Finite Group Databasebeta

← #30641972

Subgroups — #30641972

Order 16256.

696 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
3C2×#16861168C211yesmaximal, characteristic, complemented
4#22549809C211yesmaximal, characteristic, complemented
5#22549804C211yesmaximal, characteristic, complemented
6#22549799C211yesmaximal, characteristic, complemented
7#22549798C211yesmaximal, characteristic, complemented
8#22549792C211yesmaximal, characteristic, complemented
9#22549791C211yesmaximal, characteristic, complemented
10#22549790C211yesmaximal, characteristic, complemented
11#22549784C211yesmaximal, characteristic
12C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
13C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
15C2×C4×D4×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19C4×C22⋊D127—24no—
20C4×C22⋊D127—24no—
21#16861168—24no—
22C2×C4⋊Dic127C2211yescharacteristic
23C2×C4⋊Dic127C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×Dic127⋊C4C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C22⋊Dic127C2211yescharacteristic
30C2×C22⋊Dic127C2211yescharacteristic
31C2×C22⋊Dic127C2211yescharacteristic
32C2×C22⋊Dic127C2211yescharacteristic
33C2×C22⋊Dic127C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×(C2×D127)⋊C4C2211yescharacteristic
37C2×(C2×D127)⋊C4C2211yescharacteristic
38C2×C4×Dic127C2211yescharacteristic
39C2×C4×Dic127C2211yescharacteristic
40C2×C4×Dic127C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C127⋊#95C2211yescharacteristic
47C22×D4×C127C2211yescharacteristic
48C23×C4×C127C2211yescharacteristic
49C23×C4×C127C2211yescharacteristic
50C4×D4×C127—24no—
51C2×C4⋊C4×C127C2211yescharacteristic
52C2×C22⋊C4×C127C2211yescharacteristic
53C2×C22⋊C4×C127C2211yescharacteristic
54C2×C42×C127C2211yescharacteristic
55C23×D127C2311yescharacteristic
56C2×C22⋊D127—22no—
57C2×C22⋊D127—22no—
58C2×C22⋊D127—22no—
59C2×C22⋊D127—22no—
60C2×C22⋊D127—22no—
61C2×C22⋊D127—22no—
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C2×C4×D127—22no—
70C2×C4×D127—22no—
71C2×C4×D127—22no—
72C4⋊Dic127—22no—
73C4⋊Dic127—22no—
74Dic127⋊C4—22no—
75Dic127⋊C4—22no—
76Dic127⋊C4—22no—
77Dic127⋊C4—22no—
78C22⋊Dic127—22no—
79C22⋊Dic127—22no—
80C22⋊Dic127—22no—
81C22⋊Dic127—22no—
82C22⋊Dic127—22no—
83C22⋊Dic127—22no—
84(C2×D127)⋊C4—22no—
85(C2×D127)⋊C4—22no—
86(C2×D127)⋊C4—22no—
87(C2×D127)⋊C4—22no—
88C4×Dic127—22no—
89C4×Dic127—22no—
90C4×Dic127D414yescomplemented
91C24×C127C2311yescharacteristic
92C24×C127C2311yescharacteristic
93C2×D4×C127—22no—
94C2×D4×C127—22no—
95C2×D4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127D414yescomplemented
102C22×C4×C127C2311yescharacteristic
103C22×C4×C127C2311yescharacteristic
104C22×C4×C127C2311yescharacteristic
105C22×C4×C127C2311yescharacteristic
106C22×C4×C127C2311yescharacteristic
107C4⋊C4×C127—22no—
108C22⋊C4×C127—22no—
109C22⋊C4×C127—22no—
110C42×C127—22no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—21no—
115C22×D127—21no—
116C22×D127—21no—
117C22×D127#4112yes—
118C22⋊D127—28no—
119C22⋊D127—28no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127C2×D412yes—
139C2×Dic127C2×D412yes—
140C2×Dic127#4112yes—
141C4×D127—44no—
142C23×C127—21no—
143C23×C127—21no—
144C23×C127—21no—
145C23×C127—21no—
146C23×C127—21no—
147C23×C127—21no—
148C23×C127—21no—
149C23×C127—21no—
150C23×C127—21no—
151C23×C127—21no—
152C23×C127—21no—
153C23×C127—21no—
154C23×C127C2×D412yes—
155C23×C127#4112yes—
156C23×C127C2411yescharacteristic
157D4×C127—28no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—28no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—44no—
170C2×C4×C127C2×D412yes—
171C2×C4×C127C2×D412yes—
172C2×C4×C127C2×D412yes—
173C2×C4×C127#4112yes—
174C2×C4×C127#4112yes—
175C22×C127—22no—
176C22×C127—22no—
177C22×C127—22no—
178C22×C127—22no—
179C22×C127—22no—
180C22×C127—22no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127C22×D411yescharacteristic
190C22×C127C22×D411yescharacteristic
191C22×C127C2×#4111yescharacteristic
192C22×C127C2×#4111yescharacteristic
193C22×C127C2×#4111yescharacteristic
194C22×C127C2×#4111yescharacteristic
195C22×C127C2×#4111yescharacteristic
196C2×D127—22no—
197C2×D127—22no—
198C2×D127—22no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—41no—
203C4×C127—22no—
204C4×C127—22no—
205C4×C127—41no—
206C4×C127—41no—
207C4×C127#12114yes—
208Dic127—22no—
209Dic127—22no—
210Dic127—22no—
211Dic127—41no—
212Dic127—41no—
213Dic127—41no—
214Dic127—41no—
215C2×C127—42no—
216C2×C127—42no—
217C2×C127C2×#12111yescharacteristic
218C2×C127#52511yescharacteristic
219C2×C127#52511yescharacteristic
220C2×C127#52511yescharacteristic
221C2×C127#52911yescharacteristic
222C2×C127#52911yescharacteristic
223C2×C127#53211yescharacteristic
224D127—42no—
225#2392—1271nomaximal, Sylow
226C127#239211yescharacteristic, complemented, Sylow
227C2×C4×D4—1271no—
228C2×C4×D4—1271no—
229C2×C4×D4D12711yescharacteristic, complemented
230C2×#117—1271no—
231#398—1271no—
232#397—1271no—
233#396—1271no—
234#396—1271no—
235#394—1271no—
236#394—1271no—
237#394—1271no—
238#392—1271no—
239C4×C4⋊C4—1271no—
240C4×C22⋊C4—1271no—
241C4×C22⋊C4—1271no—
242C22×D4—1271no—
243C22×D4—1271no—
244C22×D4C2×D12711yescharacteristic
245C23×C4—1271no—
246C23×C4C2×D12711yescharacteristic
247C23×C4C2×D12711yescharacteristic
248C4×D4—24no—
249C4×D4—2544no—
250C4×D4—2544no—
251#117—2544no—
252C2×C4⋊C4—1271no—
253C2×C4⋊C4—1271no—
254C2×C4⋊C4—1271no—
255C2×C4⋊C4—1271no—
256C2×C4⋊C4—1271no—
257C2×C4⋊C4—1271no—
258C2×C4⋊C4C2×D12711yescharacteristic
259C2×C22⋊C4—1271no—
260C2×C22⋊C4—1271no—
261C2×C22⋊C4—1271no—
262C2×C22⋊C4—1271no—
263C2×C22⋊C4—1271no—
264C2×C22⋊C4—1271no—
265C2×C22⋊C4—1271no—
266C2×C22⋊C4—1271no—
267C2×C22⋊C4—1271no—
268C2×C22⋊C4—1271no—
269C2×C22⋊C4C2×D12711yescharacteristic
270C2×C22⋊C4C2×D12711yescharacteristic
271C2×C42—1271no—
272C2×C42—1271no—
273C2×C42—1271no—
274C2×C42C2×D12711yescharacteristic
275#95—1271no—
276#95—1271no—
277#95—1271no—
278#95—1271no—
279#95—1271no—
280#95—1271no—
281C24—1271no—
282C24C22×D12711yescharacteristic
283C24C22×D12711yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—2542no—
288C2×D4—2542no—
289C2×D4—2542no—
290C2×D4—2542no—
291C2×D4—2542no—
292C2×D4—2542no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—1271no—
299C22×C4—1271no—
300C22×C4—1271no—
301C22×C4—1271no—
302C22×C4—1271no—
303C22×C4—1271no—
304C22×C4—1271no—
305C22×C4—2542no—
306C22×C4—2542no—
307C22×C4—2542no—
308C22×C4C22⋊D12714yescomplemented
309C22×C4C22×D12711yescharacteristic
310C22×C4C22×D12711yescharacteristic
311C22×C4C22×D12711yescharacteristic
312C22×C4C22×D12711yescharacteristic
313C22×C4C22×D12711yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—2542no—
316C4⋊C4—2542no—
317C4⋊C4—2542no—
318C4⋊C4—2542no—
319C4⋊C4—2542no—
320C4⋊C4—2542no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—2542no—
324C22⋊C4—2542no—
325C22⋊C4—2542no—
326C22⋊C4—2542no—
327C22⋊C4—2542no—
328C22⋊C4—2542no—
329C22⋊C4—2542no—
330C22⋊C4—2542no—
331C22⋊C4—2542no—
332C22⋊C4—2542no—
333C42—22no—
334C42—1274no—
335C42—2542no—
336C42—2542no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—1272no—
350C23—2541no—
351C23—2541no—
352C23—2541no—
353C23—2541no—
354C23—2541no—
355C23—2541no—
356C23C2×C22⋊D12712yes—
357C23#1452324812yes—
358C23C23×D12711yescharacteristic
359D4—28no—
360D4—2548no—
361D4—2548no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—28no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—1272no—
375C2×C4—1272no—
376C2×C4—1272no—
377C2×C4—5084no—
378C2×C4—2541no—
379C2×C4—2541no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4—2541no—
385C2×C4—2541no—
386C2×C4—2541no—
387C2×C4—2541no—
388C2×C4—2541no—
389C2×C4—2541no—
390C2×C4—2541no—
391C2×C4—2541no—
392C2×C4—2541no—
393C2×C4—2541no—
394C2×C4—2541no—
395C2×C4—2541no—
396C2×C4D4×D12712yes—
397C2×C4C2×C22⋊D12712yes—
398C2×C4C2×C22⋊D12712yes—
399C2×C4#1452324712yes—
400C2×C4#1452324812yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—5081no—
416C22—5081no—
417C22—5081no—
418C22—5081no—
419C22—2542no—
420C22—2542no—
421C22—2542no—
422C22C2×D4×D12711yescharacteristic
423C22C22×C22⋊D12711yescharacteristic
424C22#41×D12711yescharacteristic
425C22#41×D12711yescharacteristic
426C22C2×#1452324711yescharacteristic
427C22C2×#1452324811yescharacteristic
428C22C2×#1452324811yescharacteristic
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—5081no—
434C4—5081no—
435C4—5081no—
436C4—5081no—
437C4—2542no—
438C4—2542no—
439C4—2542no—
440C4#1686118114yes—
441C2—42no—
442C2—42no—
443C2—5082no—
444C2C2×#1686118111yescharacteristic
445C2#2255038311yescharacteristic
446C2#2255038511yescharacteristic
447C2#2255038711yescharacteristic
448C2#2255040611yescharacteristic
449C2#2255040811yescharacteristic
450C2#2255042311yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC2×C4×D4×C127C2
prime radical#22549784C2
hypercenterC2×C4×D4D127
hypocenterC127#2392
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D127C2semidirect
2. C2×C4×C22⋊D127C2semidirect
3. C2×#16861168C2semidirect
4. #22549809C2semidirect
5. #22549804C2semidirect
6. #22549799C2semidirect
7. #22549798C2semidirect
8. #22549792C2semidirect
9. #22549791C2semidirect
10. #22549790C2semidirect
12. C4×Dic127⋊C4C2semidirect
13. C4×(C2×D127)⋊C4C2semidirect
14. C4×C22⋊Dic127C2semidirect
15. C2×C4×D4×C127C2semidirect
90. C4×Dic127D4semidirect
101. C22×C4×C127D4semidirect
226. C127#2392semidirect
229. C2×C4×D4D127semidirect
308. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1272
C2×#168611682
#225498092
#225498042
#225497992
#225497982
#225497922
#225497912
#225497902
#225497842
C4×Dic127⋊C42
C4×(C2×D127)⋊C42
C4×C22⋊Dic1272
C2×C4×D4×C1272
#2392127