Finite Group Databasebeta

← #10691274

Subgroups — #10691274

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83296C211yesmaximal, characteristic
2#79276C211yesmaximal, characteristic, complemented
3C3×#1502C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic, complemented
5#69494C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9208—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#950—21no—
11C4×#235C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5233—21no—
15#5090C411yescharacteristic, complemented
16#5083C411yescharacteristic, complemented
17#5075C411yescharacteristic, complemented
18#5072C411yescharacteristic, complemented
19#5057—21no—
20#5054—21no—
21C2×#805—21no—
22#4924C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4834—21no—
26#4830—21no—
27#67⋊C8—22no—
28#4685—21no—
29C4×D3×C8—21no—
30C3×#327—22no—
31D4⋊(C3⋊C8)—21no—
32D4⋊(C3⋊C8)—21no—
33#4523—21no—
34#3035—31no—
35C8×#41—31no—
36#1806—31no—
37#1502D311yescharacteristic, complemented
38#1492—31no—
39#1301—31no—
40#1279—31no—
41C2×C3×#41—21no—
42C2×#235D411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×#127C2×C411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53#950—42no—
54#950—42no—
55C2×#216—21no—
56C2×#215—21no—
57C2×#214—21no—
58C2×#213—21no—
59C4×#67—21no—
60C4×#67—21no—
61#926D411yescharacteristic
62C2×#208—21no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65C2×D3×C8—22no—
66C4×D12C2×C411yescharacteristic
67C4×D12D411yescharacteristic
68#872—21no—
69C42×D3—21no—
70C4×Dic6C2×C411yescharacteristic
71C4×Dic6D411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C22⋊C8D412yes—
74C3×C4×C8—21no—
75C3×#95—21no—
76#835—21no—
77#835—21no—
78#832—21no—
79#832—21no—
80(C2×D3)⋊C8—22no—
81Dic3⋊C8—22no—
82C8×Dic3—22no—
83#805—42no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)C2×C411yescharacteristic
89C4×C3⋊C8—21no—
90C4×C3⋊C8D411yescharacteristic
91C4×C3⋊C8D411yescharacteristic
92C2×#135—61no—
93#532C2×D311yescharacteristic
94C4×#41—31no—
95#464—61no—
96#463—61no—
97#462—31no—
98#461—31no—
99#460—31no—
100#459—31no—
101C8×Q8—61no—
102C8×D4—61no—
103C8×D4—62no—
104#431—61no—
105#431C2×D311yescharacteristic
106#423—31no—
107C2×C4⋊C8—31no—
108C2×#104—61no—
109#416—61no—
110C2×C4×C8—31no—
111C2×C4×C8—61no—
112C4×C4⋊C4C2×D311yescharacteristic
113#339—61no—
114#327—22no—
115D4⋊C8—61no—
116D4⋊C8—61no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—21no—
132#235—21no—
133#235—21no—
134#235D812yes—
135#235SD1612yes—
136C2×D12C22⋊C411yescharacteristic
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×Dic6C22⋊C411yescharacteristic
141C2×C3×C8—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42—41no—
150C3×C42C22×C411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#217—21no—
155#216—42no—
156#215—42no—
157#214—42no—
158#213—42no—
159(C2×D3)⋊C4—21no—
160(C2×D3)⋊C4—21no—
161C4⋊Dic3C22⋊C411yescharacteristic
162C4⋊Dic3C22⋊C411yescharacteristic
163Dic3⋊C4—21no—
164Dic3⋊C4—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168#208—41no—
169#208—41no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178D3×C8—44no—
179C2×#41—21no—
180C2×#41—31no—
181#135—122no—
182#135—122no—
183C2×Q16—61no—
184C2×SD16—61no—
185C2×SD16—61no—
186C2×D8—61no—
187C2×M16—61no—
188C22×C8—62no—
189C22×C8—61no—
190C22×C8—61no—
191C4⋊Q8C4×D311yescharacteristic
192#127C4×D311yescharacteristic
193#124—21no—
194#121—21no—
195#121—21no—
196C4×Q8—31no—
197C4×Q8—31no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—31no—
201C4×D4—31no—
202C4×D4—61no—
203C4×D4—61no—
204#117—31no—
205#117—61no—
206C2×C4⋊C4—21no—
207C2×C42—21no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—31no—
212C4⋊C8—31no—
213C4⋊C8—62no—
214C4⋊C8—61no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217#104—122no—
218#103—61no—
219#103—61no—
220#102—61no—
221#102—61no—
222C22⋊C8—62no—
223C22⋊C8#6712yes—
224C4×C8—21no—
225C4×C8—31no—
226C4×C8—31no—
227C4×C8—62no—
228C4×C8—61no—
229#95—21no—
230C23×C3—41no—
231C23×C3#12111yescharacteristic
232C22×D3—21no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—81no—
240C3×D4—81no—
241C3×D4—81no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4C2×C22⋊C411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12311yescharacteristic
257C2×C3×C4C2×D811yescharacteristic
258C2×C3×C4C2×SD1611yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4#13511yescharacteristic
261#67—22no—
262#67—22no—
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266D12—21no—
267D12#10212yes—
268C4×D3—22no—
269C4×D3—22no—
270C4×D3—42no—
271C4×D3—41no—
272C4×D3—41no—
273Dic6—21no—
274Dic6#10212yes—
275C3×C8—42no—
276C3⋊C8—22no—
277C3⋊C8—41no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C3⋊C8—81no—
281#41—32no—
282#41—32no—
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—61no—
288#41—61no—
289C2×Q8—21no—
290C2×Q8—31no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—31no—
295C2×D4—41no—
296C2×D4—61no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4D3×D411yescharacteristic
303C22×C4#23711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305Q16—122no—
306SD16—122no—
307SD16—122no—
308D8—122no—
309M16—121no—
310M16—121no—
311C2×C8—22no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—124no—
317C2×C8—122no—
318C2×C8—121no—
319C2×C8—121no—
320C2×C8—121no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—41no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C42—41no—
335C42—41no—
336C42—61no—
337C42—61no—
338C42—61no—
339C42C2×C4×D311yescharacteristic
340C42#23511yescharacteristic
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3#38611yescharacteristic
349C22×C3#46511yescharacteristic
350C22×C3#46611yescharacteristic
351C2×D3—22no—
352C2×D3—41no—
353C2×D3—41no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—82no—
358C3×C4—81no—
359C3×C4C2×#10211yescharacteristic
360C3×C4#41611yescharacteristic
361C3×C4#44311yescharacteristic
362C3×C4#44411yescharacteristic
363C3×C4#46511yescharacteristic
364C3×C4#46611yescharacteristic
365Dic3—22no—
366Dic3—41no—
367Dic3—41no—
368C23—41no—
369C23—61no—
370C23#93811yescharacteristic
371Q8—32no—
372Q8—41no—
373Q8—41no—
374Q8—61no—
375D4—32no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—62no—
381D4—62no—
382D4—61no—
383D4—81no—
384D4—81no—
385D4—81no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—61no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—122no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4D3×C22⋊C411yescharacteristic
404C2×C4#67⋊C411yescharacteristic
405C2×C4#88311yescharacteristic
406C2×C4#88411yescharacteristic
407C2×C4D3×D811yescharacteristic
408C2×C4#91211yescharacteristic
409C2×C4D3×SD1611yescharacteristic
410C2×C4#91611yescharacteristic
411C2×C4C4×#6711yescharacteristic
412C2×C4#93711yescharacteristic
413C8—42no—
414C8—62no—
415C8—121no—
416C8—121no—
417C8—121no—
418C8—241no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—161no—
422C2×C3#187611yescharacteristic
423C2×C3#189311yescharacteristic
424C2×C3#190611yescharacteristic
425D3—42no—
426C22—21no—
427C22—21no—
428C22—62no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—121no—
433C22—121no—
434C22#495611yescharacteristic
435C22#515711yescharacteristic
436C22#517111yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—62no—
441C4—82no—
442C4—81no—
443C4—121no—
444C4—121no—
445C4D3×#10211yescharacteristic
446C4#477211yescharacteristic
447C4#477511yescharacteristic
448C4#477811yescharacteristic
449C4#482311yescharacteristic
450C4#482411yescharacteristic
451C3#920811yescharacteristic, complemented, Sylow
452C2—21no—
453C2—21no—
454C2—122no—
455C2—161no—
456C2#7005511yescharacteristic
457C2#7020211yescharacteristic
458C2#7039011yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#386
prime radical#83296C2
cosocleC2×C3×C42C23
hypercenter#1502D3
hypocenterC3#9208
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #79276C2semidirect
3. C3×#1502C2semidirect
4. #71802C2semidirect
5. #69494C2semidirect
6. #67745C2semidirect
7. #67619C2semidirect
15. #5090C4semidirect
16. #5083C4semidirect
17. #5075C4semidirect
18. #5072C4semidirect
37. #1502D3semidirect
451. C3#9208semidirect

Maximal subgroups

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

subgroupindex
#832962
#792762
C3×#15022
#718022
#694942
#677452
#676192
#92083