4×M16—31no—
36#1986D311yescharacteristic, complemented
37C4×#102—31no—
38#1493—31no—
39#1303—31no—
40#1278—31no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16D411yescharacteristic
44C2×C3×M16D411yescharacteristic
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×C4⋊Q8C2×C411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#215—21no—
57C2×#215—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×#208—21no—
61C22×C3⋊C8—21no—
62D3×M16—42no—
63D3×M16—42no—
64C2×ΓL(1,𝔽25)—22no—
65C2×D3×C8—22no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12C2×C411yescharacteristic
69C4×D12C2×C411yescharacteristic
70C4×D12D411yescharacteristic
71C4×D12D411yescharacteristic
72C3×#97—21no—
73C3×C4×C8—21no—
74C3×#95—21no—
75#835—21no—
76#835—21no—
77#830—21no—
78(C2×D3)⋊C8—22no—
79(C2×D3)⋊C8—22no—
80#809—24no—
81#809—21no—
82#809—21no—
83#808—21no—
84#808—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)D412yes—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92C22×M16—61no—
93#533C2×D311yescharacteristic
94C2×C4×D4—31no—
95#483—61no—
96#476—31no—
97#476—31no—
98#461—31no—
99#461—31no—
100#450—61no—
101C4×SD16—61no—
102C4×SD16—61no—
103#436—61no—
104#435—62no—
105C8×D4—61no—
106#431—31no—
107#423—31no—
108C2×#102—61no—
109#407—61no—
110C4×M16C2×D311yescharacteristic
111C2×C4×C8—31no—
112C4×C4⋊C4C2×D311yescharacteristic
113#340—61no—
114#339—21no—
115#339—21no—
116Q8⋊C8—62no—
117C23×D3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×Dic3—21no—
125C2×D12—22no—
126C2×D12—21no—
127C2×D12—41no—
128C2×D12C22⋊C411yescharacteristic
129C2×D12C22⋊C411yescharacteristic
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C3×M16—21no—
139C3×M16—21no—
140C3×M16—21no—
141C3×M16SD1612yescomplemented
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42—41no—
156C3×C42C22×C411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160#215—42no—
161#215—42no—
162(C2×D3)⋊C4—21no—
163(C2×D3)⋊C4—21no—
164(C2×D3)⋊C4—21no—
165(C2×D3)⋊C4—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3C22⋊C411yescharacteristic
168C4⋊Dic3C22⋊C411yescharacteristic
169#208—41no—
170#208—41no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177ΓL(1,𝔽25)—44no—
178D3×C8—44no—
179C22×D4—31no—
180C23×C4—61no—
181C2×SD16—61no—
182C2×SD16—61no—
183C2×M16—62no—
184C2×M16—61no—
185C2×M16—122no—
186C2×M16—122no—
187C2×M16#6711yescharacteristic
188C2×M16#6711yescharacteristic
189C22×C8—62no—
190C22×C8—61no—
191C4⋊Q8C4×D311yescharacteristic
192C4⋊Q8C4×D311yescharacteristic
193#125—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×D4—31no—
199C4×D4—31no—
200C4×D4—31no—
201C4×D4—31no—
202C4×D4—61no—
203C4×D4—61no—
204#117—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C22⋊C4—61no—
208C2×C42—21no—
209C2×C42C22×D311yescharacteristic
210#106—61no—
211#106—61no—
212C4⋊C8—31no—
213C4⋊C8—31no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216#103—61no—
217#103—61no—
218#102—61no—
219#102—61no—
220#102—64no—
221C22⋊C8—62no—
222C22⋊C8—62no—
223C22⋊C8—61no—
224#97—21no—
225C4×C8—21no—
226C4×C8—61no—
227C4×C8—61no—
228C4×C8—61no—
229#95—21no—
230C23×C3#12111yescharacteristic
231C22×D3—21no—
232C22×D3—21no—
233C22×D3—42no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—81no—
237C3×Q8—42no—
238C3×Q8—41no—
239C3×Q8—41no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4C2×C22⋊C411yescharacteristic
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12311yescharacteristic
254C2×C3×C4C2×SD1611yescharacteristic
255C2×C3×C4#13511yescharacteristic
256C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
257C2×C3×C4#13711yescharacteristic
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262D12—21no—
263D12—21no—
264D12—21no—
265D12—42no—
266D12#10412yes—
267C4×D3—41no—
268C4×D3—41no—
269C4×D3—41no—
270C4×D3—41no—
271C4×D3—41no—
272C4×D3—41no—
273C4×D3—41no—
274C4×D3—41no—
275C3×C8—22no—
276C3×C8—41no—
277C3⋊C8—42no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C3⋊C8—42no—
281C24—61no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×D4—31no—
285C2×D4—31no—
286C2×D4—62no—
287C2×D4—61no—
288C2×D4—121no—
289C22×C4—41no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—121no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—122no—
303SD16—122no—
304M16—21no—
305M16—21no—
306M16—21no—
307M16—124no—
308M16—121no—
309M16—121no—
310M16#21512yescomplemented
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—62no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—62no—
317C2×C8—124no—
318C2×C8—121no—
319C2×C8—121no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—61no—
331C22⋊C4—41no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42C2×C4×D311yescharacteristic
340C42#23511yescharacteristic
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3#38611yescharacteristic
346C22×C3#46111yescharacteristic
347C22×C3#46311yescharacteristic
348C2×D3—22no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—82no—
355C2×D3—82no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—82no—
360C3×C4—81no—
361C3×C4—81no—
362C3×C4C2×#10411yescharacteristic
363C3×C4#42111yescharacteristic
364C3×C4C4×SD1611yescharacteristic
365C3×C4#44011yescharacteristic
366C3×C4#45011yescharacteristic
367C3×C4#45211yescharacteristic
368Dic3—41no—
369Dic3—41no—
370C23—61no—
371C23—61no—
372C23—122no—
373C23—121no—
374C23—121no—
375C23—241no—
376C23#93811yescharacteristic
377Q8—42no—
378Q8—41no—
379Q8—41no—
380D4—32no—
381D4—61no—
382D4—61no—
383D4—61no—
384D4—122no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4D3×C22⋊C411yescharacteristic
406C2×C4#67⋊C411yescharacteristic
407C2×C4#88311yescharacteristic
408C2×C4#88411yescharacteristic
409C2×C4C4×#6711yescharacteristic
410C2×C4#93711yescharacteristic
411C2×C4C2×#21511yescharacteristic
412C2×C4#94211yescharacteristic
413C2×C4#94911yescharacteristic
414C2×C4#95011yescharacteristic
415C8—22no—
416C8—41no—
417C8—122no—
418C8—122no—
419C8—121no—
420C8—121no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3#187111yescharacteristic
424C2×C3#187711yescharacteristic
425C2×C3#188111yescharacteristic
426D3—42no—
427D3—81no—
428C22—21no—
429C22—21no—
430C22—62no—
431C22—121no—
432C22—121no—
433C22—121no—
434C22—121no—
435C22—121no—
436C22—242no—
437C22—242no—
438C22#495611yescharacteristic
439C22#505711yescharacteristic
440C22#508311yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—82no—
445C4—81no—
446C4—81no—
447C4—121no—
448C4—121no—
449C4D3×#10411yescharacteristic
450C4#482111yescharacteristic
451C4C4×#21511yescharacteristic
452C4#502611yescharacteristic
453C4#504611yescharacteristic
454C4#504711yescharacteristic
455C3#898911yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—122no—
459C2—241no—
460C2#6968711yescharacteristic
461C2#7038111yescharacteristic
462C2#7041611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1986D3
hypocenterC3#8989
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
3. C3×#1986C2semidirect
4. #71826C2semidirect
6. #67667C2semidirect
7. #67620C2semidirect
15. #5084C4semidirect
16. #5084C4semidirect
17. #5083C4semidirect
18. #5083C4semidirect
36. #1986D3semidirect
141. C3×M16SD16semidirect
310. M16#215semidirect
455. C3#8989semidirect

Maximal subgroups

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

subgroupindex
#833512
M16×D122
C3×#19862
#718262
C4×#8092
#676672
#676202
#89893