Finite Group Databasebeta

← #28302247

Subgroups — #28302247

Order 13568.

836 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#20290569C211yesmaximal, characteristic
2C4×C22⋊#88324C211yesmaximal, characteristic
3C2×#16811473C212yesmaximal, complemented
4C2×Q8⋊#88324C212yesmaximal, complemented
5C2×#16811175C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9C2×C4×#88324C2212yes—
10#16811489—21no—
11#16811489—21no—
12#16811473—24no—
13Q8⋊#88324—24no—
14#16811175—24no—
15#16811175—24no—
16C2×#13133216C411yescharacteristic
17C2×#13133215C411yescharacteristic
18C2×#13133207—21no—
19C2×#13133196C411yescharacteristic
20C2×#13133194—21no—
21C2×Q8×Dic53C2211yescharacteristic
22C2×#13133182—21no—
23C2×#13133182C2211yescharacteristic
24C2×#13133173—21no—
25C22×C4×Dic53C2211yescharacteristic
26C2×C22⋊Q8×C53C411yescharacteristic
27C22×#88324—22no—
28C22×#88324—22no—
29C4⋊#88324—22no—
30C4⋊#88324—22no—
31C22⋊#88324—24no—
32C22⋊#88324—42no—
33C4×#88324—22no—
34C4×#88324—22no—
35C22×Dic106—21no—
36C22×Dic106C2×C411yescharacteristic
37C23×Dic53D411yescharacteristic
38C23×Dic53D411yescharacteristic
39#13133216—22no—
40#13133216—22no—
41#13133215—22no—
42#13133215—22no—
43#13133207—28no—
44#13133196—22no—
45#13133196—22no—
46#13133194—28no—
47Q8×Dic53—22no—
48Q8×Dic53D414yes—
49#13133182—28no—
50#13133182—22no—
51#13133182D414yes—
52#13133173—28no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2×C411yescharacteristic
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×#11719919—21no—
58C2×#11719919—21no—
59C2×#11719919—21no—
60C2×C22⋊Dic53—21no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53—22no—
63C2×C4×Dic53—21no—
64C2×C4×Dic53—21no—
65C2×C4×Dic53C2×C411yescharacteristic
66C2×C4×Dic53D411yescharacteristic
67C2×C4×Dic53D411yescharacteristic
68C2×C4×Dic53C2311yescharacteristic
69C22×Q8×C53C2×C411yescharacteristic
70C23×C4×C53C2×C411yescharacteristic
71C22⋊Q8×C53—22no—
72C22⋊Q8×C53—22no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53C2×C411yescharacteristic
75C2×C22⋊C4×C53—21no—
76C24×C53C22⋊C411yescharacteristic
77C2×Q8×C53—21no—
78C2×Q8×C53—22no—
79C2×Q8×C53—22no—
80C2×Q8×C53C22⋊C412yes—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22×C411yescharacteristic
89C22×Dic53—22no—
90C22×Dic53—22no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—22no—
94C22×Dic53—22no—
95C22×Dic53—41no—
96C22×Dic53C22⋊C411yescharacteristic
97C22×Dic53C22⋊C411yescharacteristic
98C22×Dic53SD1612yes—
99C22×Dic53Q1612yes—
100C22×Dic53C2×D411yescharacteristic
101C22×Dic53C2×D411yescharacteristic
102C2×Dic106—22no—
103C2×Dic106—22no—
104C2×Dic106—24no—
105C2×Dic106—21no—
106C2×Dic106—42no—
107C2×Dic106—42no—
108C2×Dic106C22⋊C412yes—
109C2×#88324—24no—
110C2×#88324—42no—
111C2×#88324—42no—
112C2×#88324—42no—
113C2×#88324—42no—
114C2×#88324—42no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53C22⋊C412yes—
118C22⋊C4×C53—24no—
119C22⋊Dic53—24no—
120C4⋊Dic53—21no—
121C4⋊Dic53—24no—
122C4⋊Dic53C22⋊C412yes—
123#11719919—24no—
124#11719919—24no—
125#11719919—24no—
126#11719919—24no—
127#11719919—24no—
128C4×Dic53—21no—
129C4×Dic53—21no—
130C4×Dic53—22no—
131C4×Dic53—22no—
132C4×Dic53—24no—
133C4×Dic53—24no—
134C4×Dic53C2×D412yes—
135C4×Dic53C2×D412yes—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53#10312yes—
143C23×C53C2×C22⋊C411yescharacteristic
144Q8×C53—24no—
145Q8×C53—42no—
146C2×C4×C53—22no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53C2×C22⋊C411yescharacteristic
158C2×C4×C53C2×C22⋊C411yescharacteristic
159C2×C4×C53#13112yes—
160C2×C4×C53#13112yes—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—24no—
168C2×Dic53—24no—
169C2×Dic53—22no—
170C2×Dic53—22no—
171C2×Dic53—44no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53#10312yes—
177C2×Dic53#10412yes—
178C2×Dic53#12012yes—
179C2×Dic53#12012yes—
180C2×Dic53C2×SD1611yescharacteristic
181C2×Dic53C2×Q1611yescharacteristic
182C2×Dic53#13712yes—
183Dic106—24no—
184Dic106—48no—
185Dic106—42no—
186#88324—44no—
187#88324—82no—
188#8314—531nomaximal, Sylow
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—22no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53#37911yescharacteristic
197C22×C53#40811yescharacteristic
198C22×C53#40811yescharacteristic
199C22×C53C2×#10311yescharacteristic
200C22×C53#41911yescharacteristic
201C22×C53C2×#10411yescharacteristic
202C22×C53#42111yescharacteristic
203C4×C53—41no—
204C4×C53—42no—
205C4×C53—42no—
206C4×C53#40812yes—
207C4×C53#40812yes—
208Dic53—22no—
209Dic53—22no—
210Dic53—22no—
211Dic53—42no—
212Dic53—42no—
213Dic53#44812yes—
214Dic53#45112yes—
215C2×#533—531no—
216#1797—531no—
217C4×C22⋊C8—531no—
218C2×Q8⋊C8—532no—
219C2×Q8⋊C8—532no—
220C2×C53—42no—
221C2×C53#148111yescharacteristic
222C2×C53#148511yescharacteristic
223C2×C53#149311yescharacteristic
224C2×C53#149711yescharacteristic
225C2×C53#178011yescharacteristic
226C2×C53#178711yescharacteristic
227C2×C53#178911yescharacteristic
228#533—1064no—
229#533—1064no—
230C2×C4⋊Q8—531no—
231C2×C4⋊Q8—531no—
232C2×#125—1061no—
233C2×C22⋊Q8—531no—
234C2×C22⋊Q8—1061no—
235C2×C22⋊Q8#615711yescharacteristic
236C2×C4×Q8—531no—
237C2×C4×Q8—531no—
238C2×C4×Q8—1061no—
239C2×#117—1061no—
240C22×C42—531no—
241C2×C4⋊C8—532no—
242C2×C22⋊C8—1061no—
243C2×C22⋊C8—1061no—
244C2×C4×C8—532no—
245#336—1061no—
246#336—1061no—
247Q8⋊C8—1064no—
248Q8⋊C8—1064no—
249C53#831411yescharacteristic, complemented, Sylow
250C22×Q8—531no—
251C22×Q8—1061no—
252C22×Q8C2×#615711yescharacteristic
253C23×C4—531no—
254C23×C4—531no—
255C23×C4C2×#615711yescharacteristic
256C22×C8—1062no—
257C22×C8—1062no—
258C4⋊Q8—1062no—
259C4⋊Q8—1062no—
260C4⋊Q8—1062no—
261C4⋊Q8—1062no—
262#125—1068no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—1068no—
266C22⋊Q8—1062no—
267C22⋊Q8—1062no—
268C4×Q8—534no—
269C4×Q8—534no—
270C4×Q8—1062no—
271C4×Q8—1062no—
272C4×Q8—1068no—
273#117—1068no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4C2×#615711yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—1061no—
285C2×C42—531no—
286C2×C42—531no—
287C2×C42—531no—
288C2×C42—531no—
289C2×C42—1062no—
290C2×C42—1062no—
291C2×C42—1061no—
292C2×C42—1061no—
293C4⋊C8—1062no—
294C4⋊C8—1062no—
295C22⋊C8—1064no—
296C22⋊C8—2122no—
297C4×C8—1062no—
298C4×C8—1062no—
299C24C22⋊#615711yescharacteristic
300C2×Q8—21no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—532no—
304C2×Q8—1064no—
305C2×Q8—1061no—
306C2×Q8—1062no—
307C2×Q8—1062no—
308C2×Q8—2122no—
309C2×Q8—2122no—
310C2×Q8C22⋊#615712yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—531no—
318C22×C4—532no—
319C22×C4—532no—
320C22×C4—531no—
321C22×C4—531no—
322C22×C4—531no—
323C22×C4—1062no—
324C22×C4—1062no—
325C22×C4—1062no—
326C22×C4—1062no—
327C22×C4—1061no—
328C22×C4—1061no—
329C22×C4—2121no—
330C22×C4C22⋊#615711yescharacteristic
331C22×C4C22×#615711yescharacteristic
332C2×C8—1064no—
333C2×C8—2122no—
334C2×C8—2122no—
335C2×C8—2122no—
336C2×C8—2122no—
337C2×C8—2122no—
338C4⋊C4—24no—
339C4⋊C4—21no—
340C4⋊C4—532no—
341C4⋊C4—1064no—
342C4⋊C4—1064no—
343C4⋊C4—1064no—
344C4⋊C4—1064no—
345C4⋊C4—1064no—
346C4⋊C4—1064no—
347C4⋊C4—1061no—
348C4⋊C4C22⋊#615712yes—
349C22⋊C4—24no—
350C22⋊C4—1064no—
351C42—532no—
352C42—532no—
353C42—1062no—
354C42—1062no—
355C42—1064no—
356C42—1064no—
357C42—1061no—
358C42—1061no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23#1313328012yes—
366C23C2×C22⋊#615711yescharacteristic
367Q8—24no—
368Q8—42no—
369Q8—1064no—
370Q8—2128no—
371Q8—2122no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—532no—
384C2×C4—531no—
385C2×C4—531no—
386C2×C4—532no—
387C2×C4—532no—
388C2×C4—532no—
389C2×C4—532no—
390C2×C4—1064no—
391C2×C4—1064no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—1062no—
399C2×C4—1062no—
400C2×C4—2124no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4C2×C22⋊#615711yescharacteristic
406C2×C4C2×C22⋊#615711yescharacteristic
407C2×C4#1313330412yes—
408C2×C4#1313330412yes—
409C8—4242no—
410C8—2124no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22#1681156411yescharacteristic
419C22#1681159611yescharacteristic
420C22#1681159611yescharacteristic
421C22C2×#1313328011yescharacteristic
422C22#1681161611yescharacteristic
423C22D53⋊#10411yescharacteristic
424C22#1681162011yescharacteristic
425C4—41no—
426C4—42no—
427C4—42no—
428C4—532no—
429C4—532no—
430C4—1062no—
431C4—1062no—
432C4—1062no—
433C4—2122no—
434C4—2122no—
435C4#1681159612yes—
436C4#1681159712yes—
437C2—42no—
438C2#2029002311yescharacteristic
439C2#2029003711yescharacteristic
440C2#2029006311yescharacteristic
441C2#2029007711yescharacteristic
442C2#2029053911yescharacteristic
443C2#2029055411yescharacteristic
444C2#2029055711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#8314
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16811473C2semidirect
4. C2×Q8⋊#88324C2semidirect
249. C53#8314semidirect

Maximal subgroups

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

subgroupindex
#202905692
C4×C22⋊#883242
C2×#168114732
C2×Q8⋊#883242
C2×#168111752
#831453