Finite Group Databasebeta

← #27179430

Subgroups — #27179430

Order 12928.

696 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692241C211yesmaximal, characteristic, complemented
2C2×#15692233C211yesmaximal, characteristic, complemented
3C2×#15692228C211yesmaximal, characteristic
4C2×C42×D101C211yesmaximal, characteristic, complemented
5#20265081C211yesmaximal, characteristic, complemented
6#20265077C211yesmaximal, characteristic, complemented
7#20265071C211yesmaximal, characteristic, complemented
8#20265068C211yesmaximal, characteristic, complemented
9#20265067C211yesmaximal, characteristic, complemented
10#20265055C211yesmaximal, characteristic, complemented
11#20265055C211yesmaximal, characteristic, complemented
12#20265053C211yesmaximal, characteristic, complemented
13#20265053C211yesmaximal, characteristic, complemented
14#20265041C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C22×Dic202C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C22×C22⋊D101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22C22×C4×D101C2211yescharacteristic
23#15692241—24no—
24#15692233—24no—
25#15692228—24no—
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41C42×D101—24no—
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47#15692162C2211yescharacteristic
48C2×C4⋊C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101C2211yescharacteristic
52C2×C42×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54#95×C101C2211yescharacteristic
55C23×D101C2311yescharacteristic
56C23×D101C2311yescharacteristic
57C2×Dic202—22no—
58C2×Dic202—22no—
59C2×Dic202—22no—
60C2×D404—22no—
61C2×D404—22no—
62C2×D404—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—
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C22×Dic101C2311yescharacteristic
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C2×C4×D101—22no—
76C2×C4×D101—22no—
77C2×C4×D101—22no—
78C2×C4×D101—22no—
79C2×C4×D101—22no—
80C2×C4×D101—22no—
81C2×C4×D101—22no—
82C2×C4×D101—22no—
83C2×C4×D101D414yes—
84#13108053—22no—
85#13108053—22no—
86#13108053—22no—
87C22⋊Dic101—22no—
88C22⋊Dic101—22no—
89#13108051—22no—
90#13108051—22no—
91#13108051—22no—
92#13108051—22no—
93#13108051—22no—
94#13108051—22no—
95#13108051—22no—
96C4×Dic101—22no—
97C4×Dic101—22no—
98C4×Dic101D414yescomplemented
99C24×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101C22×C4×C101C2311yescharacteristic
102C22×C4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104C22×C4×C101C2311yescharacteristic
105C22×C4×C101C2311yescharacteristic
106C4⋊C4×C101—22no—
107C22⋊C4×C101—22no—
108C22⋊C4×C101—22no—
109C22⋊C4×C101—22no—
110C42×C101—22no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101—21no—
114C23×C101—21no—
115C23×C101—21no—
116C23×C101—21no—
117C23×C101—21no—
118C23×C101C2411yescharacteristic
119C22×D101—21no—
120C22×D101—21no—
121C22×D101—21no—
122C22×D101—21no—
123C22×D101—21no—
124C22×D101—21no—
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101C2×D412yes—
131C22×D101#4112yes—
132C22×D101#4112yes—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
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×C101C2×D412yes—
149C2×C4×C101C2×D412yes—
150C2×C4×C101#4112yes—
151C22⋊D101—44no—
152C22⋊D101—44no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—21no—
160C2×Dic101—21no—
161C2×Dic101—21no—
162C2×Dic101—21no—
163C2×Dic101—21no—
164C2×Dic101—21no—
165C2×Dic101—21no—
166C2×Dic101C2×D412yes—
167C2×Dic101C2×D412yes—
168C2×Dic101C2×D412yes—
169C2×Dic101#4112yes—
170C2×Dic101#4112yes—
171D404—44no—
172C4×D101—28no—
173C4×D101—28no—
174C4×D101—28no—
175Dic202—44no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101C22×D411yescharacteristic
184C22×C101C22×D411yescharacteristic
185C22×C101C2×#4111yescharacteristic
186C22×C101C2×#4111yescharacteristic
187C22×C101C2×#4111yescharacteristic
188C22×C101C2×#4111yescharacteristic
189C22×C101C2×#4111yescharacteristic
190C2×D101—22no—
191C2×D101—22no—
192C2×D101—22no—
193C2×D101—22no—
194C2×D101—22no—
195C2×D101—22no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101#12414yescomplemented
204C4×C101—22no—
205C4×C101—22no—
206C4×C101—22no—
207C4×C101—41no—
208C4×C101—41no—
209C4×C101—41no—
210Dic101—22no—
211Dic101—22no—
212Dic101—41no—
213Dic101—41no—
214Dic101—41no—
215Dic101#12114yes—
216C2×C101—81no—
217C2×C101C2×#12111yescharacteristic
218C2×C101#52511yescharacteristic
219C2×C101#52511yescharacteristic
220C2×C101C2×#12411yescharacteristic
221C2×C101#52911yescharacteristic
222C2×C101#52911yescharacteristic
223C2×C101#53211yescharacteristic
224D101—24no—
225D101—81no—
226#2850—1011nomaximal, Sylow
227C101#285011yescharacteristic, complemented, Sylow
228C2×#124—1011no—
229C2×C22⋊Q8—1011no—
230C2×#121—1011no—
231C22×C42—1011no—
232#391—1011no—
233#390—1011no—
234#388—1011no—
235#388—1011no—
236#388—1011no—
237#388D10111yescharacteristic, complemented
238#386—1011no—
239#386—1011no—
240#386—1011no—
241#386—1011no—
242#382—1011no—
243C22×Q8—1011no—
244C22×D4—1011no—
245C22×D4—1011no—
246C22×D4—1011no—
247C23×C4—1011no—
248C23×C4—1011no—
249C23×C4—1011no—
250#124—2024no—
251C22⋊Q8—2024no—
252#121—2024no—
253C2×C4⋊C4—1011no—
254C2×C4⋊C4—1011no—
255C2×C4⋊C4—1011no—
256C2×C4⋊C4C2×D10111yescharacteristic
257C2×C22⋊C4—1011no—
258C2×C22⋊C4—1011no—
259C2×C22⋊C4—1011no—
260C2×C22⋊C4—1011no—
261C2×C22⋊C4—1011no—
262C2×C22⋊C4—1011no—
263C2×C22⋊C4—1011no—
264C2×C22⋊C4—1011no—
265C2×C22⋊C4—1011no—
266C2×C22⋊C4C2×D10111yescharacteristic
267C2×C22⋊C4C2×D10111yescharacteristic
268C2×C22⋊C4C2×D10111yescharacteristic
269C2×C42—1011no—
270C2×C42—1011no—
271C2×C42—1011no—
272C2×C42—2024no—
273C2×C42C2×D10111yescharacteristic
274#95—1011no—
275#95—1011no—
276#95—1011no—
277#95—1011no—
278#95—1011no—
279#95—1011no—
280#95C2×D10111yescharacteristic
281#95C2×D10111yescharacteristic
282C24—1011no—
283C24—1011no—
284C24C22×D10111yescharacteristic
285C2×Q8—2022no—
286C2×Q8—2022no—
287C2×Q8—2022no—
288C2×D4—2022no—
289C2×D4—2022no—
290C2×D4—2022no—
291C2×D4—2022no—
292C2×D4—2022no—
293C2×D4—2022no—
294C2×D4—2022no—
295C2×D4—2022no—
296C2×D4—2022no—
297C22×C4—1011no—
298C22×C4—1011no—
299C22×C4—1011no—
300C22×C4—1014no—
301C22×C4—1011no—
302C22×C4—1011no—
303C22×C4—1011no—
304C22×C4—2022no—
305C22×C4—2022no—
306C22×C4—2022no—
307C22×C4—2022no—
308C22×C4—2022no—
309C22×C4—2022no—
310C22×C4—2022no—
311C22×C4—2022no—
312C22×C4C22×D10111yescharacteristic
313C22×C4C22×D10111yescharacteristic
314C22×C4C22×D10111yescharacteristic
315C22×C4C22×D10111yescharacteristic
316C22×C4C22×D10111yescharacteristic
317C22×C4C22×D10111yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—2022no—
320C4⋊C4—2022no—
321C4⋊C4—2022no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—2022no—
326C22⋊C4—2022no—
327C22⋊C4—2022no—
328C22⋊C4—2022no—
329C22⋊C4—2022no—
330C22⋊C4—2022no—
331C22⋊C4—2022no—
332C22⋊C4—2022no—
333C22⋊C4—2022no—
334C42—22no—
335C42—1014no—
336C42—2022no—
337C42—2022no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—1012no—
346C23—1012no—
347C23—1012no—
348C23—2021no—
349C23—2021no—
350C23—2021no—
351C23—2021no—
352C23—2021no—
353C23—2021no—
354C23—2021no—
355C23—2021no—
356C23—2021no—
357C23—2021no—
358C23—2021no—
359C23C23×D10111yescharacteristic
360Q8—4044no—
361D4—4044no—
362D4—4044no—
363D4—4044no—
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—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—1012no—
380C2×C4—1012no—
381C2×C4—1012no—
382C2×C4—1012no—
383C2×C4—1012no—
384C2×C4—2028no—
385C2×C4—2028no—
386C2×C4—2028no—
387C2×C4—2021no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4—2021no—
391C2×C4—2021no—
392C2×C4—2021no—
393C2×C4—2021no—
394C2×C4—2021no—
395C2×C4—2021no—
396C2×C4—2021no—
397C2×C4—2021no—
398C2×C4—2021no—
399C2×C4—2021no—
400C2×C4D4×D10112yes—
401C2×C4D4×D10112yes—
402C2×C4#1310807412yes—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—1014no—
411C22—4041no—
412C22—4041no—
413C22—4041no—
414C22—4041no—
415C22—4041no—
416C22—4041no—
417C22—4041no—
418C22—2022no—
419C22—2022no—
420C22—2022no—
421C22—2022no—
422C22—2022no—
423C22—2022no—
424C22C2×D4×D10111yescharacteristic
425C22C2×D4×D10111yescharacteristic
426C22#41×D10111yescharacteristic
427C22#41×D10111yescharacteristic
428C22#41×D10111yescharacteristic
429C22#41×D10111yescharacteristic
430C22C2×#1310807411yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—1014no—
438C4—4041no—
439C4—4041no—
440C4—4041no—
441C4—2022no—
442C4—2022no—
443C2—81no—
444C2—8081no—
445C2—2024no—
446C2#121×D10111yescharacteristic
447C2#2026568511yescharacteristic
448C2#2026568611yescharacteristic
449C2#124×D10111yescharacteristic
450C2#2026570511yescharacteristic
451C2#2026571011yescharacteristic
452C2#2026572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#388×C101C2
prime radicalC2×#15692228C2
hypercenter#388D101
hypocenterC101#2850
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#15692241C2semidirect
2. C2×#15692233C2semidirect
4. C2×C42×D101C2semidirect
5. #20265081C2semidirect
6. #20265077C2semidirect
7. #20265071C2semidirect
8. #20265068C2semidirect
9. #20265067C2semidirect
10. #20265055C2semidirect
11. #20265055C2semidirect
12. #20265053C2semidirect
13. #20265053C2semidirect
14. #20265041C2semidirect
15. #388×C101C2semidirect
98. C4×Dic101D4semidirect
203. C2×D101#124semidirect
227. C101#2850semidirect
237. #388D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922412
C2×#156922332
C2×#156922282
C2×C42×D1012
#202650812
#202650772
#202650712
#202650682
#202650672
#202650552
#202650532
#202650412
#388×C1012
#2850101