Finite Group Databasebeta

← #10690283

Subgroups — #10690283

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#82401C211yesmaximal, characteristic
2#79048C211yesmaximal, characteristic, complemented
3C3×#1486C211yesmaximal, characteristic, complemented
4#71598C211yesmaximal, characteristic, complemented
5#70782C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9035—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×C4×#67—21no—
11C2×D3×M16—21no—
12C2×#872C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5223C411yescharacteristic
16#5222C411yescharacteristic
17C4×#217—21no—
18#5118—21no—
19#4992C411yescharacteristic
20#4983C411yescharacteristic
21#4961—21no—
22#4948—21no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4839—22no—
27#4839—22no—
28C3×D4⋊C8—22no—
29D4⋊(C3⋊C8)—22no—
30#4469—21no—
31#4468—21no—
32#2905—31no—
33#2392—31no—
34#1486—31no—
35#1486D311yescharacteristic, complemented
36#1458—31no—
37#1303—31no—
38#1303—31no—
39C22×C3×D4C2×C411yescharacteristic
40C23×C3×C4—21no—
41C22×#67—21no—
42C22×C4×D3D411yescharacteristic
43C2×C3×M16—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C4×#67—42no—
55C4×#67—42no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4C2×C411yescharacteristic
58C2×(C2×D3)⋊C4D411yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4C2×C411yescharacteristic
62C2×Dic3⋊C4D411yescharacteristic
63C2×C4×Dic3—21no—
64C2×C4×Dic3D411yescharacteristic
65C2×#208—21no—
66D3×M16—42no—
67D3×M16—42no—
68C2×ΓL(1,𝔽25)—22no—
69C2×D3×C8—22no—
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⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95C2×C411yescharacteristic
82C3⋊#95C2×C411yescharacteristic
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—61no—
91C2×C4×D4C2×D311yescharacteristic
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⋊C4—31no—
107C4×C22⋊C4—61no—
108D4⋊C8—22no—
109D4⋊C8—62no—
110#324—61no—
111#323—61no—
112C24×C3—21no—
113C23×D3C22⋊C411yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—42no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C2×#67—42no—
128C2×#67—42no—
129C2×#67—42no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3C22⋊C411yescharacteristic
133C22×Dic3C22⋊C411yescharacteristic
134C22×Dic3C22⋊C411yescharacteristic
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3D812yes—
140C2×C4×D3SD1612yes—
141C3×M16—41no—
142C3×M16—41no—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152C3×C42#4111yescharacteristic
153#217—42no—
154#217—42no—
155#217—42no—
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—21no—
158(C2×D3)⋊C4—21no—
159(C2×D3)⋊C4—21no—
160(C2×D3)⋊C4—42no—
161C4⋊Dic3—42no—
162Dic3⋊C4—21no—
163Dic3⋊C4—21no—
164Dic3⋊C4—21no—
165Dic3⋊C4—21no—
166Dic3⋊C4—42no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—41no—
170C4×Dic3—41no—
171#208—41no—
172#208—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175ΓL(1,𝔽25)—44no—
176D3×C8—44no—
177C22×D4—61no—
178C22×D4C4×D311yescharacteristic
179C23×C4—21no—
180C23×C4—31no—
181C2×M16—21no—
182C2×M16—62no—
183C2×M16—61no—
184C2×M16—122no—
185C2×M16—122no—
186C22×C8—62no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—122no—
192C4×D4—122no—
193#117—61no—
194#117—61no—
195#117—61no—
196#117—61no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4C4×D311yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—31no—
204C2×C22⋊C4—31no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C42—31no—
210C2×C42—61no—
211C2×C42C22×D311yescharacteristic
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—32no—
215C4⋊C8—64no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218C4⋊C8#6712yescomplemented
219C22⋊C8—21no—
220C22⋊C8—61no—
221C4×C8—21no—
222C4×C8—61no—
223#95—31no—
224#95—31no—
225#95—61no—
226#95—61no—
227C23×C3—22no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3C2×C22⋊C411yescharacteristic
235C22×D3—21no—
236C22×D3—21no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3#10212yes—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—41no—
243C3×D4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4C4×D411yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12311yescharacteristic
259C2×C3×C4C2×D811yescharacteristic
260C2×C3×C4C2×SD1611yescharacteristic
261C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
262C2×C3×C4#13711yescharacteristic
263#67—48no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—22no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3#10212yes—
277C4×D3—22no—
278C4×D3—22no—
279C4×D3—22no—
280C4×D3—22no—
281C3×C8—42no—
282C3×C8—81no—
283C3⋊C8—42no—
284C3⋊C8—81no—
285C24—21no—
286C24—31no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—42no—
290C2×D4—122no—
291C2×D4—122no—
292C2×D4—122no—
293C22×C4—21no—
294C22×C4—31no—
295C22×C4—32no—
296C22×C4—32no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
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—21no—
325C4⋊C4—21no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—122no—
331C4⋊C4—122no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C22⋊C4—122no—
339C22⋊C4—122no—
340C22⋊C4—122no—
341C22⋊C4—122no—
342C42—61no—
343C42—61no—
344C42—121no—
345C42—121no—
346C42#23511yescharacteristic
347C42D3×D411yescharacteristic
348C42#23711yescharacteristic
349C42C2×#6711yescharacteristic
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3#38611yescharacteristic
360C22×C3C2×#10211yescharacteristic
361C22×C3#41711yescharacteristic
362C2×D3—22no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—42no—
366C2×D3—81no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—81no—
372C3×C4#44711yescharacteristic
373C3×C4#44811yescharacteristic
374C3×C4#48011yescharacteristic
375C3×C4#48111yescharacteristic
376Dic3—22no—
377Dic3—42no—
378Dic3—41no—
379Dic3—41no—
380Dic3—81no—
381C23—22no—
382C23—32no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—61no—
390C23—61no—
391C23—121no—
392C23—121no—
393C23D3×C22⋊C411yescharacteristic
394D4—22no—
395D4—22no—
396D4—41no—
397D4—41no—
398D4—128no—
399C2×C4—32no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4#67⋊C411yescharacteristic
426C2×C4#88311yescharacteristic
427C2×C4#88411yescharacteristic
428C2×C4D3×D811yescharacteristic
429C2×C4#91111yescharacteristic
430C2×C4D3×SD1611yescharacteristic
431C2×C4#91511yescharacteristic
432C2×C4C4×#6711yescharacteristic
433C2×C4#93711yescharacteristic
434C2×C4#93811yescharacteristic
435C8—42no—
436C8—81no—
437C8—122no—
438C8—241no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—82no—
442C2×C3#187411yescharacteristic
443C2×C3#188411yescharacteristic
444C2×C3#189111yescharacteristic
445D3—42no—
446C22—21no—
447C22—21no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—62no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—122no—
457C22—121no—
458C22—121no—
459C22—241no—
460C22D3×#10211yescharacteristic
461C22#477111yescharacteristic
462C22#495611yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—62no—
468C4—81no—
469C4—122no—
470C4—121no—
471C4—121no—
472C4—241no—
473C4#477411yescharacteristic
474C4#477711yescharacteristic
475C4#515611yescharacteristic
476C4#517011yescharacteristic
477C3#903511yescharacteristic, complemented, Sylow
478C2—21no—
479C2—21no—
480C2—82no—
481C2—122no—
482C2#6999211yescharacteristic
483C2#7020011yescharacteristic
484C2#7041711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#386
prime radical#5223C4
cosocleC2×C3×C42C23
hypercenter#1486D3
hypocenterC3#9035
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #79048C2semidirect
3. C3×#1486C2semidirect
4. #71598C2semidirect
5. #70782C2semidirect
6. #67746C2semidirect
7. #67715C2semidirect
35. #1486D3semidirect
76. C3×C4⋊C8D4semidirect
87. C4⋊(C3⋊C8)D4semidirect
218. C4⋊C8#67semidirect
477. C3#9035semidirect

Maximal subgroups

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

subgroupindex
#824012
#790482
C3×#14862
#715982
#707822
#677462
#677152
#90353