Finite Group Databasebeta

← #27174465

Subgroups — #27174465

Order 12928.

864 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108074C211yesmaximal, characteristic
2C2×#15692240C211yesmaximal, characteristic, complemented
3C2×#15692237C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
5#20265098C211yesmaximal, characteristic, complemented
6#20265097C211yesmaximal, characteristic, complemented
7#20265093C211yesmaximal, characteristic, complemented
8#20265092C211yesmaximal, characteristic, complemented
9#20265055C211yesmaximal, characteristic, complemented
10#20265053C211yesmaximal, characteristic, complemented
11#20265051C211yesmaximal, characteristic, complemented
12#20265050C211yesmaximal, characteristic, complemented
13#20265048C211yesmaximal, characteristic, complemented
14#20265039C211yesmaximal, characteristic, complemented
15#385×C101C211yesmaximal, characteristic, complemented
16C2×#13108074—24no—
17C2×#13108074—24no—
18C2×#13108074—24no—
19C22×Dic202C2211yescharacteristic
20C22×D404C2211yescharacteristic
21C22×C22⋊D101C2211yescharacteristic
22C22×C22⋊D101C2211yescharacteristic
23C22×C4×D101C2211yescharacteristic
24C22×C4×D101C2211yescharacteristic
25C22×C4×D101C2211yescharacteristic
26C22×C4×D101C2211yescharacteristic
27#15692240—24no—
28#15692237—24no—
29C2×C4⋊Dic101C2211yescharacteristic
30C2×C4⋊Dic101C2211yescharacteristic
31C2×#13108053C2211yescharacteristic
32C2×#13108053C2211yescharacteristic
33C2×#13108053C2211yescharacteristic
34C2×#13108053C2211yescharacteristic
35C4⋊C4×D101—24no—
36C2×C22⋊Dic101C2211yescharacteristic
37C2×C22⋊Dic101C2211yescharacteristic
38C2×#13108051C2211yescharacteristic
39C2×#13108051C2211yescharacteristic
40C2×#13108051C2211yescharacteristic
41C2×#13108051C2211yescharacteristic
42C2×#13108051C2211yescharacteristic
43C2×#13108051C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47#15692162C2211yescharacteristic
48#15692162C2211yescharacteristic
49#15692162C2211yescharacteristic
50C23×C4×C101C2211yescharacteristic
51C2×C4⋊C4×C101C2211yescharacteristic
52C2×C4⋊C4×C101C2211yescharacteristic
53C2×C22⋊C4×C101C2211yescharacteristic
54C2×C22⋊C4×C101C2211yescharacteristic
55#95×C101C2211yescharacteristic
56#95×C101C2211yescharacteristic
57C23×D101C2311yescharacteristic
58C23×D101C2311yescharacteristic
59#13108074—232no—
60C2×Dic202—22no—
61C2×Dic202—22no—
62C2×Dic202—22no—
63C2×D404—22no—
64C2×D404—22no—
65C2×D404D414yes—
66C2×C22⋊D101—22no—
67C2×C22⋊D101—22no—
68C2×C22⋊D101—22no—
69C2×C22⋊D101—22no—
70C2×C22⋊D101—22no—
71C2×C22⋊D101D414yes—
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C2×C4×D101—22no—
79C2×C4×D101—22no—
80C2×C4×D101—22no—
81C2×C4×D101—22no—
82C2×C4×D101—22no—
83C2×C4×D101—22no—
84C2×C4×D101—22no—
85C2×C4×D101—22no—
86C2×C4×D101—22no—
87C2×C4×D101—22no—
88C2×C4×D101—22no—
89C2×C4×D101D414yes—
90C4⋊Dic101—22no—
91C4⋊Dic101—22no—
92#13108053—22no—
93#13108053—22no—
94#13108053—22no—
95#13108053—22no—
96C22⋊Dic101—22no—
97C22⋊Dic101—22no—
98#13108051—22no—
99#13108051—22no—
100#13108051—22no—
101#13108051—22no—
102#13108051—22no—
103#13108051—22no—
104C24×C101C2311yescharacteristic
105C22×C4×C101—22no—
106C22×C4×C101—22no—
107C22×C4×C101—22no—
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113C22×C4×C101C2311yescharacteristic
114C4⋊C4×C101—22no—
115C4⋊C4×C101—22no—
116C22⋊C4×C101—22no—
117C22⋊C4×C101—22no—
118C23×C101—21no—
119C23×C101—21no—
120C23×C101—21no—
121C23×C101—21no—
122C23×C101—21no—
123C23×C101—21no—
124C23×C101C2×D412yes—
125C23×C101C2411yescharacteristic
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101—21no—
131C22×D101—21no—
132C22×D101—21no—
133C22×D101—21no—
134C22×D101—21no—
135C22×D101—21no—
136C22×D101C2×D412yes—
137C22×D101C2×D412yes—
138C22×D101C2×D412yes—
139C22×D101C2×D412yes—
140C2×C4×C101—28no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
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×C101C2×D412yes—
157C2×C4×C101C2×D412yes—
158C2×C4×C101#4112yescomplemented
159C22⋊D101—28no—
160C22⋊D101—28no—
161C2×Dic101—21no—
162C2×Dic101—21no—
163C2×Dic101—21no—
164C2×Dic101—21no—
165C2×Dic101—21no—
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×Dic101C2×D412yes—
177C2×Dic101C2×D412yes—
178C2×Dic101#4112yescomplemented
179D404—28no—
180C4×D101—28no—
181C4×D101—28no—
182C4×D101—28no—
183C4×D101—44no—
184Dic202—28no—
185C22×C101—22no—
186C22×C101—22no—
187C22×C101—22no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101—41no—
191C22×C101—41no—
192C22×C101C22×D411yescharacteristic
193C22×C101C22×D411yescharacteristic
194C22×C101C22×D411yescharacteristic
195C22×C101C2×#4111yescharacteristic
196C22×C101C2×#4111yescharacteristic
197C22×C1012−1+411yescharacteristic
198C22×C1012−1+411yescharacteristic
199C2×D101—22no—
200C2×D101—22no—
201C2×D101—22no—
202C2×D101—22no—
203C2×D101—22no—
204C2×D101—22no—
205C2×D101—22no—
206C2×D101—22no—
207C2×D101—22no—
208C2×D101—41no—
209C2×D101—41no—
210C2×D101—41no—
211C2×D101—41no—
212C2×D101#12014yes—
213C4×C101—22no—
214C4×C101—22no—
215C4×C101—22no—
216C4×C101—41no—
217C4×C101—41no—
218C4×C101—41no—
219Dic101—22no—
220Dic101—22no—
221Dic101—22no—
222Dic101—41no—
223Dic101—41no—
224Dic101—41no—
225C2×C101—42no—
226C2×C101C2×#12011yescharacteristic
227C2×C101#52511yescharacteristic
228C2×C101#53611yescharacteristic
229C2×C101#54711yescharacteristic
230C2×C101#54711yescharacteristic
231C2×C101#54711yescharacteristic
232C2×C101#54711yescharacteristic
233D101—24no—
234D101—42no—
235#2406—1011nomaximal, Sylow
236C101#240611yescharacteristic, complemented, Sylow
237C22×#41—1011no—
238C2×#123—1011no—
239C2×#123—1011no—
240C22×C4⋊C4—1011no—
241#395—1011no—
242#395—1011no—
243#394—1011no—
244#394—1011no—
245#386—1011no—
246#386—1011no—
247#385—1011no—
248#385—1011no—
249#385—1011no—
250#385D10111yescharacteristic, complemented
251#381—1011no—
252C2×#41—2024no—
253C2×#41—2024no—
254C2×#41—2024no—
255C22×Q8—1011no—
256C22×D4—1011no—
257C22×D4—1011no—
258C22×D4—1011no—
259C23×C4—1011no—
260C23×C4—1011no—
261C23×C4—1011no—
262C23×C4—1011no—
263C23×C4C2×D10111yescharacteristic
264#123—2024no—
265#123—2024no—
266C2×C4⋊C4—1011no—
267C2×C4⋊C4—1011no—
268C2×C4⋊C4—1011no—
269C2×C4⋊C4—1011no—
270C2×C4⋊C4—1011no—
271C2×C4⋊C4—1011no—
272C2×C4⋊C4—2024no—
273C2×C4⋊C4C2×D10111yescharacteristic
274C2×C4⋊C4C2×D10111yescharacteristic
275C2×C22⋊C4—1011no—
276C2×C22⋊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⋊C4C2×D10111yescharacteristic
284C2×C22⋊C4C2×D10111yescharacteristic
285#95—1011no—
286#95—1011no—
287#95—1011no—
288#95—1011no—
289#95—1011no—
290#95—1011no—
291#95C2×D10111yescharacteristic
292#95C2×D10111yescharacteristic
293C24—1011no—
294C24—1011no—
295C24C22×D10111yescharacteristic
296#41—20232no—
297C2×Q8—2022no—
298C2×Q8—2022no—
299C2×Q8—2022no—
300C2×D4—1014no—
301C2×D4—1014no—
302C2×D4—2022no—
303C2×D4—2022no—
304C2×D4—2022no—
305C2×D4—2022no—
306C2×D4—2022no—
307C2×D4—2022no—
308C2×D4—2022no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—1011no—
313C22×C4—1011no—
314C22×C4—1011no—
315C22×C4—1014no—
316C22×C4—1011no—
317C22×C4—1011no—
318C22×C4—1011no—
319C22×C4—2022no—
320C22×C4—2022no—
321C22×C4—2022no—
322C22×C4—2022no—
323C22×C4—2022no—
324C22×C4—2022no—
325C22×C4—2022no—
326C22×C4—2022no—
327C22×C4—2022no—
328C22×C4—2022no—
329C22×C4—2022no—
330C22×C4C22×D10111yescharacteristic
331C22×C4C22×D10111yescharacteristic
332C22×C4C22×D10111yescharacteristic
333C22×C4C22×D10111yescharacteristic
334C22×C4C22×D10111yescharacteristic
335C22×C4C22×D10111yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—2022no—
339C4⋊C4—2022no—
340C4⋊C4—2022no—
341C4⋊C4—2022no—
342C4⋊C4—2022no—
343C4⋊C4—2022no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—2022no—
347C22⋊C4—2022no—
348C22⋊C4—2022no—
349C22⋊C4—2022no—
350C22⋊C4—2022no—
351C22⋊C4—2022no—
352C22⋊C4—2022no—
353C22⋊C4—2022no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—1012no—
361C23—1012no—
362C23—1012no—
363C23—1012no—
364C23—2021no—
365C23—2021no—
366C23—2021no—
367C23—2021no—
368C23—2021no—
369C23—2021no—
370C23—2021no—
371C23—2021no—
372C23—2021no—
373C23—2021no—
374C23D4×D10112yes—
375C23C23×D10111yescharacteristic
376Q8—2028no—
377D4—2028no—
378D4—2028no—
379D4—2028no—
380C2×C4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—1012no—
397C2×C4—1012no—
398C2×C4—1012no—
399C2×C4—4044no—
400C2×C4—2028no—
401C2×C4—2028no—
402C2×C4—2028no—
403C2×C4—2021no—
404C2×C4—2021no—
405C2×C4—2021no—
406C2×C4—2021no—
407C2×C4—2021no—
408C2×C4—2021no—
409C2×C4—2021no—
410C2×C4—2021no—
411C2×C4—2021no—
412C2×C4—2021no—
413C2×C4—2021no—
414C2×C4—2021no—
415C2×C4—2021no—
416C2×C4—2021no—
417C2×C4—2021no—
418C2×C4D4×D10112yes—
419C2×C4D4×D10112yes—
420C2×C4#1310807412yescomplemented
421C22—22no—
422C22—22no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—1014no—
429C22—4041no—
430C22—4041no—
431C22—4041no—
432C22—4041no—
433C22—2022no—
434C22—2022no—
435C22—2022no—
436C22—2022no—
437C22—2022no—
438C22—2022no—
439C22—2022no—
440C22—2022no—
441C22—2022no—
442C22C2×D4×D10111yescharacteristic
443C22C2×D4×D10111yescharacteristic
444C22C2×D4×D10111yescharacteristic
445C22#41×D10111yescharacteristic
446C22C2×#1310807411yescharacteristic
447C22#1569230211yescharacteristic
448C22#1569230311yescharacteristic
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—4041no—
456C4—4041no—
457C4—4041no—
458C4—2022no—
459C4—2022no—
460C4—2022no—
461C2—42no—
462C2—4042no—
463C2—2024no—
464C2#120×D10111yescharacteristic
465C2#2026568611yescharacteristic
466C2#2026575011yescharacteristic
467C2#2026582511yescharacteristic
468C2#2026582811yescharacteristic
469C2#2026582911yescharacteristic
470C2#2026583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#385×C101C2
prime radicalC22×#13108074C2
hypercenter#385D101
hypocenterC101#2406
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#15692240C2semidirect
3. C2×#15692237C2semidirect
4. C2×C4⋊C4×D101C2semidirect
5. #20265098C2semidirect
6. #20265097C2semidirect
7. #20265093C2semidirect
8. #20265092C2semidirect
9. #20265055C2semidirect
10. #20265053C2semidirect
11. #20265051C2semidirect
12. #20265050C2semidirect
13. #20265048C2semidirect
14. #20265039C2semidirect
15. #385×C101C2semidirect
158. C2×C4×C101#41semidirect
178. C2×Dic101#41semidirect
236. C101#2406semidirect
250. #385D101semidirect
420. C2×C4#13108074semidirect

Maximal subgroups

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

subgroupindex
C22×#131080742
C2×#156922402
C2×#156922372
C2×C4⋊C4×D1012
#202650982
#202650972
#202650932
#202650922
#202650552
#202650532
#202650512
#202650502
#202650482
#202650392
#385×C1012
#2406101