Finite Group Databasebeta

← #30650289

Subgroups — #30650289

Order 16256.

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×#16861238C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550536C211yesmaximal, characteristic, complemented
4#22550525C211yesmaximal, characteristic, complemented
5#22550269C211yesmaximal, characteristic, complemented
6#22550125C211yesmaximal, characteristic, complemented
7#22550115C211yesmaximal, characteristic, complemented
8#22550093C211yesmaximal, characteristic, complemented
9#22550085C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550030C211yesmaximal, characteristic, complemented
12#22550022C211yesmaximal, characteristic, complemented
13#22549973C211yesmaximal, characteristic, complemented
14#22549944C211yesmaximal, characteristic, complemented
15#452×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861238—24no—
20#16861229—24no—
21C2×Dic508C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523237C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29C22×C127⋊C8C2211yescharacteristic
30#16861207C2211yescharacteristic
31#16861198C2211yescharacteristic
32#16861195—21no—
33#16861192—21no—
34#16861190—21no—
35#16861188C2211yescharacteristic
36#16861187—21no—
37#16861185C2211yescharacteristic
38#16861184C2211yescharacteristic
39#16861182—21no—
40#16861181—21no—
41#16861174C2211yescharacteristic
42D4×Dic127C2211yescharacteristic
43#16861171—21no—
44C4×C22⋊D127C2211yescharacteristic
45C2×Dic127⋊C4—21no—
46#16861142C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861133C2211yescharacteristic
50#16861132C2211yescharacteristic
51#16861131C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53#16861119C2211yescharacteristic
54C2×#41×C127C2211yescharacteristic
55C2×Q16×C127C2211yescharacteristic
56C2×SD16×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61#14523248—22no—
62#14523248—41no—
63#14523247—22no—
64#14523247—41no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—41no—
70C2×D508—21no—
71C2×D508C2311yescharacteristic
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127D411yescharacteristic
75C22×Dic127—21no—
76C22×Dic127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79C2×C4×D127C2311yescharacteristic
80Dic508—22no—
81#14523238—22no—
82#14523238—22no—
83#14523237—41no—
84#14523236D414yescomplemented
85#14523235—22no—
86#14523233—22no—
87#14523231—22no—
88C2×C127⋊C8—22no—
89C2×C127⋊C8C2311yescharacteristic
90C2×C127⋊C8C2311yescharacteristic
91C4⋊Dic127C2311yescharacteristic
92Dic127⋊C4—21no—
93Dic127⋊C4—21no—
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4D411yescharacteristic
98C22⋊Dic127—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127D411yescharacteristic
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4D411yescharacteristic
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108#41×C127—22no—
109#41×C127—41no—
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116Q16×C127—41no—
117SD16×C127—22no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127—21no—
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129Dic254C2×D412yes—
130D508—21no—
131D508—21no—
132D508—42no—
133C22⋊D127—41no—
134C22⋊D127—42no—
135C22⋊D127—42no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—42no—
140C2×Dic127—42no—
141C2×Dic127C2×D411yescharacteristic
142C2×Dic127C2×D411yescharacteristic
143C2×Dic127C2×D411yescharacteristic
144C4×D127—21no—
145C4×D127—21no—
146C4×D127—42no—
147C4×D127—42no—
148C127⋊C8—21no—
149C127⋊C8C2×D412yes—
150C23×C127—21no—
151C23×C127C2×D411yescharacteristic
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127—21no—
155Q8×C127—21no—
156D4×C127—21no—
157D4×C127—42no—
158D4×C127C2×D412yes—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—42no—
165C2×C4×C127C2×D411yescharacteristic
166C2×C4×C127C2411yescharacteristic
167C8×C127—21no—
168C8×C127—21no—
169C22×C127—21no—
170C22×C127—22no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127C22×D411yescharacteristic
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C4×C127—21no—
183C4×C127—41no—
184C4×C127—41no—
185C4×C127—41no—
186C4×C127C22×D411yescharacteristic
187C4×C127H(5,𝔽2)11yescharacteristic
188Dic127—21no—
189Dic127—22no—
190Dic127—21no—
191Dic127—21no—
192Dic127—41no—
193Dic127—41no—
194C2×C127—41no—
195C2×C127—42no—
196C2×C127D4211yescharacteristic
197C2×C127#57511yescharacteristic
198C2×C127#57811yescharacteristic
199D127—41no—
200D127—81no—
201#3269—1271nomaximal, Sylow
202C127#326911yescharacteristic, complemented, Sylow
203C2×#137—1271no—
204C2×#135—1271no—
205#548—1271no—
206#547—1271no—
207#497—1271no—
208#468—1271no—
209#467—1271no—
210#463—1271no—
211#462—1271no—
212#452—1271no—
213#452D12711yescharacteristic, complemented
214#451—1271no—
215#449—1271no—
216#440—1271no—
217#435—1271no—
218C2×#41—1271no—
219C2×#41—1271no—
220C2×#41C2×D12711yescharacteristic
221C22×Q8—1271no—
222#137—2544no—
223#135—2544no—
224C2×Q16—1271no—
225C2×Q16—1271no—
226C2×Q16—1271no—
227C2×Q16C2×D12711yescharacteristic
228C2×SD16—1271no—
229C2×SD16—1271no—
230C2×SD16—1271no—
231C2×SD16C2×D12711yescharacteristic
232C2×D8—1271no—
233C2×M16—1271no—
234C22×C8—1271no—
235C4⋊Q8—1271no—
236#124—1271no—
237#124—2541no—
238#123—2541no—
239#123—2541no—
240C22⋊Q8—1271no—
241C22⋊Q8—1271no—
242C22⋊Q8—2541no—
243C22⋊Q8C2×D12711yescharacteristic
244#121—1271no—
245#121—2541no—
246#121—2541no—
247C4×Q8—1271no—
248C4×D4—1271no—
249C4×D4—1271no—
250C4×D4—2541no—
251C2×C4⋊C4—2541no—
252#107—1271no—
253C4⋊C8—1271no—
254#103—1271no—
255#103—1271no—
256#103—1271no—
257#103C2×D12711yescharacteristic
258#102—1271no—
259#102C2×D12711yescharacteristic
260C22⋊C8—1271no—
261C22⋊C8C2×D12711yescharacteristic
262#97—1271no—
263#41—22no—
264#41—41no—
265#41—5081no—
266#41—5081no—
267#41—2542no—
268#41—2542no—
269C2×Q8—1271no—
270C2×Q8—1271no—
271C2×Q8—5081no—
272C2×Q8—2541no—
273C2×Q8—2542no—
274C2×Q8C22×D12711yescharacteristic
275C2×Q8C22×D12711yescharacteristic
276C2×D4—21no—
277C2×D4—1271no—
278C2×D4—1271no—
279C2×D4—2541no—
280C2×D4—2541no—
281C2×D4—2541no—
282C2×D4C22×D12711yescharacteristic
283C22×C4—21no—
284C22×C4—1271no—
285C22×C4—2541no—
286C22×C4—2541no—
287C22×C4—2541no—
288C22×C4—2541no—
289C22×C4C22×D12711yescharacteristic
290Q16—41no—
291Q16—2542no—
292Q16—2542no—
293Q16—2542no—
294SD16—22no—
295SD16—1274no—
296SD16—5081no—
297SD16—2542no—
298D8—2542no—
299M16—2542no—
300C2×C8—1271no—
301C2×C8—1271no—
302C2×C8—2542no—
303C2×C8C22×D12711yescharacteristic
304C2×C8C22×D12711yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—1271no—
307C4⋊C4—1271no—
308C4⋊C4—2541no—
309C4⋊C4—2541no—
310C4⋊C4—2541no—
311C4⋊C4—2541no—
312C4⋊C4—2541no—
313C4⋊C4C22×D12711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—1271no—
316C22⋊C4—1271no—
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C42—1271no—
324C42—2541no—
325C23—21no—
326C23—1271no—
327C23—2541no—
328C23D4×D12711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—1272no—
334Q8—5082no—
335Q8—2541no—
336Q8—2541no—
337Q8—2541no—
338D4—21no—
339D4—42no—
340D4—5082no—
341D4—5082no—
342D4—5082no—
343D4—5081no—
344D4—2541no—
345D4—2541no—
346D4D4×D12712yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—1271no—
354C2×C4—1271no—
355C2×C4—1271no—
356C2×C4—5082no—
357C2×C4—5082no—
358C2×C4—5082no—
359C2×C4—5082no—
360C2×C4—2541no—
361C2×C4—2541no—
362C2×C4—2541no—
363C2×C4—2541no—
364C2×C4—2541no—
365C2×C4D4×D12711yescharacteristic
366C2×C4C23×D12711yescharacteristic
367C8—21no—
368C8—21no—
369C8—1272no—
370C8—2541no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—5081no—
378C22—5081no—
379C22—5081no—
380C22—2541no—
381C22—2541no—
382C22—2541no—
383C22C2×D4×D12711yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—5081no—
389C4—5081no—
390C4—2541no—
391C4—2541no—
392C4—2541no—
393C4—2542no—
394C4C2×D4×D12711yescharacteristic
395C4#1686125411yescharacteristic
396C2—41no—
397C2—42no—
398C2—10161no—
399C2—5081no—
400C2#2255050811yescharacteristic
401C2#2255067311yescharacteristic
402C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#452×C127C2
socleC22×C127C22×D4
hypercenter#452D127
hypocenterC127#3269
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. C2×#16861229C2semidirect
3. #22550536C2semidirect
4. #22550525C2semidirect
5. #22550269C2semidirect
6. #22550125C2semidirect
7. #22550115C2semidirect
8. #22550093C2semidirect
9. #22550085C2semidirect
10. #22550040C2semidirect
11. #22550030C2semidirect
12. #22550022C2semidirect
13. #22549973C2semidirect
14. #22549944C2semidirect
15. #452×C127C2semidirect
84. #14523236D4semidirect
202. C127#3269semidirect
213. #452D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C2×#168612292
#225505362
#225505252
#225502692
#225501252
#225501152
#225500932
#225500852
#225500402
#225500302
#225500222
#225499732
#225499442
#452×C1272
#3269127