Finite Group Databasebeta

← #30650281

Subgroups — #30650281

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×#16861239C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550535C211yesmaximal, characteristic, complemented
4#22550268C211yesmaximal, characteristic, complemented
5#22550124C211yesmaximal, characteristic, complemented
6#22550115C211yesmaximal, characteristic, complemented
7#22550091C211yesmaximal, characteristic, complemented
8#22550083C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550036C211yesmaximal, characteristic, complemented
11#22550031C211yesmaximal, characteristic, complemented
12#22550021C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15#547×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18#16861239—24no—
19#16861229—24no—
20C2×#14523238C2211yescharacteristic
21C2×#14523238C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523235C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523232C2211yescharacteristic
30C22×C127⋊C8C2211yescharacteristic
31#16861197C2211yescharacteristic
32#16861195—21no—
33#16861188C2211yescharacteristic
34#16861187—21no—
35#16861183C2211yescharacteristic
36#16861181—21no—
37C4×Dic254C2211yescharacteristic
38(C22⋊D127)⋊C4—21no—
39#16861142C2211yescharacteristic
40C4⋊(C127⋊C8)C2211yescharacteristic
41#16861133C2211yescharacteristic
42#16861133C2211yescharacteristic
43#16861132C2211yescharacteristic
44#16861132C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861129C2211yescharacteristic
47C22⋊(C127⋊C8)C2211yescharacteristic
48C22⋊(C127⋊C8)C2211yescharacteristic
49#16861120C2211yescharacteristic
50C2×#41×C127C2211yescharacteristic
51C2×#41×C127C2211yescharacteristic
52C4⋊Q8×C127C2211yescharacteristic
53#123×C127—21no—
54#123×C127—21no—
55C22⋊Q8×C127C2211yescharacteristic
56C22⋊Q8×C127C2211yescharacteristic
57#121×C127—21no—
58C4×D4×C127C2211yescharacteristic
59C4×D4×C127C2211yescharacteristic
60C2×C4⋊C4×C127—21no—
61#14523248—22no—
62#14523248—41no—
63C2×Dic254—22no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73#14523238—22no—
74#14523238—22no—
75#14523238—22no—
76#14523238—41no—
77#14523236—22no—
78#14523236D414yescomplemented
79#14523235—22no—
80#14523235—41no—
81#14523233—22no—
82#14523232—22no—
83C2×C127⋊C8—22no—
84C2×C127⋊C8C2311yescharacteristic
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C2×C127⋊C8C2311yescharacteristic
88C4⋊Dic127C2311yescharacteristic
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95C4×Dic127—21no—
96#41×C127—22no—
97#41×C127—22no—
98#41×C127—41no—
99#41×C127—41no—
100C2×Q8×C127C2311yescharacteristic
101C2×Q8×C127C2311yescharacteristic
102C2×D4×C127—21no—
103C2×D4×C127—21no—
104C2×D4×C127D411yescharacteristic
105C2×D4×C127C2311yescharacteristic
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111C4⋊C4×C127—21no—
112C4⋊C4×C127—21no—
113C4⋊C4×C127—21no—
114C4⋊C4×C127—21no—
115C4⋊C4×C127D411yescharacteristic
116C4⋊C4×C127C2311yescharacteristic
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—42no—
128Dic254C2×D412yes—
129D508—21no—
130D508—21no—
131C22⋊D127—42no—
132C22⋊D127—42no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—42no—
137C4×D127—42no—
138C127⋊C8—21no—
139C127⋊C8—21no—
140C127⋊C8—21no—
141C127⋊C8C2×D412yes—
142C23×C127—21no—
143C23×C127C2×D411yescharacteristic
144C23×C127C2×D411yescharacteristic
145Q8×C127—21no—
146Q8×C127—21no—
147Q8×C127—21no—
148Q8×C127—21no—
149D4×C127—21no—
150D4×C127—41no—
151D4×C127—42no—
152D4×C127—42no—
153D4×C127C2×D412yes—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—42no—
162C2×C4×C127—42no—
163C2×C4×C127—42no—
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2×D411yescharacteristic
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2411yescharacteristic
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—22no—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—21no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127C22×D411yescharacteristic
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C4×C127—21no—
183C4×C127—21no—
184C4×C127—21no—
185C4×C127—21no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127—41no—
189C4×C127C22×D411yescharacteristic
190C4×C127H(5,𝔽2)11yescharacteristic
191Dic127—22no—
192Dic127—41no—
193Dic127—41no—
194C2×C127—41no—
195C2×C127—41no—
196C2×C127—42no—
197C2×C127D4211yescharacteristic
198C2×C127#57511yescharacteristic
199C2×C127#57811yescharacteristic
200D127—81no—
201#3269—1271nomaximal, Sylow
202C127#326911yescharacteristic, complemented, Sylow
203C2×#137—1271no—
204C2×#135—1271no—
205#548—1271no—
206#547D12711yescharacteristic, complemented
207#497—1271no—
208#468—1271no—
209#467—1271no—
210#463—1271no—
211#462—1271no—
212#452—1271no—
213#452—1271no—
214#451—1271no—
215#449—1271no—
216#440—1271no—
217#435—1271no—
218C2×#41—1271no—
219C2×#41C2×D12711yescharacteristic
220C2×#41C2×D12711yescharacteristic
221C22×Q8—1271no—
222#137—2544no—
223#135—2544no—
224C2×Q16—1271no—
225C2×Q16—1271no—
226C2×Q16—1271no—
227C2×Q16—1271no—
228C2×SD16—1271no—
229C2×SD16—1271no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×D8—1271no—
233C2×M16—1271no—
234C22×C8—1271no—
235C4⋊Q8C2×D12711yescharacteristic
236#124—1271no—
237#124—2541no—
238#123—21no—
239#123—21no—
240C22⋊Q8—1271no—
241C22⋊Q8—2541no—
242C22⋊Q8C2×D12711yescharacteristic
243C22⋊Q8C2×D12711yescharacteristic
244#121—21no—
245#121—1271no—
246#121—2541no—
247C4×Q8—1271no—
248C4×D4—2541no—
249C4×D4C2×D12711yescharacteristic
250C4×D4C2×D12711yescharacteristic
251C2×C4⋊C4—21no—
252#107—1271no—
253C4⋊C8—1271no—
254#103—1271no—
255#103—1271no—
256#103—1271no—
257#103—1271no—
258#102—1271no—
259#102—1271no—
260C22⋊C8—1271no—
261C22⋊C8—1271no—
262#97—1271no—
263#41—22no—
264#41—22no—
265#41—41no—
266#41—41no—
267#41—5081no—
268#41—2542no—
269C2×Q8—1271no—
270C2×Q8—1271no—
271C2×Q8—5081no—
272C2×Q8—2542no—
273C2×Q8—2541no—
274C2×Q8C22×D12711yescharacteristic
275C2×Q8C22×D12711yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—1271no—
279C2×D4—2541no—
280C2×D4—2541no—
281C2×D4C22⋊D12711yescharacteristic
282C2×D4C22×D12711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—2541no—
287C22×C4—2541no—
288C22×C4C22×D12711yescharacteristic
289C22×C4C22×D12711yescharacteristic
290Q16—5081no—
291Q16—2542no—
292Q16—2542no—
293Q16—2542no—
294SD16—1274no—
295SD16—5081no—
296SD16—2542no—
297SD16—2542no—
298D8—2542no—
299M16—2542no—
300C2×C8—1271no—
301C2×C8—1271no—
302C2×C8—1271no—
303C2×C8—1271no—
304C2×C8—2542no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—1271no—
310C4⋊C4—2541no—
311C4⋊C4—2541no—
312C4⋊C4C22⋊D12711yescharacteristic
313C4⋊C4C22×D12711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C22⋊C4C22⋊D12711yescharacteristic
322C22⋊C4C22⋊D12711yescharacteristic
323C42—2541no—
324C42C22×D12711yescharacteristic
325C23—21no—
326C23—2541no—
327C23C2×C22⋊D12711yescharacteristic
328C23C2×C22⋊D12711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—1272no—
334Q8—5082no—
335Q8—2541no—
336Q8—2541no—
337Q8—2541no—
338D4—21no—
339D4—41no—
340D4—42no—
341D4—42no—
342D4—5082no—
343D4—5082no—
344D4—2541no—
345D4—2541no—
346D4D4×D12712yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—5082no—
358C2×C4—5082no—
359C2×C4—2541no—
360C2×C4—2541no—
361C2×C4—2541no—
362C2×C4C2×C22⋊D12711yescharacteristic
363C2×C4C2×C22⋊D12711yescharacteristic
364C2×C4C2×C22⋊D12711yescharacteristic
365C2×C4C2×C22⋊D12711yescharacteristic
366C2×C4C23×D12711yescharacteristic
367C8—1272no—
368C8—2541no—
369C8—2541no—
370C8—2541no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—5081no—
381C22—5081no—
382C22—5081no—
383C22C22×C22⋊D12711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—5081no—
392C4—5081no—
393C4—2542no—
394C4C2×D4×D12711yescharacteristic
395C4#1686125411yescharacteristic
396C2—41no—
397C2—41no—
398C2—42no—
399C2—10161no—
400C2D4×C22⋊D12711yescharacteristic
401C2#2255067411yescharacteristic
402C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#547×C127C2
socleC22×C127C22×D4
hypercenter#547D127
hypocenterC127#3269
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861229C2semidirect
3. #22550535C2semidirect
4. #22550268C2semidirect
5. #22550124C2semidirect
6. #22550115C2semidirect
7. #22550091C2semidirect
8. #22550083C2semidirect
9. #22550038C2semidirect
10. #22550036C2semidirect
11. #22550031C2semidirect
12. #22550021C2semidirect
13. #22549976C2semidirect
14. #22549948C2semidirect
15. #547×C127C2semidirect
78. #14523236D4semidirect
202. C127#3269semidirect
206. #547D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612292
#225505352
#225502682
#225501242
#225501152
#225500912
#225500832
#225500382
#225500362
#225500312
#225500212
#225499762
#225499482
#547×C1272
#3269127