Finite Group Databasebeta

← #30641974

Subgroups — #30641974

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×D508C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
4C2×#16861166C211yesmaximal, characteristic, complemented
5#22549810C211yesmaximal, characteristic, complemented
6#22549805C211yesmaximal, characteristic, complemented
7#22549800C211yesmaximal, characteristic, complemented
8#22549799C211yesmaximal, characteristic, complemented
9#22549794C211yesmaximal, characteristic, complemented
10#22549791C211yesmaximal, characteristic, complemented
11#22549791C211yesmaximal, characteristic, complemented
12#22549785C211yesmaximal, characteristic
13C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
14C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C22⋊D127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21C4×D508—24no—
22C4×C22⋊D127—24no—
23C4×C22⋊D127—24no—
24#16861166—24no—
25C2×C4⋊Dic127C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×Dic127⋊C4C2211yescharacteristic
28C2×Dic127⋊C4C2211yescharacteristic
29C2×Dic127⋊C4C2211yescharacteristic
30C2×Dic127⋊C4C2211yescharacteristic
31C2×Dic127⋊C4C2211yescharacteristic
32C2×C22⋊Dic127C2211yescharacteristic
33C2×C22⋊Dic127C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×(C2×D127)⋊C4C2211yescharacteristic
37C2×(C2×D127)⋊C4C2211yescharacteristic
38C2×(C2×D127)⋊C4C2211yescharacteristic
39C2×(C2×D127)⋊C4C2211yescharacteristic
40C2×(C2×D127)⋊C4C2211yescharacteristic
41C2×(C2×D127)⋊C4C2211yescharacteristic
42C2×C4×Dic127C2211yescharacteristic
43C2×C4×Dic127C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C127⋊#95C2211yescharacteristic
47C127⋊#95C2211yescharacteristic
48C23×C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C22⋊C4×C127C2211yescharacteristic
51C2×C42×C127C2211yescharacteristic
52C2×C42×C127C2211yescharacteristic
53#95×C127C2211yescharacteristic
54#95×C127C2211yescharacteristic
55C23×D127C2311yescharacteristic
56C23×D127C2311yescharacteristic
57C2×D508—22no—
58C2×D508—22no—
59C2×D508—22no—
60C2×C22⋊D127—22no—
61C2×C22⋊D127—22no—
62C2×C22⋊D127—22no—
63C2×C22⋊D127—22no—
64C2×C22⋊D127—22no—
65C2×C22⋊D127—22no—
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C22×Dic127C2311yescharacteristic
70C22×Dic127C2311yescharacteristic
71C22×Dic127C2311yescharacteristic
72C2×C4×D127—22no—
73C2×C4×D127—22no—
74C2×C4×D127—22no—
75C2×C4×D127—22no—
76C2×C4×D127—22no—
77C2×C4×D127D414yescomplemented
78C4⋊Dic127—22no—
79Dic127⋊C4—22no—
80Dic127⋊C4—22no—
81Dic127⋊C4—22no—
82Dic127⋊C4—22no—
83Dic127⋊C4—22no—
84Dic127⋊C4—22no—
85C22⋊Dic127—22no—
86C22⋊Dic127—22no—
87(C2×D127)⋊C4—22no—
88(C2×D127)⋊C4—22no—
89(C2×D127)⋊C4—22no—
90(C2×D127)⋊C4—22no—
91(C2×D127)⋊C4—22no—
92(C2×D127)⋊C4—22no—
93(C2×D127)⋊C4—22no—
94(C2×D127)⋊C4—22no—
95C4×Dic127—22no—
96C4×Dic127D414yescomplemented
97C24×C127C2311yescharacteristic
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127C2311yescharacteristic
102C22×C4×C127C2311yescharacteristic
103C22×C4×C127C2311yescharacteristic
104C22×C4×C127C2311yescharacteristic
105C22×C4×C127C2311yescharacteristic
106C22×C4×C127C2311yescharacteristic
107C22⋊C4×C127—22no—
108C22⋊C4×C127—22no—
109C42×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—21no—
118C22×D127—21no—
119C22×D127—21no—
120C22×D127—21no—
121C22×D127—21no—
122C22×D127—21no—
123C22×D127C2×D412yes—
124C22×D127#4112yes—
125D508—28no—
126C22⋊D127—28no—
127C22⋊D127—28no—
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×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127C2×D412yes—
143C2×Dic127C2×D412yes—
144C2×Dic127C2×D412yes—
145C2×Dic127#4112yes—
146C4×D127—28no—
147C4×D127—44no—
148C23×C127—21no—
149C23×C127—21no—
150C23×C127—21no—
151C23×C127—21no—
152C23×C127—21no—
153C23×C127—21no—
154C23×C127#4112yes—
155C23×C127C2411yescharacteristic
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
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—21no—
170C2×C4×C127—44no—
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—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127C22×D411yescharacteristic
183C22×C127C22×D411yescharacteristic
184C22×C127C2×#4111yescharacteristic
185C22×C127C2×#4111yescharacteristic
186C22×C127C2×#4111yescharacteristic
187C22×C127C2×#4111yescharacteristic
188C22×C127C2×#4111yescharacteristic
189C2×D127—22no—
190C2×D127—22no—
191C2×D127—22no—
192C2×D127—22no—
193C2×D127—22no—
194C2×D127—22no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—41no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—41no—
203C4×C127—22no—
204C4×C127—41no—
205C4×C127—41no—
206C4×C127—41no—
207C4×C127—41no—
208C4×C127#12114yes—
209Dic127—22no—
210Dic127—22no—
211Dic127—22no—
212Dic127—22no—
213Dic127—41no—
214Dic127—41no—
215C2×C127—42no—
216C2×C127C2×#12111yescharacteristic
217C2×C127#52511yescharacteristic
218C2×C127#52511yescharacteristic
219C2×C127#52511yescharacteristic
220C2×C127#52911yescharacteristic
221C2×C127#52911yescharacteristic
222C2×C127#53211yescharacteristic
223D127—42no—
224D127—42no—
225#2392—1271nomaximal, Sylow
226C127#239211yescharacteristic, complemented, Sylow
227C2×C4×D4—1271no—
228C2×C4×D4—1271no—
229C2×C4×D4—1271no—
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⋊C4D12711yescharacteristic, complemented
242C22×D4—1271no—
243C22×D4—1271no—
244C22×D4—1271no—
245C23×C4—1271no—
246C23×C4—1271no—
247C23×C4C2×D12711yescharacteristic
248C4×D4—2544no—
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⋊C4—1271no—
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×C42C2×D12711yescharacteristic
274C2×C42C2×D12711yescharacteristic
275#95—1271no—
276#95—1271no—
277#95—1271no—
278#95—1271no—
279#95C2×D12711yescharacteristic
280#95C2×D12711yescharacteristic
281C24—1271no—
282C24—1271no—
283C24C22×D12711yescharacteristic
284C2×D4—2542no—
285C2×D4—2542no—
286C2×D4—2542no—
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—1271no—
297C22×C4—1271no—
298C22×C4—1271no—
299C22×C4—1271no—
300C22×C4—1274no—
301C22×C4—1271no—
302C22×C4—1271no—
303C22×C4—2542no—
304C22×C4—2542no—
305C22×C4—2542no—
306C22×C4—2542no—
307C22×C4—2542no—
308C22×C4C22×D12711yescharacteristic
309C22×C4C22×D12711yescharacteristic
310C22×C4C22×D12711yescharacteristic
311C22×C4C22×D12711yescharacteristic
312C22×C4C22×D12711yescharacteristic
313C22×C4C22×D12711yescharacteristic
314C4⋊C4—2542no—
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—22no—
335C42—1274no—
336C42—2542no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—1272no—
344C23—1272no—
345C23—2541no—
346C23—2541no—
347C23—2541no—
348C23—2541no—
349C23—2541no—
350C23—2541no—
351C23—2541no—
352C23—2541no—
353C23—2541no—
354C23—2541no—
355C23—2541no—
356C23—2541no—
357C23#1452324812yes—
358C23C23×D12711yescharacteristic
359D4—2548no—
360D4—2548no—
361D4—2548no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—44no—
377C2×C4—1272no—
378C2×C4—1272no—
379C2×C4—1272no—
380C2×C4—1272no—
381C2×C4—5084no—
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×C4—2548no—
397C2×C4D4×D12712yes—
398C2×C4D4×D12712yes—
399C2×C4#1452324812yes—
400C2×C4#1452324812yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—5081no—
409C22—5081no—
410C22—5081no—
411C22—5081no—
412C22—5081no—
413C22—5081no—
414C22—5081no—
415C22—5081no—
416C22—2542no—
417C22—2542no—
418C22—2542no—
419C22—2542no—
420C22—2542no—
421C22—2542no—
422C22C2×D4×D12711yescharacteristic
423C22C2×D4×D12711yescharacteristic
424C22#41×D12711yescharacteristic
425C22#41×D12711yescharacteristic
426C22C2×#1452324811yescharacteristic
427C22C2×#1452324811yescharacteristic
428C22C2×#1452324811yescharacteristic
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—5081no—
435C4—5081no—
436C4—2542no—
437C4—2542no—
438C4—2542no—
439C4—2542no—
440C4#1686118214yes—
441C2—42no—
442C2—5082no—
443C2—5082no—
444C2C2×#1686118211yescharacteristic
445C2#2255038511yescharacteristic
446C2#2255038511yescharacteristic
447C2#2255038511yescharacteristic
448C2#2255040411yescharacteristic
449C2#2255041011yescharacteristic
450C2#2255042311yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC4×C22⋊C4×C127C2
prime radical#22549785C2
hypercenterC4×C22⋊C4D127
hypocenterC127#2392
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D508C2semidirect
2. C2×C4×C22⋊D127C2semidirect
3. C2×C4×C22⋊D127C2semidirect
4. C2×#16861166C2semidirect
5. #22549810C2semidirect
6. #22549805C2semidirect
7. #22549800C2semidirect
8. #22549799C2semidirect
9. #22549794C2semidirect
10. #22549791C2semidirect
11. #22549791C2semidirect
13. C4×Dic127⋊C4C2semidirect
14. C4×(C2×D127)⋊C4C2semidirect
15. C4×C22⋊C4×C127C2semidirect
77. C2×C4×D127D4semidirect
96. C4×Dic127D4semidirect
226. C127#2392semidirect
241. C4×C22⋊C4D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D5082
C2×C4×C22⋊D1272
C2×#168611662
#225498102
#225498052
#225498002
#225497992
#225497942
#225497912
#225497852
C4×Dic127⋊C42
C4×(C2×D127)⋊C42
C4×C22⋊C4×C1272
#2392127