Finite Group Databasebeta

← #10690285

Subgroups — #10690285

Order 768.

612 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79048C211yesmaximal, characteristic, complemented
2#73776C211yesmaximal, characteristic
3#70782C211yesmaximal, characteristic, complemented
4#70762C211yesmaximal, characteristic, complemented
5#70496C211yesmaximal, characteristic, complemented
6C3×#1303C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9035—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×C4×D12—21no—
11C2×C4×D12C2211yescharacteristic
12C2×#872C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5118—21no—
17#4994—21no—
18#4991—21no—
19C4×(C2×D3)⋊C4—21no—
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4839—22no—
23#4839—22no—
24#4674C411yescharacteristic
25#4672C411yescharacteristic
26#4671C411yescharacteristic
27#4661C411yescharacteristic
28#4483—22no—
29#4469—21no—
30#4468—21no—
31#4459—22no—
32#2905—31no—
33#2392—31no—
34#1486—31no—
35#1486—31no—
36#1458—31no—
37#1303—31no—
38#1303D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3—21no—
42C22×C4×D3D411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4C2×C411yescharacteristic
53C2×(C2×D3)⋊C4D411yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3C2×C411yescharacteristic
56C2×Dic3⋊C4C2×C411yescharacteristic
57C2×Dic3⋊C4D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C2×#208—21no—
60D3×M16—42no—
61D3×M16—42no—
62C2×ΓL(1,𝔽25)—22no—
63C2×D3×C8—22no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—42no—
69C4×D12—42no—
70#872—21no—
71#872—21no—
72#872—21no—
73#872—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yescomplemented
77C3×C22⋊C8—21no—
78C3×C4×C8—21no—
79C3×#95C2×C411yescharacteristic
80C3×#95C2×C411yescharacteristic
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84Dic3⋊C8—24no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yescomplemented
88C4×C3⋊C8—21no—
89C22×M16—61no—
90C2×C4×D4—31no—
91C2×C4×D4—61no—
92C2×#117—31no—
93#431—31no—
94#431C2×D311yescharacteristic
95#424—62no—
96#424—62no—
97#423—31no—
98#423—61no—
99#423C2×D311yescharacteristic
100#398—31no—
101#397—31no—
102#396—61no—
103#394—31no—
104#394—61no—
105#392—31no—
106C4×C4⋊C4C2×D311yescharacteristic
107C4×C22⋊C4—61no—
108D4⋊C8—62no—
109D4⋊C8—62no—
110#324—61no—
111#323—61no—
112C23×D3—21no—
113C23×D3C22⋊C411yescharacteristic
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—21no—
120C22×Dic3C22⋊C411yescharacteristic
121C22×Dic3C22⋊C411yescharacteristic
122C22×Dic3C22⋊C411yescharacteristic
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—42no—
126C2×D12—42no—
127C2×D12—42no—
128C2×D12—42no—
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×C4×D3—21no—
133C2×C4×D3—41no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3D812yes—
140C2×C4×D3SD1612yes—
141C3×M16—41no—
142C3×M16—41no—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C3×C4⋊C4—42no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151C3×C42#4111yescharacteristic
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—21no—
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—42no—
157(C2×D3)⋊C4—42no—
158(C2×D3)⋊C4—42no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—42no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—42no—
165Dic3⋊C4—21no—
166Dic3⋊C4—21no—
167Dic3⋊C4—21no—
168Dic3⋊C4—21no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171#208—41no—
172#208—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175ΓL(1,𝔽25)—44no—
176D3×C8—44no—
177C22×D4—31no—
178C22×D4—61no—
179C23×C4—31no—
180C23×C4—61no—
181C2×M16—21no—
182C2×M16—62no—
183C2×M16—61no—
184C2×M16—122no—
185C2×M16—122no—
186C22×C8—62no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C4×D4—122no—
192C4×D4—122no—
193#117—61no—
194#117—61no—
195#117—61no—
196#117—61no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—61no—
202C2×C22⋊C4—31no—
203C2×C22⋊C4—31no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C42—21no—
210C2×C42—31no—
211C2×C42C22×D311yescharacteristic
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—32no—
215C4⋊C8—64no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218C4⋊C8D1212yescomplemented
219C22⋊C8—21no—
220C22⋊C8—61no—
221C4×C8—21no—
222C4×C8—61no—
223#95—61no—
224#95—61no—
225#95C4×D311yescharacteristic
226#95C4×D311yescharacteristic
227C23×C3C2×C22⋊C411yescharacteristic
228C22×D3—21no—
229C22×D3—21no—
230C22×D3—22no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3#10212yes—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12311yescharacteristic
254C2×C3×C4C2×D811yescharacteristic
255C2×C3×C4C2×SD1611yescharacteristic
256C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
257C2×C3×C4#13711yescharacteristic
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3#10212yes—
268D12—22no—
269D12—22no—
270D12—41no—
271D12—41no—
272D12—48no—
273C4×D3—22no—
274C4×D3—22no—
275C4×D3—22no—
276C4×D3—22no—
277C4×D3—81no—
278C4×D3—81no—
279C4×D3—81no—
280C4×D3—81no—
281C3×C8—42no—
282C3×C8—81no—
283C3⋊C8—42no—
284C3⋊C8—81no—
285C24—31no—
286C24—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—122no—
290C2×D4—122no—
291C2×D4—122no—
292C2×D4—122no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—31no—
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—121no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4C2×C4×D311yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4D3×D411yescharacteristic
314M16—41no—
315M16—41no—
316M16—124no—
317M16—121no—
318M16—121no—
319C2×C8—22no—
320C2×C8—41no—
321C2×C8—62no—
322C2×C8—124no—
323C2×C8—121no—
324C4⋊C4—42no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—122no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—122no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C22⋊C4—122no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C42—41no—
343C42—41no—
344C42—61no—
345C42—61no—
346C42C2×D1211yescharacteristic
347C42#23511yescharacteristic
348C42D3×D411yescharacteristic
349C42#23911yescharacteristic
350C22×C3—21no—
351C22×C3—21no—
352C22×C3#38611yescharacteristic
353C22×C3C2×#10211yescharacteristic
354C22×C3#41711yescharacteristic
355C2×D3—22no—
356C2×D3—41no—
357C2×D3—41no—
358C2×D3—42no—
359C2×D3—42no—
360C2×D3—42no—
361C2×D3—42no—
362C2×D3—81no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—81no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—42no—
372C3×C4—81no—
373C3×C4#44711yescharacteristic
374C3×C4#44811yescharacteristic
375C3×C4#48011yescharacteristic
376C3×C4#48111yescharacteristic
377Dic3—22no—
378Dic3—41no—
379Dic3—41no—
380Dic3—81no—
381C23—32no—
382C23—62no—
383C23—61no—
384C23—61no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23D3×C22⋊C411yescharacteristic
394D4—62no—
395D4—62no—
396D4—121no—
397D4—121no—
398D4—128no—
399C2×C4—22no—
400C2×C4—32no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—241no—
422C2×C4—241no—
423C2×C4—241no—
424C2×C4—241no—
425C2×C4C4×D1211yescharacteristic
426C2×C4#88211yescharacteristic
427C2×C4#88411yescharacteristic
428C2×C4#89311yescharacteristic
429C2×C4#89411yescharacteristic
430C2×C4#89511yescharacteristic
431C2×C4D3×D811yescharacteristic
432C2×C4D3×SD1611yescharacteristic
433C2×C4#91411yescharacteristic
434C2×C4#91811yescharacteristic
435C8—42no—
436C8—81no—
437C8—122no—
438C8—241no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3#187411yescharacteristic
442C2×C3#188411yescharacteristic
443C2×C3#189111yescharacteristic
444D3—42no—
445D3—82no—
446C22—21no—
447C22—21no—
448C22—62no—
449C22—122no—
450C22—122no—
451C22—122no—
452C22—121no—
453C22—121no—
454C22—122no—
455C22—241no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22#466511yescharacteristic
461C22D3×#10211yescharacteristic
462C22#480311yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—42no—
468C4—62no—
469C4—81no—
470C4—121no—
471C4—121no—
472C4—241no—
473C4#477311yescharacteristic
474C4#480611yescharacteristic
475C4#486311yescharacteristic
476C4#488211yescharacteristic
477C3#903511yescharacteristic, complemented, Sylow
478C2—21no—
479C2—21no—
480C2—122no—
481C2—242no—
482C2#6877711yescharacteristic
483C2#6888011yescharacteristic
484C2#6903611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radical#4671C4
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9035
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
1. #79048C2semidirect
3. #70782C2semidirect
4. #70762C2semidirect
5. #70496C2semidirect
6. C3×#1303C2semidirect
7. #67620C2semidirect
38. #1303D3semidirect
76. C3×C4⋊C8D4semidirect
87. C4⋊(C3⋊C8)D4semidirect
218. C4⋊C8D12semidirect
477. C3#9035semidirect

Maximal subgroups

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

subgroupindex
#790482
#737762
#707822
#707622
#704962
C3×#13032
#676202
#90353