Finite Group Databasebeta

← #27173853

Subgroups — #27173853

Order 12928.

740 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692251C211yesmaximal, characteristic
2C2×C4×C22⋊D101C211yesmaximal, characteristic, complemented
3C2×#15692215C211yesmaximal, characteristic, complemented
4#20265077C211yesmaximal, characteristic, complemented
5#20265077C211yesmaximal, characteristic, complemented
6#20265057C211yesmaximal, characteristic, complemented
7#20265056C211yesmaximal, characteristic, complemented
8#20265055C211yesmaximal, characteristic, complemented
9#20265055C211yesmaximal, characteristic, complemented
10#20265054C211yesmaximal, characteristic, complemented
11#20265053C211yesmaximal, characteristic, complemented
12#20265053C211yesmaximal, characteristic, complemented
13#20265041C211yesmaximal, characteristic, complemented
14#20265022C211yesmaximal, characteristic, complemented
15#386×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C22×C22⋊D101C2211yescharacteristic
20C22×C22⋊D101C2211yescharacteristic
21C23×Dic101C2211yescharacteristic
22C22×C4×D101C2211yescharacteristic
23C22×C4×D101C2211yescharacteristic
24#15692251C418yescomplemented
25C4×C22⋊D101—24no—
26#15692215—24no—
27C2×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×C22⋊Dic101C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47C22×D4×C101C2211yescharacteristic
48C23×C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54C23×D101C2311yescharacteristic
55C23×D101C2311yescharacteristic
56C2×D404—41no—
57C2×D404—41no—
58C2×D404C2×C414yescomplemented
59C2×C22⋊D101—22no—
60C2×C22⋊D101—22no—
61C2×C22⋊D101—22no—
62C2×C22⋊D101—22no—
63C2×C22⋊D101—41no—
64C2×C22⋊D101—41no—
65C2×C22⋊D101—41no—
66C2×C22⋊D101—41no—
67C2×C22⋊D101C2×C414yescomplemented
68C2×C22⋊D101C2×C414yescomplemented
69C2×C22⋊D101C2×C414yescomplemented
70C2×C22⋊D101C2×C414yescomplemented
71C22×Dic101—22no—
72C22×Dic101—41no—
73C22×Dic101—41no—
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C22×Dic101C2311yescharacteristic
79C22×Dic101C2311yescharacteristic
80C2×C4×D101—22no—
81C2×C4×D101—22no—
82C2×C4×D101—41no—
83C2×C4×D101—41no—
84C2×C4×D101—41no—
85C2×C4×D101—41no—
86#13108053—22no—
87#13108053—22no—
88C22⋊Dic101—22no—
89C22⋊Dic101—41no—
90C22⋊Dic101—41no—
91#13108051—22no—
92#13108051—22no—
93#13108051—41no—
94#13108051—41no—
95#13108051—41no—
96#13108051—41no—
97C4×Dic101—22no—
98C4×Dic101C2×C414yescomplemented
99C4×Dic101D414yescomplemented
100C24×C101C2311yescharacteristic
101C24×C101C2311yescharacteristic
102C2×D4×C101—41no—
103C2×D4×C101—41no—
104C2×D4×C101C2×C414yescomplemented
105C22×C4×C101—22no—
106C22×C4×C101—41no—
107C22×C4×C101—41no—
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113C22⋊C4×C101—22no—
114C22⋊C4×C101—41no—
115C22⋊C4×C101—41no—
116C23×C101—21no—
117C23×C101—21no—
118C23×C101—21no—
119C23×C101—21no—
120C23×C101—21no—
121C23×C101—21no—
122C23×C101—21no—
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—21no—
128C23×C101C22×C412yes—
129C23×C101C22×C412yes—
130C23×C101C2411yescharacteristic
131C22×D101—21no—
132C22×D101—21no—
133C22×D101—21no—
134C22×D101—21no—
135C22×D101—21no—
136C22×D101—21no—
137C22×D101—21no—
138C22×D101—21no—
139C22×D101—21no—
140C22×D101—21no—
141C22×D101—21no—
142C22×D101—21no—
143C22×D101C22×C412yes—
144C22×D101C22×C412yes—
145D4×C101—44no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—21no—
150C2×C4×C101—21no—
151C2×C4×C101—21no—
152C2×C4×C101—21no—
153C2×C4×C101—21no—
154C2×C4×C101—21no—
155C2×C4×C101—21no—
156C2×C4×C101—21no—
157C2×C4×C101—21no—
158C2×C4×C101—21no—
159C2×C4×C101—44no—
160C2×C4×C101C22×C412yes—
161C2×C4×C101C2×D412yes—
162C22⋊D101—28no—
163C22⋊D101—28no—
164C22⋊D101—44no—
165C22⋊D101—44no—
166C2×Dic101—21no—
167C2×Dic101—21no—
168C2×Dic101—21no—
169C2×Dic101—21no—
170C2×Dic101—21no—
171C2×Dic101—21no—
172C2×Dic101—21no—
173C2×Dic101—21no—
174C2×Dic101—21no—
175C2×Dic101—21no—
176C2×Dic101—21no—
177C2×Dic101—21no—
178C2×Dic101—21no—
179C2×Dic101—44no—
180C2×Dic101C22×C412yes—
181C2×Dic101C22×C412yes—
182C2×Dic101C2×D412yes—
183C2×Dic101C2×D412yes—
184C2×Dic101#4112yes—
185D404—44no—
186C4×D101—44no—
187C4×D101—44no—
188C22×C101—22no—
189C22×C101—22no—
190C22×C101—22no—
191C22×C101—22no—
192C22×C101—22no—
193C22×C101—22no—
194C22×C101—41no—
195C22×C101—41no—
196C22×C101—41no—
197C22×C101—41no—
198C22×C101—41no—
199C22×C101—41no—
200C22×C101—41no—
201C22×C101—41no—
202C22×C101C23×C411yescharacteristic
203C22×C101C22×D411yescharacteristic
204C22×C101C2×#4111yescharacteristic
205C22×C101H(5,𝔽2)11yescharacteristic
206C22×C101H(5,𝔽2)11yescharacteristic
207C22×C101H(5,𝔽2)11yescharacteristic
208C22×C101H(5,𝔽2)11yescharacteristic
209C2×D101—22no—
210C2×D101—22no—
211C2×D101—22no—
212C2×D101—22no—
213C2×D101—22no—
214C2×D101—22no—
215C2×D101—41no—
216C2×D101—41no—
217C2×D101—41no—
218C2×D101—41no—
219C2×D101—41no—
220C2×D101—41no—
221C2×D101—41no—
222C2×D101—41no—
223C4×C101—22no—
224C4×C101—22no—
225C4×C101—41no—
226C4×C101—41no—
227C4×C101—41no—
228Dic101—22no—
229Dic101—22no—
230Dic101—41no—
231Dic101—41no—
232Dic101—41no—
233Dic101C4×D414yes—
234C2×C101—42no—
235C2×C101—42no—
236C2×C101C2×C4×D411yescharacteristic
237C2×C101#51811yescharacteristic
238C2×C101#51811yescharacteristic
239C2×C101#53511yescharacteristic
240C2×C101#53511yescharacteristic
241C2×C101#54011yescharacteristic
242C2×C101#54011yescharacteristic
243D101—42no—
244D101—42no—
245#2328—1011nomaximal, Sylow
246C101#232811yescharacteristic, complemented, Sylow
247C2×#127—1011no—
248C2×C4×D4—1011no—
249C2×C4×D4—1011no—
250#390—1011no—
251#390—1011no—
252#386—1011no—
253#386—1011no—
254#386—1011no—
255#386—1011no—
256#386—1011no—
257#386—1011no—
258#386—1011no—
259#386D10111yescharacteristic, complemented
260#382—1011no—
261#376—1011no—
262C22×D4—1011no—
263C22×D4—1011no—
264C22×D4—1011no—
265C22×D4—1011no—
266C22×D4—1011no—
267C22×D4C2×D10111yescharacteristic
268C23×C4—1011no—
269C23×C4—1011no—
270C23×C4—1011no—
271C23×C4C2×D10111yescharacteristic
272#127—1018no—
273C4×D4—2024no—
274C4×D4—2024no—
275C2×C4⋊C4—1011no—
276C2×C4⋊C4—1011no—
277C2×C22⋊C4—1011no—
278C2×C22⋊C4—1011no—
279C2×C22⋊C4—1011no—
280C2×C22⋊C4—1011no—
281C2×C22⋊C4—1011no—
282C2×C22⋊C4—1011no—
283C2×C22⋊C4—1011no—
284C2×C22⋊C4—1011no—
285C2×C22⋊C4—1011no—
286C2×C22⋊C4C2×D10111yescharacteristic
287C2×C22⋊C4C2×D10111yescharacteristic
288C2×C22⋊C4C2×D10111yescharacteristic
289C2×C42—1011no—
290C2×C42—1011no—
291C2×C42—1011no—
292#95—1011no—
293#95—1011no—
294#95—1011no—
295#95—1011no—
296#95—1011no—
297#95—1011no—
298#95C2×D10111yescharacteristic
299#95C2×D10111yescharacteristic
300C24—1011no—
301C24—1011no—
302C24C22×D10111yescharacteristic
303C24C22×D10111yescharacteristic
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—1014no—
307C2×D4—1014no—
308C2×D4—1014no—
309C2×D4—1014no—
310C2×D4—1014no—
311C2×D4—4041no—
312C2×D4—4041no—
313C2×D4—4041no—
314C2×D4—4041no—
315C2×D4—4041no—
316C2×D4—4041no—
317C2×D4—2022no—
318C2×D4—2022no—
319C2×D4—2022no—
320C2×D4—2022no—
321C2×D4C4×D10114yescomplemented
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—1011no—
326C22×C4—1011no—
327C22×C4—1011no—
328C22×C4—1011no—
329C22×C4—1011no—
330C22×C4—1011no—
331C22×C4—4041no—
332C22×C4—4041no—
333C22×C4—4041no—
334C22×C4—4041no—
335C22×C4—4041no—
336C22×C4—4041no—
337C22×C4—2022no—
338C22×C4—2022no—
339C22×C4—2022no—
340C22×C4C22×D10111yescharacteristic
341C22×C4C22×D10111yescharacteristic
342C22×C4C22×D10111yescharacteristic
343C22×C4C22×D10111yescharacteristic
344C22×C4C22×D10111yescharacteristic
345C4⋊C4—2022no—
346C4⋊C4—2022no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—4041no—
351C22⋊C4—4041no—
352C22⋊C4—4041no—
353C22⋊C4—4041no—
354C22⋊C4—4041no—
355C22⋊C4—4041no—
356C22⋊C4—2022no—
357C22⋊C4—2022no—
358C22⋊C4—2022no—
359C42—1014no—
360C42—1014no—
361C42—2022no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—1012no—
375C23—1012no—
376C23—2021no—
377C23—2021no—
378C23—2021no—
379C23—2021no—
380C23—2021no—
381C23—2021no—
382C23—2021no—
383C23—2021no—
384C23—2021no—
385C23—2021no—
386C23—2021no—
387C23—2021no—
388C23C2×C4×D10112yes—
389C23C2×C4×D10112yes—
390C23C23×D10111yescharacteristic
391D4—44no—
392D4—4044no—
393D4—4044no—
394D4—4044no—
395D4—2028no—
396D4—2028no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—44no—
411C2×C4—1012no—
412C2×C4—1012no—
413C2×C4—1012no—
414C2×C4—1012no—
415C2×C4—1012no—
416C2×C4—4044no—
417C2×C4—4044no—
418C2×C4—4044no—
419C2×C4—2021no—
420C2×C4—2021no—
421C2×C4—2021no—
422C2×C4—2021no—
423C2×C4—2021no—
424C2×C4—2021no—
425C2×C4—2021no—
426C2×C4—2021no—
427C2×C4—2021no—
428C2×C4—2021no—
429C2×C4—2021no—
430C2×C4—2021no—
431C2×C4—2021no—
432C2×C4C2×C4×D10112yes—
433C2×C4D4×D10112yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—4041no—
449C22—4041no—
450C22—4041no—
451C22—4041no—
452C22—4041no—
453C22—4041no—
454C22—4041no—
455C22—4041no—
456C22—2022no—
457C22—2022no—
458C22—2022no—
459C22—2022no—
460C22—2022no—
461C22—2022no—
462C22C22×C4×D10111yescharacteristic
463C22C2×D4×D10111yescharacteristic
464C22#41×D10111yescharacteristic
465C22#1569230011yescharacteristic
466C22#1569230011yescharacteristic
467C22#1569230011yescharacteristic
468C22#1569230111yescharacteristic
469C4—22no—
470C4—22no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—1014no—
475C4—4041no—
476C4—4041no—
477C4—4041no—
478C4—2022no—
479C4—2022no—
480C2—42no—
481C2—42no—
482C2—4042no—
483C2—4042no—
484C2C4×D4×D10111yescharacteristic
485C2#2026564911yescharacteristic
486C2#2026565011yescharacteristic
487C2#2026574211yescharacteristic
488C2#2026574311yescharacteristic
489C2#2026577711yescharacteristic
490C2#2026577911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#386×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692251C2
hypercenter#386D101
hypocenterC101#2328
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D101C2semidirect
3. C2×#15692215C2semidirect
4. #20265077C2semidirect
5. #20265077C2semidirect
6. #20265057C2semidirect
7. #20265056C2semidirect
8. #20265055C2semidirect
9. #20265055C2semidirect
10. #20265054C2semidirect
11. #20265053C2semidirect
12. #20265053C2semidirect
13. #20265041C2semidirect
14. #20265022C2semidirect
15. #386×C101C2semidirect
24. #15692251C4semidirect
58. C2×D404C2×C4semidirect
67. C2×C22⋊D101C2×C4semidirect
68. C2×C22⋊D101C2×C4semidirect
69. C2×C22⋊D101C2×C4semidirect
70. C2×C22⋊D101C2×C4semidirect
98. C4×Dic101C2×C4semidirect
99. C4×Dic101D4semidirect
104. C2×D4×C101C2×C4semidirect
246. C101#2328semidirect
259. #386D101semidirect
321. C2×D4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922512
C2×C4×C22⋊D1012
C2×#156922152
#202650772
#202650572
#202650562
#202650552
#202650542
#202650532
#202650412
#202650222
#386×C1012
#2328101