Finite Group Databasebeta

← #10824875

Subgroups — #10824875

Order 768.

752 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79200C212yesmaximal
2C2×#4873C211yesmaximal, characteristic, complemented
3#79070C211yesmaximal, characteristic
4C3×#2089C211yesmaximal, characteristic, complemented
5#70679C212yesmaximal, complemented
6#70647C211yesmaximal, characteristic
7#70543C211yesmaximal, characteristic, complemented
8#68987C212yesmaximal
9#68973C212yesmaximal, complemented
10#68944C211yesmaximal, characteristic
11#68916C211yesmaximal, characteristic, complemented
12#23717—31nomaximal, Sylow
13C2×#909C2212yes—
14C22×Dic12C2211yescharacteristic
15#5522C2211yescharacteristic
16C2×#887C2212yes—
17C3×#428C2212yes—
18C2×C3×#107C2211yescharacteristic
19C3×#424C2211yescharacteristic
20C2×#824C2212yes—
21C2×#824C2212yes—
22#5129C2211yescharacteristic
23C2×#822C2212yes—
24Dic3×M16C2212yes—
25Dic3×M16C2211yescharacteristic
26#5110C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#5004C2212yes—
29#4999C2211yescharacteristic
30C2×#810C2212yes—
31#4966C2212yes—
32#4889—22no—
33#4873—22no—
34#4873—22no—
35#4873—22no—
36#4853—22no—
37#4852—22no—
38#4840—22no—
39#4833—22no—
40C3×#343C2212yes—
41C3×#341C2211yescharacteristic
42#4552C2211yescharacteristic
43#4530C2212yes—
44C4×#137—32no—
45C2×C4×Q16—31no—
46#2922—31no—
47#2089D311yescharacteristic, complemented
48#2078—31no—
49#2064—32no—
50#2052—32no—
51#2047—31no—
52#2020—31no—
53C4×#108—31no—
54C4×#101—32no—
55C2×#235C2311yescharacteristic
56C22×Dic6C2312yes—
57C2×C3×M16C2311yescharacteristic
58C2×C3×M16C2312yes—
59C22×C3×C8C2311yescharacteristic
60C3×#117C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×Dic3⋊C4—22no—
63C2×C4×Dic3—22no—
64C2×C4×Dic3C2311yescharacteristic
65C2×#208C2312yes—
66C2×#208C2311yescharacteristic
67C22×C3⋊C8C2311yescharacteristic
68#909—24no—
69#909—24no—
70C2×Dic12—22no—
71C2×Dic12—24no—
72C2×Dic12—22no—
73C2×#204—22no—
74#893D411yescharacteristic
75#893D411yescharacteristic
76#887—22no—
77#887—22no—
78#887—22no—
79#887—22no—
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#824—24no—
91#824—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#810—22no—
103#810—22no—
104#810—24no—
105#809—22no—
106C4⋊(C3⋊C8)—22no—
107C2×#137—32no—
108C22×Q16—31no—
109C4×#41—31no—
110C2×C4×Q8—32no—
111#441—62no—
112#440—62no—
113C4×Q16—62no—
114C4×Q16—62no—
115C4×Q16—62no—
116C4×SD16—62no—
117#433—62no—
118#431—62no—
119C2×#108—32no—
120#428C2×D312yes—
121C2×#107C2×D311yescharacteristic
122#424—31no—
123#424C2×D311yescharacteristic
124#419—32no—
125#417—32no—
126C2×#103—32no—
127C2×#101—32no—
128C2×#101—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
138C22×C3×C4—22no—
139C22×C3×C4C2411yescharacteristic
140C2×#67—21no—
141C22×Dic3—22no—
142C22×Dic3#4111yescharacteristic
143C22×Dic3#4111yescharacteristic
144#235—22no—
145#235—22no—
146C2×D12C2×D411yescharacteristic
147C2×C4×D3—21no—
148C2×C4×D3—21no—
149C2×C4×D3—21no—
150C2×Dic6—22no—
151C2×Dic6—22no—
152C2×Dic6—22no—
153C2×Dic6—22no—
154C2×Dic6—22no—
155C2×Dic6C2×D411yescharacteristic
156C2×Dic6#4114yes—
157C3×M16—22no—
158C3×M16—22no—
159C3×M16#4114yes—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—21no—
163C2×C3×C8C2×D412yes—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4C2×D411yescharacteristic
167C3×C4⋊C4C2×D411yescharacteristic
168C3×C22⋊C4—21no—
169C3×C42—21no—
170(C2×D3)⋊C4—21no—
171(C2×D3)⋊C4—21no—
172Dic3⋊C4—21no—
173Dic3⋊C4—21no—
174Dic3⋊C4—42no—
175Dic3⋊C4—42no—
176C4×Dic3—21no—
177C4×Dic3—21no—
178C4×Dic3—42no—
179C4×Dic3—42no—
180C4×Dic3C2×D411yescharacteristic
181C4×Dic3C2×D411yescharacteristic
182C4×Dic3C2×D411yescharacteristic
183C4×Dic3C2×D411yescharacteristic
184#208—22no—
185#208—22no—
186#208—22no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8C2×D412yes—
191Dic12—28no—
192Dic12—28no—
193#204—28no—
194C2×#41—31no—
195C22×Q8—32no—
196#137—64no—
197#137—64no—
198C2×Q16—64no—
199C2×Q16—62no—
200C2×Q16—62no—
201C2×SD16—62no—
202C2×M16—32no—
203C2×M16—31no—
204C2×M16C22×D311yescharacteristic
205C2×M16C22×D312yes—
206C22×C8—31no—
207C22×C8C22×D311yescharacteristic
208C4×Q8—31no—
209C4×Q8—31no—
210C4×Q8—62no—
211C4×Q8—62no—
212C4×Q8—62no—
213C4×Q8—62no—
214C4×D4—31no—
215C4×D4—31no—
216C4×D4—61no—
217C4×D4—61no—
218#117—61no—
219#117C22×D311yescharacteristic
220C2×C4⋊C4—62no—
221C2×C4⋊C4C22×D312yes—
222C2×C42—31no—
223C2×C42—61no—
224C2×C42—62no—
225#108—64no—
226#107—22no—
227#107—22no—
228#106—22no—
229C4⋊C8—22no—
230C4⋊C8—62no—
231#103—64no—
232#103—62no—
233#103—62no—
234#103—62no—
235#102—62no—
236#101—64no—
237#101—64no—
238C22⋊C8—62no—
239C22⋊C8—62no—
240#97—61no—
241#97—62no—
242C4×C8—32no—
243C4×C8—32no—
244C4×C8—61no—
245C4×C8—62no—
246C23×C3C2×#4111yescharacteristic
247C22×D3—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—42no—
251C2×C3×C4—82no—
252C2×C3×C4C22×D411yescharacteristic
253C2×C3×C4C22×D411yescharacteristic
254C2×C3×C4C2×#4112yes—
255C2×C3×C4C2×#4112yes—
256#67—42no—
257C2×Dic3—22no—
258C2×Dic3—21no—
259C2×Dic3—24no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3#12111yescharacteristic
264C2×Dic3#12111yescharacteristic
265C2×Dic3#12111yescharacteristic
266C2×Dic3#12111yescharacteristic
267D12—22no—
268C4×D3—42no—
269C4×D3—41no—
270C4×D3—41no—
271C4×D3—42no—
272Dic6—24no—
273Dic6—24no—
274Dic6—22no—
275Dic6—42no—
276Dic6—42no—
277C3×C8—24no—
278C3×C8—22no—
279C3×C8—41no—
280C3⋊C8—22no—
281C3⋊C8—42no—
282C3⋊C8—41no—
283#41—62no—
284#41—62no—
285C2×Q8—31no—
286C2×Q8—34no—
287C2×Q8—62no—
288C2×Q8—62no—
289C2×Q8—62no—
290C2×Q8—62no—
291C2×Q8—62no—
292C2×D4—31no—
293C2×D4—61no—
294C22×C4—22no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—62no—
300C22×C4—61no—
301C22×C4C23×D311yescharacteristic
302Q16—68no—
303Q16—68no—
304SD16—68no—
305M16—22no—
306M16—22no—
307M16—62no—
308M16—62no—
309M16—62no—
310M16#23914yes—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—32no—
315C2×C8—62no—
316C2×C8—61no—
317C2×C8—62no—
318C2×C8D3×D412yes—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—122no—
324C4⋊C4—122no—
325C4⋊C4D3×D411yescharacteristic
326C4⋊C4D3×D411yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C42—21no—
331C42—31no—
332C42—31no—
333C42—31no—
334C42—31no—
335C42—61no—
336C42—61no—
337C42—122no—
338C42—122no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3C2×#12111yescharacteristic
342C22×C3#52912yes—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—82no—
349C3×C4#52511yescharacteristic
350C3×C4#52512yes—
351C3×C4#53211yescharacteristic
352Dic3—21no—
353Dic3—21no—
354Dic3—21no—
355Dic3—44no—
356Dic3—41no—
357Dic3#47312yes—
358C23—61no—
359C23D3×#4111yescharacteristic
360Q8—64no—
361Q8—64no—
362Q8—62no—
363Q8—122no—
364Q8—122no—
365D4—62no—
366D4—122no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—42no—
374C2×C4—62no—
375C2×C4—64no—
376C2×C4—61no—
377C2×C4—82no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4C2×D3×D411yescharacteristic
386C2×C4C2×D3×D411yescharacteristic
387C2×C4C2×#23912yes—
388C2×C4D3×#4112yes—
389C8—22no—
390C8—24no—
391C8—41no—
392C8—62no—
393C8—121no—
394C8—122no—
395C2×C3—22no—
396C2×C3#239211yescharacteristic
397C2×C3C2×#47311yescharacteristic
398C2×C3#306711yescharacteristic
399D3—81no—
400C22—21no—
401C22—21no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22D3×#12111yescharacteristic
406C22#569112yes—
407C4—32no—
408C4—41no—
409C4—41no—
410C4—61no—
411C4—61no—
412C4—61no—
413C4—82no—
414C4—121no—
415C4—124no—
416C4#562712yes—
417C4#565611yescharacteristic
418C4#572511yescharacteristic
419C3#2371711yescharacteristic, complemented, Sylow
420C2—22no—
421C2—241no—
422C2#7773711yescharacteristic
423C2#8093711yescharacteristic
424C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2089C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#2089D3
hypocenterC3#23717
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
2. C2×#4873C2semidirect
4. C3×#2089C2semidirect
5. #70679C2semidirect
7. #70543C2semidirect
9. #68973C2semidirect
11. #68916C2semidirect
47. #2089D3semidirect
419. C3#23717semidirect

Maximal subgroups

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

subgroupindex
#792002
C2×#48732
#790702
C3×#20892
#706792
#706472
#705432
#689872
#689732
#689442
#689162
#237173