Finite Group Databasebeta

← #27278867

Subgroups — #27278867

Order 13568.

660 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810823C211yesmaximal, characteristic
2#20275642C211yesmaximal, characteristic
3#20275520C211yesmaximal, characteristic
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5#20272598C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7Q8×C4⋊C4×C53C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133214C2211yescharacteristic
10#16810823—22no—
11#16810823C414yescomplemented
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—21no—
15C2×#13133137—22no—
16C2×#13133137—21no—
17C2×#13133137—21no—
18C2×C4×#88322C2211yescharacteristic
19Q8⋊#88322—22no—
20C2×C4×Q8×C53—21no—
21C2×C4×Q8×C53C2211yescharacteristic
22C53×#389—21no—
23C53×#389—22no—
24C53×#384—21no—
25C53×#384C2211yescharacteristic
26C4×C4⋊C4×C53—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719924—24no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C22×#88322—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214C2×C412yes—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322C2×C412yes—
45#13133137—24no—
46#13133137—44no—
47#13133137—42no—
48#13133137—42no—
49C4×#88322—21no—
50C4×#88322—21no—
51C22×Q8×C53D411yescharacteristic
52C4×Q8×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53—42no—
55C4×Q8×C53—42no—
56C4×Q8×C53C2×C412yes—
57C4×Q8×C53D412yes—
58C2×C4⋊C4×C53—22no—
59C2×C4⋊C4×C53—22no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53D412yes—
67C2×C42×C53—22no—
68C2×C42×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#95×C53—22no—
72#95×C53—21no—
73C23×D53—41no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53C22⋊C412yes—
78C2×Q8×C53D812yes—
79C2×Q8×C53SD1612yes—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—42no—
92C2×D212—42no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—44no—
111C4⋊C4×C53—42no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—82no—
117C4⋊C4×C53C22⋊C412yes—
118C4⋊C4×C53SD1612yescomplemented
119C42×C53—22no—
120C42×C53—22no—
121C42×C53—44no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53C22×C411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C42×C53#4111yescharacteristic
128#11719924—48no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C23×C53#12011yescharacteristic
139C22×D53—42no—
140C22×D53—81no—
141C22×D53—81no—
142C22×D53—81no—
143C22×D53—81no—
144C22×D53—81no—
145C22×D53—81no—
146Q8×C53—22no—
147Q8×C53—22no—
148Q8×C53—22no—
149Q8×C53#10214yescomplemented
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53C2×C22⋊C411yescharacteristic
168C2×C4×C53C4×D411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12311yescharacteristic
173C2×C4×C53C2×D811yescharacteristic
174C2×C4×C53C2×SD1611yescharacteristic
175C2×C4×C53C2×SD1611yescharacteristic
176C2×C4×C53#13511yescharacteristic
177C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C53#13711yescharacteristic
180C2×C4×C53#13711yescharacteristic
181D212—44no—
182D212—42no—
183D212—42no—
184D212—42no—
185D212—42no—
186D212—82no—
187D212—81no—
188D212—81no—
189D212—81no—
190D212—81no—
191#88322—42no—
192#88322—42no—
193#88322—81no—
194#8979—531nomaximal, Sylow
195C22×C53#38611yescharacteristic
196C22×C53#44711yescharacteristic
197C22×C53#44811yescharacteristic
198C22×C53#44811yescharacteristic
199C22×C53#44911yescharacteristic
200C22×C53#45311yescharacteristic
201C22×C53#45411yescharacteristic
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—82no—
205C2×D53—161no—
206C2×D53—161no—
207C2×D53—161no—
208C2×D53—161no—
209C4×C53—24no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—42no—
213C4×C53—42no—
214C4×C53—81no—
215C4×C53—82no—
216C4×C53C2×#10211yescharacteristic
217C4×C53#41711yescharacteristic
218C4×C53C4×SD1611yescharacteristic
219C4×C53#44011yescharacteristic
220C4×C53#46011yescharacteristic
221C4×C53#46411yescharacteristic
222C4×C53#48011yescharacteristic
223C4×C53#48111yescharacteristic
224C2×#460—531no—
225Q8×C4⋊C4D5311yescharacteristic, complemented
226#1982—531no—
227#1957—531no—
228C2×Q8⋊C8—531no—
229#1296—531no—
230#1279—531no—
231C2×C53#162111yescharacteristic
232C2×C53#162311yescharacteristic
233C2×C53#162611yescharacteristic
234C2×C53#162711yescharacteristic
235C2×C53#187411yescharacteristic
236C2×C53#188011yescharacteristic
237C2×C53#188311yescharacteristic
238D53—162no—
239C22×SD16—1061no—
240C2×#127—531no—
241C2×C4×Q8—21no—
242C2×C4×Q8C2×D5311yescharacteristic
243#460—534no—
244#460—1062no—
245C2×C4⋊C8—531no—
246C2×C4⋊C8—531no—
247C2×#102—1061no—
248C2×#102—1062no—
249C2×#102—1061no—
250C2×#102—1061no—
251C2×C4×C8—531no—
252#389—21no—
253#389—22no—
254#384—21no—
255#384C2×D5311yescharacteristic
256C4×C4⋊C4—21no—
257C4×C4⋊C4C2×D5311yescharacteristic
258Q8⋊C8—1062no—
259C53#897911yescharacteristic, complemented, Sylow
260C22×Q8C22⋊D5311yescharacteristic
261C22×D4—1061no—
262C22×D4—1061no—
263C22×D4—1061no—
264C2×SD16—1064no—
265C2×SD16—2122no—
266C2×SD16—2122no—
267C22×C8—1061no—
268C22×C8—1061no—
269C22×C8—1061no—
270#127—532no—
271#127—1061no—
272#127—1061no—
273#127—1061no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8C4×D5312yes—
279C4×Q8D21212yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C22⋊D5311yescharacteristic
286C2×C4⋊C4C22⋊D5311yescharacteristic
287C2×C4⋊C4C22⋊D5311yescharacteristic
288C2×C4⋊C4C22⋊D5312yes—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42C22×D5311yescharacteristic
293C4⋊C8—532no—
294C4⋊C8—1061no—
295C4⋊C8—1061no—
296C4⋊C8—1061no—
297#102—1064no—
298#102—2122no—
299#102—2124no—
300#102—2122no—
301C4×C8—1061no—
302C4×C8—1061no—
303#95—22no—
304#95—21no—
305C24—2121no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8#1171992112yes—
310C2×Q8#1171992212yes—
311C2×Q8#1171992412yes—
312C2×D4—1062no—
313C2×D4—1062no—
314C2×D4—1062no—
315C2×D4—2122no—
316C2×D4—2122no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4—2121no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4C2×C22⋊D5311yescharacteristic
334C22×C4C2×C22⋊D5311yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336SD16—2128no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—2121no—
340C2×C8—2121no—
341C2×C8—2121no—
342C2×C8—2121no—
343C2×C8—2121no—
344C2×C8—2121no—
345C2×C8—2121no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—82no—
359C4⋊C4#1171992112yes—
360C4⋊C4#1171992412yescomplemented
361C42—22no—
362C42—22no—
363C42—44no—
364C42—41no—
365C42—41no—
366C42C2×C4×D5311yescharacteristic
367C42C2×D21211yescharacteristic
368C42#1171994511yescharacteristic
369C42#1171994511yescharacteristic
370C23—4241no—
371C23—4241no—
372C23—4241no—
373C23—4241no—
374C23—4241no—
375C23—4241no—
376C23—2122no—
377C23#1313318611yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8#1313313714yescomplemented
382D4—4242no—
383D4—4241no—
384D4—4241no—
385D4—4241no—
386D4—4241no—
387D4—2124no—
388D4—2122no—
389D4—2122no—
390D4—2122no—
391D4—2122no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4C2×#1171992111yescharacteristic
410C2×C4C4×C22⋊D5311yescharacteristic
411C2×C4C4×C22⋊D5311yescharacteristic
412C2×C4#1313319111yescharacteristic
413C2×C4#1313319111yescharacteristic
414C2×C4#1313320011yescharacteristic
415C2×C4C2×#1171992211yescharacteristic
416C2×C4C2×#1171992411yescharacteristic
417C2×C4C2×#1171992411yescharacteristic
418C2×C4#1313323711yescharacteristic
419C2×C4#1313324211yescharacteristic
420C2×C4#1313324311yescharacteristic
421C2×C4#1313324811yescharacteristic
422C2×C4#1313324811yescharacteristic
423C8—4241no—
424C8—2122no—
425C8—2122no—
426C22—8481no—
427C22—8481no—
428C22—8481no—
429C22—8481no—
430C22—4242no—
431C22—4242no—
432C22—4242no—
433C22#1681050211yescharacteristic
434C22#1681075711yescharacteristic
435C22#1681076211yescharacteristic
436C22#1681076211yescharacteristic
437C22#1681076811yescharacteristic
438C22#1681079111yescharacteristic
439C22#1681079611yescharacteristic
440C4—24no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—42no—
445C4—81no—
446C4—82no—
447C4C2×#1313313711yescharacteristic
448C4#1681062211yescharacteristic
449C4C4×#1171992411yescharacteristic
450C4#1681072111yescharacteristic
451C4#1681082311yescharacteristic
452C4#1681084511yescharacteristic
453C4#1681093611yescharacteristic
454C4#1681094111yescharacteristic
455C2—8482no—
456C2#2027378011yescharacteristic
457C2#2027379311yescharacteristic
458C2#2027381111yescharacteristic
459C2#2027381811yescharacteristic
460C2#2027503211yescharacteristic
461C2#2027507011yescharacteristic
462C2#2027509111yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingQ8×C4⋊C4×C53C2
socleC23×C53#120
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterQ8×C4⋊C4D53
hypocenterC53#8979
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊#88322C2semidirect
5. #20272598C2semidirect
6. #20272556C2semidirect
7. Q8×C4⋊C4×C53C2semidirect
11. #16810823C4semidirect
118. C4⋊C4×C53SD16semidirect
149. Q8×C53#102semidirect
225. Q8×C4⋊C4D53semidirect
259. C53#8979semidirect
360. C4⋊C4#11719924semidirect
381. Q8#13133137semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
#202756422
#202755202
C2×Q8⋊#883222
#202725982
#202725562
Q8×C4⋊C4×C532
#897953