Finite Group Databasebeta

← #27280504

Subgroups — #27280504

Order 13568.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810823C211yesmaximal, characteristic
2#20275555C211yesmaximal, characteristic
3#20273384C211yesmaximal, characteristic
4#20273144C211yesmaximal, characteristic, complemented
5#20272589C211yesmaximal, characteristic, complemented
6#20272566C211yesmaximal, characteristic, complemented
7Q8×M16×C53C211yesmaximal, characteristic, complemented
8C22×#11719924C412yes—
9C2×#13133214C2211yescharacteristic
10#16810823—21no—
11#16810823—21no—
12#16810823—21no—
13#16810823—21no—
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137C412yes—
18C2×#13133133—21no—
19C2×#13133133—21no—
20#16810366—22no—
21Q8⋊#88322—21no—
22Q8⋊#88322—21no—
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#446—21no—
25C53×#446—22no—
26C8×Q8×C53—21no—
27C53×#423—21no—
28C53×#423C2211yescharacteristic
29C4×M16×C53—21no—
30C4×M16×C53C2211yescharacteristic
31C22×D212—21no—
32C22×D212—21no—
33C22×D212C2×C412yes—
34C2×#11719924—22no—
35C2×#11719924—22no—
36C2×#11719924—44no—
37C2×#11719917—21no—
38C22×#88322C2×C412yes—
39#13133214—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133137—22no—
49#13133137—22no—
50#13133133—42no—
51#13133133—42no—
52C22⋊#88322—21no—
53C4×#88322—21no—
54C22×Q8×C53C2×C411yescharacteristic
55C2×M16×C53—21no—
56C2×M16×C53D411yescharacteristic
57C2×M16×C53D411yescharacteristic
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53D411yescharacteristic
61C4×Q8×C53D411yescharacteristic
62C4×Q8×C53D411yescharacteristic
63C4×Q8×C53D411yescharacteristic
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53C2×C411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—22no—
69C4⋊C8×C53—22no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D412yes—
72#97×C53—22no—
73#97×C53—21no—
74C4×C8×C53—22no—
75C4×C8×C53—21no—
76C23×D53—22no—
77C2×Q8×C53—21no—
78C2×Q8×C53—22no—
79C2×Q8×C53—41no—
80C2×Q8×C53C22⋊C411yescharacteristic
81C2×Q8×C53C22⋊C411yescharacteristic
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—42no—
90C2×D212—42no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—82no—
100M16×C53—21no—
101M16×C53—21no—
102M16×C53—21no—
103M16×C53—41no—
104M16×C53—41no—
105M16×C53SD1612yescomplemented
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—22no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53C22⋊C411yescharacteristic
115C4⋊C4×C53C22⋊C411yescharacteristic
116C42×C53—21no—
117C42×C53—21no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53#4111yescharacteristic
124#11719924—44no—
125#11719924—44no—
126#11719917—41no—
127#11719917—41no—
128C2×#88322—22no—
129C2×#88322—42no—
130C2×#88322—41no—
131C2×#88322—41no—
132C23×C53C4×D411yescharacteristic
133C22×D53—42no—
134C22×D53—42no—
135C22×D53—42no—
136C22×D53—82no—
137C22×D53—82no—
138Q8×C53—21no—
139Q8×C53—21no—
140Q8×C53—21no—
141Q8×C53—42no—
142Q8×C53#10012yescomplemented
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—81no—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12311yescharacteristic
158C2×C4×C53C2×SD1611yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53#13711yescharacteristic
162D212—42no—
163D212—42no—
164D212—82no—
165D212—82no—
166D212—82no—
167D212—81no—
168D212—81no—
169D212—81no—
170D212—81no—
171C8×C53—22no—
172C8×C53—41no—
173C8×C53—42no—
174#88322—42no—
175#88322—81no—
176#88322—81no—
177#9254—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53#38611yescharacteristic
181C22×C53C4×SD1611yescharacteristic
182C22×C53#44011yescharacteristic
183C2×D53—82no—
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—162no—
187C2×D53—162no—
188C4×C53—21no—
189C4×C53—22no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53—41no—
195C4×C53C2×#10011yescharacteristic
196C4×C53#41311yescharacteristic
197C4×C53#44811yescharacteristic
198C4×C53#44911yescharacteristic
199C4×C53#46011yescharacteristic
200C4×C53#46411yescharacteristic
201C2×#460—531no—
202Q8×M16D5311yescharacteristic, complemented
203#1965—531no—
204#1545—531no—
205#1487—531no—
206#1292—531no—
207#1283—531no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53#188011yescharacteristic
211C2×C53#190011yescharacteristic
212C2×C53#190311yescharacteristic
213D53—162no—
214C22×SD16—532no—
215C2×#127—531no—
216C2×C4×Q8C2×D5311yescharacteristic
217#460—1061no—
218#460—1061no—
219#460—1061no—
220#460—1061no—
221#446—21no—
222#446—22no—
223C8×Q8—21no—
224#431—531no—
225#423—21no—
226#423—531no—
227#423C2×D5311yescharacteristic
228C2×C4⋊C8—531no—
229C2×#102—532no—
230C2×#100—1061no—
231C2×#100—1061no—
232C4×M16—21no—
233C4×M16C2×D5311yescharacteristic
234#331—1062no—
235Q8⋊C8—1061no—
236Q8⋊C8—1061no—
237C53#925411yescharacteristic, complemented, Sylow
238C22×Q8C4×D5311yescharacteristic
239C22×D4—532no—
240C22×D4—1061no—
241C22×D4—1061no—
242C2×SD16—1062no—
243C2×SD16—1062no—
244C2×SD16—2124no—
245C2×M16—21no—
246C2×M16—1061no—
247C2×M16C22⋊D5311yescharacteristic
248C2×M16C22⋊D5311yescharacteristic
249C22×C8—532no—
250#127—1061no—
251#127—1061no—
252#127—1061no—
253#127—1061no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8D21211yescharacteristic
257C4×Q8D21211yescharacteristic
258C4×Q8C22⋊D5311yescharacteristic
259C4×Q8C22⋊D5311yescharacteristic
260C2×C4⋊C4—21no—
261C2×C4⋊C4C4×D5311yescharacteristic
262C2×C42—21no—
263C2×C42C22×D5311yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270C4⋊C8—1061no—
271C4⋊C8—1061no—
272C4⋊C8C22⋊D5312yes—
273#102—1062no—
274#102—1062no—
275#100—2122no—
276#100—2122no—
277C22⋊C8—1061no—
278#97—22no—
279#97—21no—
280C4×C8—22no—
281C4×C8—21no—
282C4×C8—1061no—
283C24—1062no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8#1171992111yescharacteristic
288C2×Q8#1171992111yescharacteristic
289C2×D4—1062no—
290C2×D4—1062no—
291C2×D4—4242no—
292C2×D4—2122no—
293C2×D4—2122no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C2×C4×D5311yescharacteristic
305C22×C4#1171994511yescharacteristic
306C22×C4C2×C22⋊D5311yescharacteristic
307SD16—2124no—
308SD16—2124no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16—2121no—
315M16—2121no—
316M16#1171992412yescomplemented
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—1062no—
321C2×C8—2121no—
322C2×C8—2121no—
323C2×C8—2122no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4#1171992111yescharacteristic
330C4⋊C4#1171992111yescharacteristic
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×D21211yescharacteristic
336C42#1171994511yescharacteristic
337C42C2×C22⋊D5311yescharacteristic
338C42C2×C22⋊D5311yescharacteristic
339C23—4242no—
340C23—4242no—
341C23—2122no—
342C23—2122no—
343C23—2122no—
344C23C4×C22⋊D5311yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8#1313313312yescomplemented
350D4—4242no—
351D4—4242no—
352D4—4242no—
353D4—4241no—
354D4—4241no—
355D4—4241no—
356D4—4241no—
357D4—2122no—
358D4—2122no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4C2×#1171992111yescharacteristic
369C2×C4C4×C22⋊D5311yescharacteristic
370C2×C4#1313318611yescharacteristic
371C2×C4#1313319111yescharacteristic
372C2×C4#1313319111yescharacteristic
373C2×C4#1313320011yescharacteristic
374C2×C4C2×#1171992411yescharacteristic
375C2×C4#1313323711yescharacteristic
376C2×C4#1313324211yescharacteristic
377C2×C4#1313324811yescharacteristic
378C8—22no—
379C8—41no—
380C8—42no—
381C8—4241no—
382C8—4241no—
383C8—2122no—
384C22—21no—
385C22—21no—
386C22—8482no—
387C22—8482no—
388C22—4242no—
389C22—4242no—
390C22—4242no—
391C22#1681050211yescharacteristic
392C22C4×#1171992411yescharacteristic
393C22#1681072111yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4C2×#1313313311yescharacteristic
402C4#1681060511yescharacteristic
403C4#1681076211yescharacteristic
404C4#1681076811yescharacteristic
405C4#1681082311yescharacteristic
406C4#1681084511yescharacteristic
407C2—21no—
408C2—21no—
409C2—8482no—
410C2#2027507011yescharacteristic
411C2#2027517811yescharacteristic
412C2#2027519511yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingQ8×M16×C53C2
socleC22×C53#386
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterQ8×M16D53
hypocenterC53#9254
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
4. #20273144C2semidirect
5. #20272589C2semidirect
6. #20272566C2semidirect
7. Q8×M16×C53C2semidirect
105. M16×C53SD16semidirect
142. Q8×C53#100semidirect
202. Q8×M16D53semidirect
237. C53#9254semidirect
316. M16#11719924semidirect
349. Q8#13133133semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
#202755552
#202733842
#202731442
#202725892
#202725662
Q8×M16×C532
#925453