Finite Group Databasebeta

← #27179429

Subgroups — #27179429

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×#15692232C211yesmaximal, characteristic, complemented
3C2×#15692227C211yesmaximal, characteristic
4C22×C4×Dic101C211yesmaximal, characteristic, complemented
5#20265081C211yesmaximal, characteristic, complemented
6#20265077C211yesmaximal, characteristic, complemented
7#20265070C211yesmaximal, characteristic, complemented
8#20265069C211yesmaximal, characteristic, complemented
9#20265066C211yesmaximal, characteristic, complemented
10#20265065C211yesmaximal, characteristic, complemented
11#20265057C211yesmaximal, characteristic, complemented
12#20265056C211yesmaximal, characteristic, complemented
13#20265054C211yesmaximal, characteristic, complemented
14#20265041C211yesmaximal, characteristic, complemented
15#386×C101C211yesmaximal, characteristic, complemented
16C22×Dic202C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20C23×Dic101C2211yescharacteristic
21#15692241—24no—
22#15692232—24no—
23#15692227—24no—
24C2×C4⋊Dic101C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×C22⋊Dic101C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×C4×Dic101—24no—
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41C2×C4×Dic101C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47#15692162C2211yescharacteristic
48C22×D4×C101C2211yescharacteristic
49C23×C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101C2211yescharacteristic
52C2×C22⋊C4×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54#95×C101C2211yescharacteristic
55C23×D101C2311yescharacteristic
56C2×Dic202—22no—
57C2×Dic202—22no—
58C2×Dic202—22no—
59C2×C22⋊D101—22no—
60C2×C22⋊D101—22no—
61C2×C22⋊D101—22no—
62C2×C22⋊D101—22no—
63C2×C22⋊D101—22no—
64C2×C22⋊D101—22no—
65C22×Dic101—22no—
66C22×Dic101—22no—
67C22×Dic101—22no—
68C22×Dic101—22no—
69C22×Dic101—22no—
70C22×Dic101D414yes—
71C22×Dic101C2311yescharacteristic
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C4⋊Dic101—22no—
79#13108053—22no—
80#13108053—22no—
81#13108053—22no—
82C22⋊Dic101—22no—
83C22⋊Dic101—22no—
84C22⋊Dic101—22no—
85C22⋊Dic101—22no—
86#13108051—22no—
87#13108051—22no—
88#13108051—22no—
89#13108051—22no—
90#13108051—22no—
91C4×Dic101—22no—
92C4×Dic101—22no—
93C4×Dic101—22no—
94C4×Dic101D414yescomplemented
95C24×C101C2311yescharacteristic
96C24×C101C2311yescharacteristic
97C2×D4×C101—22no—
98C2×D4×C101—22no—
99C2×D4×C101—22no—
100C22×C4×C101—22no—
101C22×C4×C101—22no—
102C22×C4×C101—22no—
103C22×C4×C101C2311yescharacteristic
104C22×C4×C101C2311yescharacteristic
105C22×C4×C101C2311yescharacteristic
106C22×C4×C101C2311yescharacteristic
107C22×C4×C101C2311yescharacteristic
108C22⋊C4×C101—22no—
109C22⋊C4×C101—22no—
110C22⋊C4×C101—22no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101—21no—
114C23×C101—21no—
115C23×C101—21no—
116C23×C101—21no—
117C23×C101—21no—
118C23×C101—21no—
119C23×C101—21no—
120C23×C101—21no—
121C23×C101—21no—
122C23×C101C2×D412yes—
123C23×C101#4112yes—
124C23×C101#4112yes—
125C23×C101C2411yescharacteristic
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101—21no—
131C22×D101—21no—
132C22×D101—21no—
133D4×C101—44no—
134C2×C4×C101—28no—
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×C101#4112yes—
150C22⋊D101—44no—
151C22⋊D101—44no—
152C2×Dic101—28no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—28no—
160C2×Dic101—21no—
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×Dic101C2×D412yes—
170C2×Dic101C2×D412yes—
171C2×Dic101C2×D412yes—
172C2×Dic101C2×D412yes—
173C2×Dic101#4112yes—
174C2×Dic101#4112yes—
175Dic202—44no—
176C22×C101—22no—
177C22×C101—22no—
178C22×C101—22no—
179C22×C101—22no—
180C22×C101—22no—
181C22×C101—22no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101#12414yescomplemented
190C22×C101C22×D411yescharacteristic
191C22×C101C22×D411yescharacteristic
192C22×C101C2×#4111yescharacteristic
193C22×C101C2×#4111yescharacteristic
194C22×C101C2×#4111yescharacteristic
195C22×C101C2×#4111yescharacteristic
196C22×C101C2×#4111yescharacteristic
197C2×D101—41no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—41no—
204C4×C101—22no—
205C4×C101—22no—
206C4×C101—41no—
207C4×C101—41no—
208C4×C101—41no—
209Dic101—22no—
210Dic101—22no—
211Dic101—22no—
212Dic101—41no—
213Dic101—41no—
214Dic101—41no—
215Dic101#12114yes—
216C2×C101—24no—
217C2×C101—81no—
218C2×C101C2×#12111yescharacteristic
219C2×C101#52511yescharacteristic
220C2×C101#52511yescharacteristic
221C2×C101C2×#12411yescharacteristic
222C2×C101#52911yescharacteristic
223C2×C101#52911yescharacteristic
224C2×C101#53211yescharacteristic
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#388—1011no—
238#386—1011no—
239#386—1011no—
240#386—1011no—
241#386D10111yescharacteristic, complemented
242#382—1011no—
243C22×Q8—1011no—
244C22×D4—1011no—
245C22×D4—1011no—
246C22×D4C2×D10111yescharacteristic
247C23×C4—1011no—
248C23×C4—1011no—
249C23×C4C2×D10111yescharacteristic
250#124—2024no—
251C22⋊Q8—2024no—
252#121—2024no—
253C2×C4⋊C4—1011no—
254C2×C4⋊C4—1011no—
255C2×C4⋊C4—1011no—
256C2×C4⋊C4—1011no—
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—1011no—
273C2×C42—2024no—
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—
283C24C22×D10111yescharacteristic
284C24C22×D10111yescharacteristic
285C2×Q8—2022no—
286C2×Q8—2022no—
287C2×Q8—2022no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—2022no—
292C2×D4—2022no—
293C2×D4—2022no—
294C2×D4—2022no—
295C2×D4—2022no—
296C2×D4—2022no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—1011no—
301C22×C4—1014no—
302C22×C4—1011no—
303C22×C4—1011no—
304C22×C4—1011no—
305C22×C4—1011no—
306C22×C4—1011no—
307C22×C4—1011no—
308C22×C4—2022no—
309C22×C4—2022no—
310C22×C4—2022no—
311C22×C4—2022no—
312C22×C4—2022no—
313C22×C4C22×D10111yescharacteristic
314C22×C4C22×D10111yescharacteristic
315C22×C4C22×D10111yescharacteristic
316C22×C4C22×D10111yescharacteristic
317C22×C4C22×D10111yescharacteristic
318C4⋊C4—2022no—
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—1014no—
335C42—2022no—
336C42—2022no—
337C42—2022no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—2021no—
350C23—2021no—
351C23—2021no—
352C23—2021no—
353C23—2021no—
354C23—2021no—
355C23—2021no—
356C23D4×D10112yes—
357C23#1310807312yes—
358C23#1310807412yes—
359C23C23×D10111yescharacteristic
360Q8—4044no—
361D4—44no—
362D4—4044no—
363D4—4044no—
364C2×C4—28no—
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—1012no—
379C2×C4—1012no—
380C2×C4—1012no—
381C2×C4—1012no—
382C2×C4—1012no—
383C2×C4—1012no—
384C2×C4—2028no—
385C2×C4—2021no—
386C2×C4—2021no—
387C2×C4—2021no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4—2021no—
391C2×C4—2021no—
392C2×C4—2021no—
393C2×C4—2021no—
394C2×C4—2028no—
395C2×C4—2021no—
396C2×C4—2021no—
397C2×C4—2021no—
398C2×C4—2021no—
399C2×C4—2021no—
400C2×C4—2021no—
401C2×C4D4×D10112yes—
402C2×C4#1310807312yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—4041no—
417C22—4041no—
418C22—4041no—
419C22—4041no—
420C22—4041no—
421C22—4041no—
422C22—4041no—
423C22#1569224114yescomplemented
424C22C2×D4×D10111yescharacteristic
425C22C2×D4×D10111yescharacteristic
426C22#41×D10111yescharacteristic
427C22#41×D10111yescharacteristic
428C22C2×#1310807311yescharacteristic
429C22C2×#1310807311yescharacteristic
430C22C2×#1310807411yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—1014no—
437C4—4041no—
438C4—4041no—
439C4—4041no—
440C4—2022no—
441C4—2022no—
442C4—2022no—
443C2—24no—
444C2—81no—
445C2—8081no—
446C2#121×D10111yescharacteristic
447C2#2026568311yescharacteristic
448C2#2026568611yescharacteristic
449C2C2×#1569224111yescharacteristic
450C2#2026570411yescharacteristic
451C2#2026570711yescharacteristic
452C2#2026572211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#386×C101C2
prime radicalC2×#15692227C2
hypercenter#386D101
hypocenterC101#2850
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#15692241C2semidirect
2. C2×#15692232C2semidirect
4. C22×C4×Dic101C2semidirect
5. #20265081C2semidirect
6. #20265077C2semidirect
7. #20265070C2semidirect
8. #20265069C2semidirect
9. #20265066C2semidirect
10. #20265065C2semidirect
11. #20265057C2semidirect
12. #20265056C2semidirect
13. #20265054C2semidirect
14. #20265041C2semidirect
15. #386×C101C2semidirect
94. C4×Dic101D4semidirect
189. C22×C101#124semidirect
227. C101#2850semidirect
241. #386D101semidirect
423. C22#15692241semidirect

Maximal subgroups

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

subgroupindex
C2×#156922412
C2×#156922322
C2×#156922272
C22×C4×Dic1012
#202650812
#202650772
#202650702
#202650692
#202650662
#202650652
#202650572
#202650562
#202650542
#202650412
#386×C1012
#2850101