Finite Group Databasebeta

← #28304236

Subgroups — #28304236

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811639C211yesmaximal, characteristic
2#20290569C211yesmaximal, characteristic
3C2×Q8⋊#88324C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5C22⋊Q8×Dic53C211yesmaximal, characteristic
6#16811639—24no—
7C2×C4⋊#88324—21no—
8C2×C4⋊#88324C2212yes—
9C2×C22⋊#88324—21no—
10C2×C22⋊#88324—21no—
11C2×C4×#88324C2212yes—
12#16811489—21no—
13Q8⋊#88324—24no—
14C2×Q8×Dic53C2211yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16#16810530C411yescharacteristic
17#16810522C411yescharacteristic
18#16810509—21no—
19#16810498—21no—
20#16810483C411yescharacteristic
21C4⋊C4×Dic53—21no—
22C4⋊C4×Dic53C2211yescharacteristic
23C22⋊C4×Dic53—21no—
24C2×C22⋊Q8×C53C411yescharacteristic
25C22×#88324—22no—
26C22×#88324—22no—
27C4⋊#88324—24no—
28C4⋊#88324—22no—
29C4⋊#88324—22no—
30C22⋊#88324—24no—
31C22⋊#88324—42no—
32C4×#88324—22no—
33C4×#88324D414yes—
34C23×Dic53D411yescharacteristic
35C23×Dic53D411yescharacteristic
36Q8×Dic53—22no—
37Q8×Dic53—22no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—22no—
44C2×C4×Dic53—22no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53C2×C411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53C2311yescharacteristic
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56C22×Q8×C53C2×C411yescharacteristic
57C23×C4×C53C2×C411yescharacteristic
58C22⋊Q8×C53—22no—
59C22⋊Q8×C53C814yescomplemented
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C22⋊C4×C53—21no—
63C24×C53C22⋊C411yescharacteristic
64C2×Q8×C53—21no—
65C2×Q8×C53—22no—
66C2×Q8×C53—22no—
67C2×Q8×C53C2×C812yes—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C2×C812yes—
74C22×C4×C53M1612yes—
75C22×C4×C53C22×C411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—22no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—22no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—41no—
85C22×Dic53—41no—
86C22×Dic53C22⋊C411yescharacteristic
87C22×Dic53C22⋊C411yescharacteristic
88C22×Dic53C2×D411yescharacteristic
89C22×Dic53C2×D411yescharacteristic
90C2×#88324—24no—
91C2×#88324—42no—
92C2×#88324—42no—
93C2×#88324—42no—
94C2×#88324—42no—
95C2×#88324—42no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53C2×C812yes—
99C22⋊C4×C53—24no—
100C22⋊Dic53—42no—
101C4⋊Dic53—22no—
102C4⋊Dic53—22no—
103C4⋊Dic53—42no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106C4×Dic53—22no—
107C4×Dic53—21no—
108C4×Dic53—22no—
109C4×Dic53—21no—
110C4×Dic53—42no—
111C4×Dic53—41no—
112C4×Dic53—41no—
113C4×Dic53C2×D412yes—
114C4×Dic53#4112yes—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53C22⋊C812yes—
122C23×C53C2×C22⋊C411yescharacteristic
123Q8×C53—24no—
124Q8×C53—42no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—24no—
128C2×C4×C53—24no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C22⋊C812yes—
136C2×C4×C53C4×D412yes—
137C2×C4×C53C22×C811yescharacteristic
138C2×C4×C53C2×M1611yescharacteristic
139C2×C4×C53#13112yes—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53#12011yescharacteristic
159C2×Dic53#12112yes—
160C2×Dic53#12311yescharacteristic
161C2×Dic53AGL(1,ℤ/8)11yescharacteristic
162C2×Dic53#13712yes—
163C2×Dic53#13711yescharacteristic
164#88324—44no—
165#88324—82no—
166#9073—531nomaximal, Sylow
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53#38611yescharacteristic
175C22×C53C2×C22⋊C811yescharacteristic
176C22×C53#40811yescharacteristic
177C22×C53#41711yescharacteristic
178C22×C53#41911yescharacteristic
179C22×C53#42111yescharacteristic
180C22×C53#42111yescharacteristic
181C4×C53—22no—
182C4×C53—42no—
183C4×C53—42no—
184C4×C53C8×D412yes—
185C4×C53#43512yes—
186Dic53—21no—
187Dic53—21no—
188Dic53—41no—
189Dic53—41no—
190Dic53—81no—
191Dic53—81no—
192Dic53#46812yes—
193Dic53#47012yes—
194C2×#431—531no—
195C4×C22⋊Q8—531no—
196#1797—531no—
197C2×Q8⋊C8—532no—
198#1279—532no—
199C2×C53—42no—
200C2×C53#147711yescharacteristic
201C2×C53#147911yescharacteristic
202C2×C53#149511yescharacteristic
203C2×C53#149711yescharacteristic
204C2×C53#186911yescharacteristic
205C2×C53#189411yescharacteristic
206C2×C53#189711yescharacteristic
207C2×C22⋊Q8#615711yescharacteristic
208C2×C4×Q8—531no—
209C22×C42—531no—
210#431—1064no—
211C2×C4⋊C8—532no—
212C2×C4⋊C8—1061no—
213C2×C22⋊C8—1061no—
214C2×C22⋊C8—1061no—
215C2×C4×C8—532no—
216#391—531no—
217#389—531no—
218#387—1061no—
219#385—1061no—
220#380—531no—
221C4×C4⋊C4—531no—
222C4×C4⋊C4—1061no—
223C4×C22⋊C4—1061no—
224#336—1061no—
225Q8⋊C8—1064no—
226C53#907311yescharacteristic, complemented, Sylow
227C22×Q8C2×#615711yescharacteristic
228C23×C4—531no—
229C23×C4—531no—
230C23×C4C2×#615711yescharacteristic
231C22×C8—1062no—
232C22×C8—1062no—
233C22⋊Q8—22no—
234C22⋊Q8#8832414yescomplemented
235C4×Q8—1062no—
236C4×Q8—1062no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4C2×#615711yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—1061no—
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—1062no—
250C2×C42—1062no—
251C2×C42—1061no—
252C2×C42—1061no—
253C4⋊C8—1062no—
254C4⋊C8—1062no—
255C4⋊C8—1064no—
256C22⋊C8—1064no—
257C22⋊C8—2122no—
258C4×C8—534no—
259C4×C8—1062no—
260#95—1061no—
261#95—1061no—
262#95—1061no—
263#95—1061no—
264#95—1061no—
265C24C22⋊#615711yescharacteristic
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8C2×#8832412yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1062no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1062no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4C2×#8832412yes—
289C22×C4#1171994012yes—
290C22×C4C22⋊#615711yescharacteristic
291C22×C4C22×#615711yescharacteristic
292C2×C8—1064no—
293C2×C8—2122no—
294C2×C8—2122no—
295C2×C8—2122no—
296C2×C8—2122no—
297C2×C8—2122no—
298C4⋊C4—24no—
299C4⋊C4—21no—
300C4⋊C4—1062no—
301C4⋊C4—1062no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4—2122no—
305C4⋊C4C2×#8832412yes—
306C22⋊C4—24no—
307C22⋊C4—2122no—
308C42—532no—
309C42—532no—
310C42—1061no—
311C42—1062no—
312C42—1061no—
313C42—1062no—
314C42—2122no—
315C42—2121no—
316C42—2121no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23C22⋊#8832412yes—
324C23C2×C22⋊#615711yescharacteristic
325Q8—24no—
326Q8—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—532no—
338C2×C4—531no—
339C2×C4—531no—
340C2×C4—532no—
341C2×C4—531no—
342C2×C4—531no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4C22⋊#8832412yes—
362C2×C4D4×#615712yes—
363C2×C4C22×#8832411yescharacteristic
364C2×C4C2×#1171994011yescharacteristic
365C2×C4#1313330412yes—
366C8—4242no—
367C8—2124no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22#1681157111yescharacteristic
376C22C2×C22⋊#8832411yescharacteristic
377C22#1681159611yescharacteristic
378C22#1681161411yescharacteristic
379C22#1681161611yescharacteristic
380C22#1681162011yescharacteristic
381C22#1681162211yescharacteristic
382C4—22no—
383C4—42no—
384C4—42no—
385C4—532no—
386C4—532no—
387C4—1061no—
388C4—1061no—
389C4—4241no—
390C4—4241no—
391C4—2121no—
392C4—2121no—
393C4D4×#8832412yes—
394C4#1681165412yes—
395C2—42no—
396C2#2029001011yescharacteristic
397C2#2029001711yescharacteristic
398C2#2029007211yescharacteristic
399C2#2029007711yescharacteristic
400C2#2029068811yescharacteristic
401C2#2029073011yescharacteristic
402C2#2029073411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#9073
LCSG ▷ C2×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×Q8⋊#88324C2semidirect
4. #20289551C2semidirect
59. C22⋊Q8×C53C8semidirect
226. C53#9073semidirect
234. C22⋊Q8#88324semidirect

Maximal subgroups

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

subgroupindex
C2×#168116392
#202905692
C2×Q8⋊#883242
#202895512
C22⋊Q8×Dic532
#907353