Finite Group Databasebeta

← #10824864

Subgroups — #10824864

Order 768.

816 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79199C212yesmaximal, complemented
2C2×#4872C211yesmaximal, characteristic, complemented
3#79070C211yesmaximal, characteristic, complemented
4C3×#2089C211yesmaximal, characteristic, complemented
5#70676C212yesmaximal, complemented
6#70647C211yesmaximal, characteristic, complemented
7#70540C211yesmaximal, characteristic, complemented
8#68986C212yesmaximal, complemented
9#68972C212yesmaximal, complemented
10#68942C211yesmaximal, characteristic
11#68916C211yesmaximal, characteristic, complemented
12#23716—31nomaximal, Sylow
13C2×#908C2212yes—
14C22×D24C2211yescharacteristic
15#5522C2211yescharacteristic
16C2×#893C2212yes—
17C3×#428C2212yes—
18C2×C3×#107C2211yescharacteristic
19C3×#424C2211yescharacteristic
20C2×#823C2212yes—
21C2×#823C2212yes—
22#5129C2211yescharacteristic
23C2×#822C2212yes—
24Dic3×M16C2212yes—
25Dic3×M16C2211yescharacteristic
26#5110C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#5001C2212yes—
29#4999C2211yescharacteristic
30#4966C2212yes—
31C2×#809C2212yes—
32#4889—22no—
33#4872—22no—
34#4872—22no—
35#4872—22no—
36#4869—22no—
37#4852—22no—
38#4840—22no—
39#4833—22no—
40C3×#343C2212yes—
41C3×#341C2211yescharacteristic
42#4552C2211yescharacteristic
43#4530C2212yes—
44C4×AGL(1,ℤ/8)—32no—
45C2×C4×D8—31no—
46#2922—31no—
47#2089D311yescharacteristic, complemented
48#2078—31no—
49#2063—32no—
50#2051—32no—
51#2045—31no—
52#2018—31no—
53C4×#108—31no—
54C4×#100—32no—
55C2×#235C2311yescharacteristic
56C22×D12C2312yes—
57C22×C4×D3—22no—
58C2×C3×M16C2311yescharacteristic
59C2×C3×M16C2312yes—
60C22×C3×C8C2311yescharacteristic
61C3×#117C2311yescharacteristic
62C2×C3×C4⋊C4C2312yes—
63C2×(C2×D3)⋊C4—22no—
64C2×C4×Dic3C2311yescharacteristic
65C2×#208C2312yes—
66C2×#208C2311yescharacteristic
67C22×C3⋊C8C2311yescharacteristic
68#908—24no—
69#908—24no—
70C2×D24—22no—
71C2×D24—24no—
72C2×D24—22no—
73C2×#204—22no—
74#893—22no—
75#893—22no—
76#893—22no—
77#893—22no—
78#893D411yescharacteristic
79#893D411yescharacteristic
80#887D411yescharacteristic
81#887D411yescharacteristic
82#67⋊C4—21no—
83#67⋊C4—21no—
84#872—21no—
85C42×D3—21no—
86C3×#107—22no—
87C3×#107—22no—
88C3×#106—22no—
89C3×C4⋊C8—22no—
90#823—24no—
91#823—24no—
92#822—24no—
93(C2×D3)⋊C8—22no—
94(C2×D3)⋊C8—22no—
95#815—22no—
96#815—21no—
97C8×Dic3—22no—
98C8×Dic3—21no—
99C8×Dic3D412yes—
100C8×Dic3D412yes—
101#810—22no—
102#809—22no—
103#809—22no—
104#809—22no—
105#809—24no—
106C4⋊(C3⋊C8)—22no—
107C2×AGL(1,ℤ/8)—32no—
108C22×D8—31no—
109C4×#41—31no—
110C2×C4×D4—32no—
111#442—62no—
112#440—62no—
113C4×SD16—62no—
114C4×D8—62no—
115C4×D8—62no—
116C4×D8—62no—
117#433—62no—
118#431—62no—
119C2×#108—32no—
120#428C2×D312yes—
121C2×#107C2×D311yescharacteristic
122#424—31no—
123#424C2×D311yescharacteristic
124#418—32no—
125#417—32no—
126C2×#102—32no—
127C2×#100—32no—
128C2×#100—32no—
129#408—31no—
130#408—31no—
131C4×M16—31no—
132C4×M16—32no—
133C2×C4×C8—31no—
134#343—32no—
135#343C2×D312yes—
136#341—31no—
137#341C2×D311yescharacteristic
138C23×D3—22no—
139C22×C3×C4—22no—
140C22×C3×C4C2411yescharacteristic
141C2×#67—21no—
142C22×Dic3#4111yescharacteristic
143C22×Dic3#4111yescharacteristic
144#235—22no—
145#235—22no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12C2×D411yescharacteristic
152C2×D12#4114yes—
153C2×C4×D3—21no—
154C2×C4×D3—21no—
155C2×C4×D3—21no—
156C2×C4×D3—42no—
157C2×C4×D3—42no—
158C2×C4×D3—84no—
159C2×Dic6C2×D411yescharacteristic
160C3×M16—22no—
161C3×M16—22no—
162C3×M16#4114yes—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—21no—
166C2×C3×C8C2×D412yes—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4C2×D411yescharacteristic
170C3×C4⋊C4C2×D411yescharacteristic
171C3×C22⋊C4—21no—
172C3×C42—21no—
173(C2×D3)⋊C4—21no—
174(C2×D3)⋊C4—21no—
175(C2×D3)⋊C4—42no—
176(C2×D3)⋊C4—42no—
177Dic3⋊C4—21no—
178Dic3⋊C4—21no—
179C4×Dic3—21no—
180C4×Dic3—21no—
181C4×Dic3C2×D411yescharacteristic
182C4×Dic3C2×D411yescharacteristic
183C4×Dic3C2×D411yescharacteristic
184C4×Dic3C2×D411yescharacteristic
185#208—22no—
186#208—22no—
187#208—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8C2×D412yes—
192D24—28no—
193D24—28no—
194#204—28no—
195C2×#41—31no—
196C22×D4—32no—
197C23×C4—62no—
198AGL(1,ℤ/8)—64no—
199AGL(1,ℤ/8)—64no—
200C2×SD16—62no—
201C2×D8—64no—
202C2×D8—62no—
203C2×D8—62no—
204C2×M16—32no—
205C2×M16—31no—
206C2×M16C22×D311yescharacteristic
207C2×M16C22×D312yes—
208C22×C8—31no—
209C22×C8C22×D311yescharacteristic
210C4×Q8—31no—
211C4×Q8—31no—
212C4×D4—31no—
213C4×D4—31no—
214C4×D4—62no—
215C4×D4—62no—
216C4×D4—62no—
217C4×D4—62no—
218C4×D4—61no—
219C4×D4—61no—
220#117—61no—
221#117C22×D311yescharacteristic
222C2×C4⋊C4C22×D312yes—
223C2×C22⋊C4—62no—
224C2×C42—31no—
225C2×C42—61no—
226#108—64no—
227#107—22no—
228#107—22no—
229#106—22no—
230C4⋊C8—22no—
231C4⋊C8—62no—
232#103—62no—
233#102—64no—
234#102—62no—
235#102—62no—
236#102—62no—
237#100—64no—
238#100—64no—
239C22⋊C8—62no—
240C22⋊C8—62no—
241#97—61no—
242#97—62no—
243C4×C8—32no—
244C4×C8—32no—
245C4×C8—61no—
246C4×C8—62no—
247C23×C3C2×#4111yescharacteristic
248C22×D3—21no—
249C22×D3—24no—
250C22×D3—42no—
251C22×D3—42no—
252C22×D3—82no—
253C22×D3—82no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—42no—
257C2×C3×C4—82no—
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4112yes—
261C2×C3×C4C2×#4112yes—
262#67—42no—
263C2×Dic3—22no—
264C2×Dic3—21no—
265C2×Dic3—41no—
266C2×Dic3#12111yescharacteristic
267C2×Dic3#12111yescharacteristic
268C2×Dic3#12111yescharacteristic
269C2×Dic3#12111yescharacteristic
270D12—24no—
271D12—24no—
272D12—22no—
273D12—42no—
274D12—42no—
275C4×D3—42no—
276C4×D3—41no—
277C4×D3—41no—
278C4×D3—42no—
279C4×D3—84no—
280C4×D3—84no—
281Dic6—22no—
282C3×C8—24no—
283C3×C8—22no—
284C3×C8—41no—
285C3⋊C8—22no—
286C3⋊C8—42no—
287C3⋊C8—41no—
288C24—62no—
289#41—62no—
290#41—62no—
291C2×Q8—31no—
292C2×D4—31no—
293C2×D4—34no—
294C2×D4—61no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—62no—
300C22×C4—22no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—122no—
307C22×C4—122no—
308C22×C4—244no—
309C22×C4C23×D311yescharacteristic
310SD16—68no—
311D8—68no—
312D8—68no—
313M16—22no—
314M16—22no—
315M16—62no—
316M16—62no—
317M16—62no—
318M16#23914yes—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—32no—
323C2×C8—62no—
324C2×C8—61no—
325C2×C8—62no—
326C2×C8D3×D412yes—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4D3×D411yescharacteristic
332C4⋊C4D3×D411yescharacteristic
333C22⋊C4—21no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—122no—
337C22⋊C4—122no—
338C42—21no—
339C42—31no—
340C42—31no—
341C42—31no—
342C42—31no—
343C42—61no—
344C42—61no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3C2×#12111yescharacteristic
348C22×C3#52912yes—
349C2×D3—41no—
350C2×D3—44no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—84no—
354C2×D3—82no—
355C2×D3—82no—
356C2×D3—84no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—82no—
360C3×C4#52511yescharacteristic
361C3×C4#52512yes—
362C3×C4#53211yescharacteristic
363Dic3—21no—
364Dic3—21no—
365Dic3—21no—
366Dic3—41no—
367Dic3#47212yes—
368C23—61no—
369C23—64no—
370C23—122no—
371C23—122no—
372C23—242no—
373C23—242no—
374C23D3×#4111yescharacteristic
375Q8—62no—
376D4—64no—
377D4—62no—
378D4—64no—
379D4—122no—
380D4—122no—
381D4—122no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—42no—
389C2×C4—62no—
390C2×C4—61no—
391C2×C4—82no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—244no—
398C2×C4—244no—
399C2×C4C2×D3×D411yescharacteristic
400C2×C4C2×D3×D411yescharacteristic
401C2×C4C2×#23912yes—
402C2×C4D3×#4112yes—
403C8—22no—
404C8—24no—
405C8—41no—
406C8—62no—
407C8—121no—
408C8—122no—
409C2×C3—22no—
410C2×C3#239211yescharacteristic
411C2×C3C2×#47211yescharacteristic
412C2×C3#306711yescharacteristic
413D3—81no—
414D3—84no—
415C22—21no—
416C22—21no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22—124no—
421C22—244no—
422C22—244no—
423C22—242no—
424C22—242no—
425C22D3×#12111yescharacteristic
426C22#569112yes—
427C4—32no—
428C4—41no—
429C4—41no—
430C4—61no—
431C4—61no—
432C4—61no—
433C4—82no—
434C4—121no—
435C4#562712yes—
436C4#565611yescharacteristic
437C4#572511yescharacteristic
438C3#2371611yescharacteristic, complemented, Sylow
439C2—22no—
440C2—241no—
441C2—244no—
442C2#7773711yescharacteristic
443C2D3×#47211yescharacteristic
444C2#8094311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2089C2
socleC22×C3C2×#121
prime radical#68942C2
hypercenter#2089D3
hypocenterC3#23716
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2089

Complements

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

NH ≅ G/Ntype
1. #79199C2semidirect
2. C2×#4872C2semidirect
3. #79070C2semidirect
4. C3×#2089C2semidirect
5. #70676C2semidirect
6. #70647C2semidirect
7. #70540C2semidirect
8. #68986C2semidirect
9. #68972C2semidirect
11. #68916C2semidirect
47. #2089D3semidirect
438. C3#23716semidirect

Maximal subgroups

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

subgroupindex
#791992
C2×#48722
#790702
C3×#20892
#706762
#706472
#705402
#689862
#689722
#689422
#689162
#237163