Finite Group Databasebeta

← #27174470

Subgroups — #27174470

Order 12928.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108073C211yesmaximal, characteristic, complemented
2C2×#15692232C211yesmaximal, characteristic, complemented
3C2×#15692227C211yesmaximal, characteristic, complemented
4C2×D4×Dic101C211yesmaximal, characteristic, complemented
5#20265093C211yesmaximal, characteristic, complemented
6#20265091C211yesmaximal, characteristic, complemented
7#20265085C211yesmaximal, characteristic, complemented
8#20265081C211yesmaximal, characteristic, complemented
9#20265077C211yesmaximal, characteristic, complemented
10#20265057C211yesmaximal, characteristic, complemented
11#20265053C211yesmaximal, characteristic, complemented
12#20265052C211yesmaximal, characteristic, complemented
13#20265051C211yesmaximal, characteristic, complemented
14#20265035C211yesmaximal, characteristic, complemented
15#392×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—24no—
17C2×#13108073—24no—
18C2×#13108073—24no—
19C22×Dic202C2211yescharacteristic
20C22×C22⋊D101C2211yescharacteristic
21C22×C22⋊D101C2211yescharacteristic
22C22×C22⋊D101C2211yescharacteristic
23C23×Dic101C2211yescharacteristic
24C23×Dic101C2211yescharacteristic
25C23×Dic101C2211yescharacteristic
26C22×C4×D101C2211yescharacteristic
27#15692232—24no—
28#15692227—24no—
29D4×Dic101—24no—
30C2×C4⋊Dic101C2211yescharacteristic
31C2×C4⋊Dic101C2211yescharacteristic
32C2×#13108053C2211yescharacteristic
33C2×#13108053C2211yescharacteristic
34C2×#13108053C2211yescharacteristic
35C2×C22⋊Dic101C2211yescharacteristic
36C2×C22⋊Dic101C2211yescharacteristic
37C2×C22⋊Dic101C2211yescharacteristic
38C2×C22⋊Dic101C2211yescharacteristic
39C2×C22⋊Dic101C2211yescharacteristic
40C2×C22⋊Dic101C2211yescharacteristic
41C2×#13108051C2211yescharacteristic
42C2×#13108051C2211yescharacteristic
43C2×#13108051C2211yescharacteristic
44C2×C4×Dic101C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47#15692162C2211yescharacteristic
48#15692162C2211yescharacteristic
49#15692162C2211yescharacteristic
50C22×D4×C101C2211yescharacteristic
51C22×D4×C101C2211yescharacteristic
52C22×D4×C101C2211yescharacteristic
53C2×C22⋊C4×C101C2211yescharacteristic
54C2×C22⋊C4×C101C2211yescharacteristic
55C2×C22⋊C4×C101C2211yescharacteristic
56#95×C101C2211yescharacteristic
57C23×D101C2311yescharacteristic
58#13108073—232no—
59C2×Dic202—22no—
60C2×Dic202—22no—
61C2×Dic202D414yescomplemented
62C2×C22⋊D101—22no—
63C2×C22⋊D101—22no—
64C2×C22⋊D101—22no—
65C2×C22⋊D101—22no—
66C2×C22⋊D101—22no—
67C2×C22⋊D101—22no—
68C2×C22⋊D101—22no—
69C2×C22⋊D101—41no—
70C2×C22⋊D101D414yescomplemented
71C22×Dic101—22no—
72C22×Dic101—22no—
73C22×Dic101—22no—
74C22×Dic101—22no—
75C22×Dic101—22no—
76C22×Dic101—22no—
77C22×Dic101—22no—
78C22×Dic101—22no—
79C22×Dic101D414yescomplemented
80C22×Dic101C2311yescharacteristic
81C22×Dic101C2311yescharacteristic
82C22×Dic101C2311yescharacteristic
83C22×Dic101C2311yescharacteristic
84C22×Dic101C2311yescharacteristic
85C22×Dic101C2311yescharacteristic
86C22×Dic101C2311yescharacteristic
87C2×C4×D101—22no—
88C2×C4×D101—22no—
89C2×C4×D101—22no—
90C4⋊Dic101—22no—
91C4⋊Dic101—22no—
92#13108053—22no—
93#13108053—22no—
94#13108053—22no—
95C22⋊Dic101—22no—
96C22⋊Dic101—22no—
97C22⋊Dic101—22no—
98C22⋊Dic101—22no—
99C22⋊Dic101—41no—
100C22⋊Dic101—41no—
101#13108051—22no—
102#13108051—22no—
103#13108051—41no—
104C4×Dic101—22no—
105C24×C101C2311yescharacteristic
106C24×C101C2311yescharacteristic
107C24×C101C2311yescharacteristic
108C2×D4×C101—22no—
109C2×D4×C101—22no—
110C2×D4×C101—22no—
111C2×D4×C101—22no—
112C2×D4×C101—22no—
113C2×D4×C101—22no—
114C2×D4×C101—22no—
115C2×D4×C101—22no—
116C2×D4×C101—41no—
117C22×C4×C101C2311yescharacteristic
118C22×C4×C101C2311yescharacteristic
119C22×C4×C101C2311yescharacteristic
120C22×C4×C101C2311yescharacteristic
121C22⋊C4×C101—22no—
122C22⋊C4×C101—22no—
123C22⋊C4×C101—41no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—21no—
128C23×C101—21no—
129C23×C101—21no—
130C23×C101—21no—
131C23×C101—21no—
132C23×C101—21no—
133C23×C101—21no—
134C23×C101—21no—
135C23×C101—21no—
136C23×C101—21no—
137C23×C101—21no—
138C23×C101—21no—
139C23×C101—21no—
140C23×C101—21no—
141C23×C101—21no—
142C23×C101C2×D412yes—
143C23×C101C2×D412yes—
144C23×C101#4112yescomplemented
145C23×C101C2411yescharacteristic
146C22×D101—21no—
147C22×D101—21no—
148C22×D101—21no—
149C22×D101—21no—
150C22×D101—21no—
151C22×D101—21no—
152C22×D101C2×D412yes—
153D4×C101—28no—
154D4×C101—28no—
155D4×C101—44no—
156C2×C4×C101—21no—
157C2×C4×C101—21no—
158C2×C4×C101—21no—
159C2×C4×C101—21no—
160C2×C4×C101—21no—
161C2×C4×C101—21no—
162C2×C4×C101—21no—
163C2×C4×C101—21no—
164C2×C4×C101—21no—
165C2×C4×C101—21no—
166C2×C4×C101C2×D412yes—
167C2×C4×C101#4112yescomplemented
168C22⋊D101—28no—
169C22⋊D101—28no—
170C22⋊D101—44no—
171C2×Dic101—28no—
172C2×Dic101—21no—
173C2×Dic101—21no—
174C2×Dic101—21no—
175C2×Dic101—21no—
176C2×Dic101—28no—
177C2×Dic101—21no—
178C2×Dic101—21no—
179C2×Dic101—21no—
180C2×Dic101—21no—
181C2×Dic101—21no—
182C2×Dic101—21no—
183C2×Dic101—21no—
184C2×Dic101—21no—
185C2×Dic101—21no—
186C2×Dic101—21no—
187C2×Dic101—21no—
188C2×Dic101—21no—
189C2×Dic101—44no—
190C2×Dic101C2×D412yes—
191C2×Dic101C2×D412yes—
192C2×Dic101C2×D412yes—
193C2×Dic101C2×D412yes—
194C2×Dic101C2×D412yes—
195C4×D101—28no—
196Dic202—28no—
197C22×C101—22no—
198C22×C101—22no—
199C22×C101—22no—
200C22×C101—22no—
201C22×C101—22no—
202C22×C101—22no—
203C22×C101—22no—
204C22×C101—22no—
205C22×C101—22no—
206C22×C101—41no—
207C22×C101—41no—
208C22×C101—41no—
209C22×C101—41no—
210C22×C101—41no—
211C22×C101—41no—
212C22×C101—41no—
213C22×C101—41no—
214C22×C101—41no—
215C22×C101—41no—
216C22×C101—41no—
217C22×C101—41no—
218C22×C101C22×D411yescharacteristic
219C22×C101C22×D411yescharacteristic
220C22×C101C22×D411yescharacteristic
221C22×C101C2×#4111yescharacteristic
222C22×C101C2×#4111yescharacteristic
223C22×C101H(5,𝔽2)11yescharacteristic
224C22×C1012−1+411yescharacteristic
225C2×D101—22no—
226C2×D101—22no—
227C2×D101—22no—
228C2×D101—41no—
229C2×D101—41no—
230C2×D101—41no—
231C2×D101—41no—
232C4×C101—22no—
233C4×C101—22no—
234C4×C101—41no—
235C4×C101—41no—
236Dic101—22no—
237Dic101—22no—
238Dic101—41no—
239Dic101—41no—
240Dic101—41no—
241Dic101—41no—
242Dic101#12014yes—
243C2×C101—42no—
244C2×C101—42no—
245C2×C101—42no—
246C2×C101C2×#12011yescharacteristic
247C2×C101#52511yescharacteristic
248C2×C101#53711yescharacteristic
249C2×C101#54611yescharacteristic
250C2×C101#54711yescharacteristic
251C2×C101#54811yescharacteristic
252C2×C101#55011yescharacteristic
253D101—42no—
254#2407—1011nomaximal, Sylow
255C101#240711yescharacteristic, complemented, Sylow
256C22×#41—1011no—
257C2×C22⋊Q8—1011no—
258C2×#121—1011no—
259C2×C4×D4—1011no—
260#394—1011no—
261#394—1011no—
262#392—1011no—
263#392D10111yescharacteristic, complemented
264#391—1011no—
265#390—1011no—
266#386—1011no—
267#386—1011no—
268#385—1011no—
269#385—1011no—
270#380—1011no—
271C2×#41—2024no—
272C2×#41—2024no—
273C2×#41—2024no—
274C22×Q8—1011no—
275C22×D4—1011no—
276C22×D4—1011no—
277C22×D4—1011no—
278C22×D4C2×D10111yescharacteristic
279C22×D4C2×D10111yescharacteristic
280C22×D4C2×D10111yescharacteristic
281C23×C4—1011no—
282C23×C4—1011no—
283C23×C4—1011no—
284C23×C4—1011no—
285C22⋊Q8—2024no—
286#121—2024no—
287C4×D4—2024no—
288C2×C4⋊C4—1011no—
289C2×C4⋊C4—1011no—
290C2×C4⋊C4—1011no—
291C2×C4⋊C4—1011no—
292C2×C4⋊C4—1011no—
293C2×C22⋊C4—1011no—
294C2×C22⋊C4—1011no—
295C2×C22⋊C4—1011no—
296C2×C22⋊C4—1011no—
297C2×C22⋊C4—1011no—
298C2×C22⋊C4—1011no—
299C2×C22⋊C4—1011no—
300C2×C22⋊C4—1011no—
301C2×C22⋊C4—1011no—
302C2×C22⋊C4C2×D10111yescharacteristic
303C2×C22⋊C4C2×D10111yescharacteristic
304C2×C22⋊C4C2×D10111yescharacteristic
305C2×C42—1011no—
306#95—1011no—
307#95—1011no—
308#95—1011no—
309#95—1011no—
310#95—1011no—
311#95C2×D10111yescharacteristic
312C24—1011no—
313C24C22×D10111yescharacteristic
314C24C22×D10111yescharacteristic
315C24C22×D10111yescharacteristic
316#41—20232no—
317C2×Q8—1014no—
318C2×Q8—2022no—
319C2×Q8—2022no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—1014no—
330C2×D4—4041no—
331C2×D4—2022no—
332C2×D4—2022no—
333C2×D4—2022no—
334C2×D4—2022no—
335C2×D4—2022no—
336C2×D4—2022no—
337C2×D4—2022no—
338C22×C4—1011no—
339C22×C4—1014no—
340C22×C4—1011no—
341C22×C4—1011no—
342C22×C4—1011no—
343C22×C4—1011no—
344C22×C4—1011no—
345C22×C4—1011no—
346C22×C4—2022no—
347C22×C4—2022no—
348C22×C4—2022no—
349C22×C4—2022no—
350C22×C4—2022no—
351C22×C4—2022no—
352C22×C4—2022no—
353C22×C4—2022no—
354C22×C4—2022no—
355C22×C4—2022no—
356C22×C4—2022no—
357C22×C4C22×D10111yescharacteristic
358C22×C4C22×D10111yescharacteristic
359C22×C4C22×D10111yescharacteristic
360C22×C4C22×D10111yescharacteristic
361C4⋊C4—2022no—
362C4⋊C4—2022no—
363C4⋊C4—2022no—
364C4⋊C4—2022no—
365C4⋊C4—2022no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—4041no—
370C22⋊C4—4041no—
371C22⋊C4—4041no—
372C22⋊C4—2022no—
373C22⋊C4—2022no—
374C22⋊C4—2022no—
375C22⋊C4—2022no—
376C22⋊C4—2022no—
377C22⋊C4—2022no—
378C42—2022no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—1012no—
398C23—2021no—
399C23—2021no—
400C23—2021no—
401C23—2021no—
402C23—2021no—
403C23—2021no—
404C23D4×D10112yes—
405C23D4×D10112yes—
406C23#1310807312yescomplemented
407C23C23×D10111yescharacteristic
408Q8—2028no—
409D4—28no—
410D4—28no—
411D4—44no—
412D4—4044no—
413D4—2028no—
414D4—2028no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—1012no—
426C2×C4—1012no—
427C2×C4—1012no—
428C2×C4—1012no—
429C2×C4—1012no—
430C2×C4—4044no—
431C2×C4—2028no—
432C2×C4—2028no—
433C2×C4—2021no—
434C2×C4—2021no—
435C2×C4—2021no—
436C2×C4—2021no—
437C2×C4—2021no—
438C2×C4—2021no—
439C2×C4—2021no—
440C2×C4—2021no—
441C2×C4—2021no—
442C2×C4—2021no—
443C2×C4—2021no—
444C2×C4—2021no—
445C2×C4—2028no—
446C2×C4—2021no—
447C2×C4—2021no—
448C2×C4—2021no—
449C2×C4—2021no—
450C2×C4D4×D10112yes—
451C2×C4#1310807312yescomplemented
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—4041no—
474C22—4041no—
475C22—4041no—
476C22—4041no—
477C22—2022no—
478C22—2022no—
479C22—2022no—
480C22C2×D4×D10111yescharacteristic
481C22C2×D4×D10111yescharacteristic
482C22C2×D4×D10111yescharacteristic
483C22C2×#1310807311yescharacteristic
484C22C2×#1310807311yescharacteristic
485C22#1569230011yescharacteristic
486C22#1569230211yescharacteristic
487C4—22no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—1014no—
492C4—4041no—
493C4—4041no—
494C4—4041no—
495C4—4041no—
496C4—2022no—
497C4—2022no—
498C2—42no—
499C2—42no—
500C2—42no—
501C2—4042no—
502C2#120×D10111yescharacteristic
503C2#2026568311yescharacteristic
504C2#2026575411yescharacteristic
505C2#2026581411yescharacteristic
506C2#2026582711yescharacteristic
507C2#2026583311yescharacteristic
508C2#2026584711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#392×C101C2
hypercenter#392D101
hypocenterC101#2407
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#13108073C2semidirect
2. C2×#15692232C2semidirect
3. C2×#15692227C2semidirect
4. C2×D4×Dic101C2semidirect
5. #20265093C2semidirect
6. #20265091C2semidirect
7. #20265085C2semidirect
8. #20265081C2semidirect
9. #20265077C2semidirect
10. #20265057C2semidirect
11. #20265053C2semidirect
12. #20265052C2semidirect
13. #20265051C2semidirect
14. #20265035C2semidirect
15. #392×C101C2semidirect
61. C2×Dic202D4semidirect
70. C2×C22⋊D101D4semidirect
79. C22×Dic101D4semidirect
144. C23×C101#41semidirect
167. C2×C4×C101#41semidirect
255. C101#2407semidirect
263. #392D101semidirect
406. C23#13108073semidirect
451. C2×C4#13108073semidirect

Maximal subgroups

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

subgroupindex
C22×#131080732
C2×#156922322
C2×#156922272
C2×D4×Dic1012
#202650932
#202650912
#202650852
#202650812
#202650772
#202650572
#202650532
#202650522
#202650512
#202650352
#392×C1012
#2407101