Finite Group Databasebeta

← #28402129

Subgroups — #28402129

Order 14464.

478 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×#16813651C211yesmaximal, characteristic, complemented
2#20299255C211yesmaximal, characteristic, complemented
3#20299247C211yesmaximal, characteristic, complemented
4#20298989C211yesmaximal, characteristic, complemented
5#20298841C211yesmaximal, characteristic, complemented
6#20298839C211yesmaximal, characteristic, complemented
7#20298811C211yesmaximal, characteristic, complemented
8#20298806C211yesmaximal, characteristic, complemented
9#20298760C211yesmaximal, characteristic, complemented
10#20298760C211yesmaximal, characteristic, complemented
11#20298752C211yesmaximal, characteristic, complemented
12#20298740C211yesmaximal, characteristic, complemented
13#20298699C211yesmaximal, characteristic, complemented
14#20298669C211yesmaximal, characteristic, complemented
15C2×#135×C113C211yesmaximal, characteristic, complemented
16C2×#14268646C2211yescharacteristic
17C2×Q8×D113C2211yescharacteristic
18#16813651—24no—
19C2×Dic452C2211yescharacteristic
20C2×#14268637C2211yescharacteristic
21C2×#14268637C2211yescharacteristic
22C2×#14268635C2211yescharacteristic
23C2×#14268635C2211yescharacteristic
24C2×#14268630C2211yescharacteristic
25#16813620C2211yescharacteristic
26#16813609—21no—
27#16813609C2211yescharacteristic
28#16813605—21no—
29#16813603—21no—
30#16813601—21no—
31#16813601C2211yescharacteristic
32#16813601C2211yescharacteristic
33#16813598C2211yescharacteristic
34#16813594—21no—
35#16813593—21no—
36#16813593C2211yescharacteristic
37Q8×Dic113C2211yescharacteristic
38D4×Dic113C2211yescharacteristic
39C4×C22⋊D113—21no—
40C4×C22⋊D113C2211yescharacteristic
41C2×#14268626—21no—
42#16813556C2211yescharacteristic
43#16813551C2211yescharacteristic
44#16813547C2211yescharacteristic
45#16813547C2211yescharacteristic
46#16813546C2211yescharacteristic
47#16813546C2211yescharacteristic
48#16813543C2211yescharacteristic
49#16813543C2211yescharacteristic
50C22⋊#11744751C2211yescharacteristic
51#16813534C2211yescharacteristic
52#16813532C2211yescharacteristic
53C2×#41×C113C2211yescharacteristic
54C2×#41×C113C2211yescharacteristic
55#135×C113—24no—
56C2×Q16×C113C2211yescharacteristic
57C2×SD16×C113C2211yescharacteristic
58C2×SD16×C113C2211yescharacteristic
59C2×D8×C113C2211yescharacteristic
60C22×C8×C113C2211yescharacteristic
61#14268646—22no—
62#14268646—41no—
63C2×Dic226—21no—
64C2×Dic226C2311yescharacteristic
65C2×Dic226C2311yescharacteristic
66Q8×D113—22no—
67Q8×D113—41no—
68C2×C22⋊D113—21no—
69C2×C22⋊D113—21no—
70C2×C22⋊D113D411yescharacteristic
71C22×Dic113—21no—
72C22×Dic113—21no—
73C2×C4×D113—21no—
74C2×C4×D113C2311yescharacteristic
75Dic452—22no—
76#14268637—22no—
77#14268637—41no—
78#14268635—22no—
79#14268635—41no—
80#14268630—22no—
81C2×#11744751C2311yescharacteristic
82C2×#11744751C2311yescharacteristic
83C4⋊Dic113—21no—
84C4⋊Dic113C2311yescharacteristic
85C4⋊Dic113C2311yescharacteristic
86#14268626—21no—
87#14268626—21no—
88#14268626—21no—
89#14268626—21no—
90#14268626—21no—
91#14268626D411yescharacteristic
92C22⋊Dic113—21no—
93C22⋊Dic113—21no—
94C22⋊Dic113—21no—
95C22⋊Dic113—21no—
96C22⋊Dic113—21no—
97C22⋊Dic113—21no—
98C22⋊Dic113D411yescharacteristic
99#14268624—21no—
100#14268624D411yescharacteristic
101C4×Dic113—21no—
102C4×Dic113C2311yescharacteristic
103#41×C113—22no—
104#41×C113—22no—
105#41×C113—41no—
106#41×C113—41no—
107C2×Q8×C113C2311yescharacteristic
108C2×Q8×C113C2311yescharacteristic
109C2×D4×C113—21no—
110C2×D4×C113—21no—
111C2×D4×C113C2311yescharacteristic
112C2×D4×C113C2311yescharacteristic
113C22×C4×C113—21no—
114C22×C4×C113—21no—
115C22×C4×C113C2311yescharacteristic
116Q16×C113—22no—
117SD16×C113—22no—
118SD16×C113D414yescomplemented
119D8×C113—22no—
120C2×C8×C113—22no—
121C2×C8×C113C2311yescharacteristic
122C2×C8×C113C2311yescharacteristic
123C23×C113—21no—
124C23×C113—21no—
125C23×C113C2×D411yescharacteristic
126C22×D113C2×D411yescharacteristic
127Q8×C113—21no—
128Q8×C113—21no—
129Q8×C113—21no—
130Q8×C113C2×D412yes—
131D4×C113—21no—
132D4×C113—21no—
133D4×C113—21no—
134D4×C113—42no—
135D4×C113—42no—
136D4×C113C2×D412yes—
137C2×C4×C113—21no—
138C2×C4×C113—21no—
139C2×C4×C113—21no—
140C2×C4×C113—21no—
141C2×C4×C113—42no—
142C2×C4×C113—42no—
143C2×C4×C113C2×D411yescharacteristic
144C2×C4×C113C2411yescharacteristic
145C22⋊D113—41no—
146C22⋊D113—42no—
147C22⋊D113—42no—
148C2×Dic113—21no—
149C2×Dic113—21no—
150C2×Dic113—21no—
151C2×Dic113—21no—
152C2×Dic113—42no—
153C2×Dic113—42no—
154C2×Dic113C2×D411yescharacteristic
155C2×Dic113C2×D411yescharacteristic
156C2×Dic113C2×D411yescharacteristic
157C4×D113—21no—
158C4×D113—21no—
159C4×D113—42no—
160Dic226—21no—
161Dic226—21no—
162Dic226—21no—
163Dic226—21no—
164Dic226—42no—
165C8×C113—21no—
166C8×C113C2×D412yes—
167#11744751—21no—
168#11744751—21no—
169C22×C113—22no—
170C22×C113—21no—
171C22×C113—21no—
172C22×C113—21no—
173C22×C113—41no—
174C22×C113—41no—
175C22×C113—41no—
176C22×C113—41no—
177C22×C113—41no—
178C22×C113C22×D411yescharacteristic
179C2×D113—21no—
180C2×D113—21no—
181C2×D113—21no—
182C4×C113—22no—
183C4×C113—21no—
184C4×C113—41no—
185C4×C113C22×D411yescharacteristic
186C4×C113H(5,𝔽2)11yescharacteristic
187Dic113—21no—
188Dic113—21no—
189Dic113—21no—
190Dic113—41no—
191Dic113—41no—
192Dic113—41no—
193Dic113—41no—
194C2×C113—41no—
195C2×C113—42no—
196C2×C113—81no—
197C2×C113D4211yescharacteristic
198C2×C113#57511yescharacteristic
199C2×C113#57811yescharacteristic
200D113—41no—
201#3269—1131nomaximal, Sylow
202C113#326911yescharacteristic, complemented, Sylow
203C2×#137—1131no—
204C2×#135D11311yescharacteristic, complemented
205#548—1131no—
206#547—1131no—
207#497—1131no—
208#468—1131no—
209#467—1131no—
210#463—1131no—
211#462—1131no—
212#452—1131no—
213#452—1131no—
214#451—1131no—
215#449—1131no—
216#440—1131no—
217#435—1131no—
218C2×#41—1131no—
219C2×#41C2×D11311yescharacteristic
220C2×#41C2×D11311yescharacteristic
221C22×Q8—1131no—
222#137—2264no—
223#135—24no—
224C2×Q16—1131no—
225C2×Q16—1131no—
226C2×Q16—1131no—
227C2×Q16C2×D11311yescharacteristic
228C2×SD16—1131no—
229C2×SD16—1131no—
230C2×SD16C2×D11311yescharacteristic
231C2×SD16C2×D11311yescharacteristic
232C2×D8C2×D11311yescharacteristic
233C2×M16—1131no—
234C22×C8C2×D11311yescharacteristic
235C4⋊Q8—1131no—
236#124—1131no—
237#124—2261no—
238#123—2261no—
239#123—2261no—
240C22⋊Q8—1131no—
241C22⋊Q8—1131no—
242C22⋊Q8—1131no—
243C22⋊Q8—2261no—
244#121—1131no—
245#121—2261no—
246#121—2261no—
247C4×Q8—1131no—
248C4×D4—1131no—
249C4×D4—1131no—
250C4×D4—2261no—
251C2×C4⋊C4—2261no—
252#107—1131no—
253C4⋊C8—1131no—
254#103—1131no—
255#103—1131no—
256#103—1131no—
257#103—1131no—
258#102—1131no—
259#102—1131no—
260C22⋊C8—1131no—
261C22⋊C8—1131no—
262#97—1131no—
263#41—22no—
264#41—22no—
265#41—41no—
266#41—41no—
267#41—4521no—
268#41—2262no—
269C2×Q8—1131no—
270C2×Q8—1131no—
271C2×Q8—4521no—
272C2×Q8—2261no—
273C2×Q8—2262no—
274C2×Q8C22×D11311yescharacteristic
275C2×Q8C22×D11311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—1131no—
279C2×D4—2261no—
280C2×D4—2261no—
281C2×D4C22×D11311yescharacteristic
282C2×D4C22×D11311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—1131no—
286C22×C4—2261no—
287C22×C4—2261no—
288C22×C4—2261no—
289C22×C4C22×D11311yescharacteristic
290Q16—22no—
291Q16—4521no—
292Q16—2262no—
293Q16—2262no—
294SD16—22no—
295SD16—4521no—
296SD16—2262no—
297SD16C22⋊D11314yescomplemented
298D8—22no—
299M16—2262no—
300C2×C8—22no—
301C2×C8—1131no—
302C2×C8—1131no—
303C2×C8C22×D11311yescharacteristic
304C2×C8C22×D11311yescharacteristic
305C4⋊C4—1131no—
306C4⋊C4—1131no—
307C4⋊C4—1131no—
308C4⋊C4—2261no—
309C4⋊C4—2261no—
310C4⋊C4—2261no—
311C4⋊C4—2261no—
312C4⋊C4—2261no—
313C4⋊C4—2261no—
314C22⋊C4—1131no—
315C22⋊C4—1131no—
316C22⋊C4—2261no—
317C22⋊C4—2261no—
318C22⋊C4—2261no—
319C22⋊C4—2261no—
320C22⋊C4—2261no—
321C22⋊C4—2261no—
322C22⋊C4—2261no—
323C42—1131no—
324C42—2261no—
325C23—21no—
326C23—21no—
327C23—1131no—
328C23D4×D11311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—4522no—
333Q8—2261no—
334Q8—2261no—
335Q8—2261no—
336Q8—2261no—
337Q8C2×C22⋊D11312yes—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—4522no—
344D4—4522no—
345D4—4521no—
346D4C2×C22⋊D11312yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—1131no—
354C2×C4—1131no—
355C2×C4—1131no—
356C2×C4—4522no—
357C2×C4—4522no—
358C2×C4—4522no—
359C2×C4—2261no—
360C2×C4—2261no—
361C2×C4—2261no—
362C2×C4—2261no—
363C2×C4—2261no—
364C2×C4—2261no—
365C2×C4D4×D11311yescharacteristic
366C2×C4C23×D11311yescharacteristic
367C8—21no—
368C8—2261no—
369C8—2261no—
370C8C2×C22⋊D11312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—2261no—
381C22—2261no—
382C22—2261no—
383C22C2×D4×D11311yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—4521no—
388C4—4521no—
389C4—4521no—
390C4—4521no—
391C4—2261no—
392C4—2261no—
393C4—2261no—
394C4C22×C22⋊D11311yescharacteristic
395C4#1681366711yescharacteristic
396C2—41no—
397C2—42no—
398C2—81no—
399C2—4521no—
400C2D4×C22⋊D11311yescharacteristic
401C2#2029939511yescharacteristic
402C2#2029941211yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×#135×C113C2
socleC22×C113C22×D4
hypercenterC2×#135D113
hypocenterC113#3269
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16813651C2semidirect
2. #20299255C2semidirect
3. #20299247C2semidirect
4. #20298989C2semidirect
5. #20298841C2semidirect
6. #20298839C2semidirect
7. #20298811C2semidirect
8. #20298806C2semidirect
9. #20298760C2semidirect
10. #20298760C2semidirect
11. #20298752C2semidirect
12. #20298740C2semidirect
13. #20298699C2semidirect
14. #20298669C2semidirect
15. C2×#135×C113C2semidirect
118. SD16×C113D4semidirect
202. C113#3269semidirect
204. C2×#135D113semidirect
297. SD16C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136512
#202992552
#202992472
#202989892
#202988412
#202988392
#202988112
#202988062
#202987602
#202987522
#202987402
#202986992
#202986692
C2×#135×C1132
#3269113