Finite Group Databasebeta

← #27182642

Subgroups — #27182642

Order 12928.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692284C211yesmaximal, characteristic, complemented
2Q8×C22⋊D101C211yesmaximal, characteristic, complemented
3#20265821C211yesmaximal, characteristic
4#20265568C211yesmaximal, characteristic, complemented
5#20265420C211yesmaximal, characteristic, complemented
6#20265418C211yesmaximal, characteristic, complemented
7#20265397C211yesmaximal, characteristic, complemented
8#20265380C211yesmaximal, characteristic, complemented
9#20265336C211yesmaximal, characteristic, complemented
10#20265333C211yesmaximal, characteristic, complemented
11#20265328C211yesmaximal, characteristic, complemented
12#20265314C211yesmaximal, characteristic, complemented
13#20265278C211yesmaximal, characteristic, complemented
14#20265248C211yesmaximal, characteristic, complemented
15C22×SD16×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic, complemented
18#15692284—24no—
19C2×Dic404C2211yescharacteristic, complemented
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108064C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic
23C2×#13108062C2211yescharacteristic
24C2×#13108057C2211yescharacteristic, complemented
25#15692253—21no—
26#15692253C2211yescharacteristic, complemented
27#15692242C2211yescharacteristic
28#15692236—21no—
29#15692234—21no—
30#15692234C2211yescharacteristic
31#15692234C2211yescharacteristic
32#15692231—21no—
33#15692231C2211yescharacteristic, complemented
34#15692227—21no—
35#15692226C2211yescharacteristic
36#15692224—21no—
37Q8×Dic101C2211yescharacteristic, complemented
38D4×Dic101C2211yescharacteristic
39C4×C22⋊D101—21no—
40C4×C22⋊D101C2211yescharacteristic
41C2×C22⋊Dic101—21no—
42#15692189C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692180C2211yescharacteristic, complemented
46#15692179C2211yescharacteristic, complemented
47#15692179C2211yescharacteristic, complemented
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic
50C22⋊#11717579C2211yescharacteristic
51#15692167C2211yescharacteristic, complemented
52#15692165C2211yescharacteristic, complemented
53C22×Q8×C101C2211yescharacteristic
54C22×D4×C101C2211yescharacteristic
55C2×SD16×C101—24no—
56C2×SD16×C101C2211yescharacteristic
57C2×SD16×C101C2211yescharacteristic
58C2×SD16×C101C2211yescharacteristic
59C2×SD16×C101C2211yescharacteristic
60C22×C8×C101C2211yescharacteristic
61#13108073—22no—
62#13108073—41no—
63C2×Dic202—21no—
64C2×Dic202C2311yescharacteristic
65C2×Dic202C2311yescharacteristic
66Q8×D101—22no—
67Q8×D101—41no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101D411yescharacteristic
71C22×Dic101—21no—
72C22×Dic101—21no—
73C2×C4×D101—21no—
74C2×C4×D101C2311yescharacteristic
75Dic404—22no—
76#13108064—22no—
77#13108064—41no—
78#13108062—22no—
79#13108062—41no—
80#13108057—22no—
81C2×#11717579C2311yescharacteristic
82C2×#11717579C2311yescharacteristic
83C4⋊Dic101—21no—
84C4⋊Dic101C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86#13108053—21no—
87#13108053—21no—
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91#13108053D411yescharacteristic
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95C22⋊Dic101—21no—
96C22⋊Dic101—21no—
97C22⋊Dic101—21no—
98C22⋊Dic101D411yescharacteristic
99#13108051—21no—
100#13108051D411yescharacteristic
101C4×Dic101—21no—
102C4×Dic101C2311yescharacteristic
103C24×C101—21no—
104C2×Q8×C101—21no—
105C2×Q8×C101—22no—
106C2×Q8×C101—41no—
107C2×Q8×C101C2311yescharacteristic
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101—22no—
110C2×D4×C101—21no—
111C2×D4×C101—41no—
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116SD16×C101—22no—
117SD16×C101—22no—
118SD16×C101—22no—
119SD16×C101D414yescomplemented
120C2×C8×C101—22no—
121C2×C8×C101C2311yescharacteristic
122C2×C8×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—42no—
126C23×C101—42no—
127C23×C101—81no—
128C23×C101C2×D411yescharacteristic
129C22×D101C2×D411yescharacteristic
130Q8×C101—21no—
131Q8×C101—21no—
132Q8×C101—21no—
133Q8×C101—42no—
134Q8×C101C2×D412yes—
135D4×C101—21no—
136D4×C101—21no—
137D4×C101—21no—
138D4×C101—42no—
139D4×C101C2×D412yes—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—42no—
145C2×C4×C101C2×D411yescharacteristic
146C2×C4×C101C2411yescharacteristic
147C22⋊D101—22no—
148C22⋊D101—42no—
149C22⋊D101—81no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—42no—
155C2×Dic101—81no—
156C2×Dic101C2×D411yescharacteristic
157C2×Dic101C2×D411yescharacteristic
158C2×Dic101C2×D411yescharacteristic
159C4×D101—21no—
160C4×D101—21no—
161C4×D101—42no—
162Dic202—21no—
163Dic202—21no—
164Dic202—21no—
165Dic202—21no—
166Dic202—42no—
167C8×C101—21no—
168C8×C101C2×D412yes—
169#11717579—21no—
170#11717579—21no—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101—22no—
174C22×C101—41no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—84no—
180C22×C101#13712yescomplemented
181C22×C101C22×D411yescharacteristic
182C2×D101—21no—
183C2×D101—21no—
184C2×D101—21no—
185C4×C101—22no—
186C4×C101—21no—
187C4×C101—41no—
188C4×C101C22×D411yescharacteristic
189C4×C101H(5,𝔽2)11yescharacteristic
190Dic101—21no—
191Dic101—21no—
192Dic101—21no—
193Dic101—41no—
194Dic101—41no—
195Dic101—41no—
196Dic101—41no—
197C2×C101—22no—
198C2×C101—42no—
199C2×C101—81no—
200C2×C101D4211yescharacteristic
201C2×C101C2×#13711yescharacteristic
202C2×C101#57711yescharacteristic
203D101—41no—
204#3266—1011nomaximal, Sylow
205C101#326611yescharacteristic, complemented, Sylow
206C2×#137—1011no—
207C22×SD16D10111yescharacteristic, complemented
208D4×Q8—1011no—
209#546—1011no—
210#497—1011no—
211#468—1011no—
212#467—1011no—
213#464—1011no—
214#461—1011no—
215#452—1011no—
216#451—1011no—
217#450—1011no—
218#448—1011no—
219#440—1011no—
220#435—1011no—
221C2×#41—1011no—
222C22×Q8—1011no—
223C22×Q8C2×D10111yescharacteristic
224C22×D4C2×D10111yescharacteristic
225#137—2024no—
226C2×Q16—1011no—
227C2×Q16—1011no—
228C2×Q16—1011no—
229C2×SD16—24no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×SD16C2×D10111yescharacteristic
233C2×SD16C2×D10111yescharacteristic
234C2×SD16C2×D10111yescharacteristic
235C2×SD16C2×D10111yescharacteristic
236C2×M16—1011no—
237C22×C8C2×D10111yescharacteristic
238C4⋊Q8—1011no—
239C4⋊Q8—2021no—
240#124—1011no—
241#123—2021no—
242C22⋊Q8—1011no—
243C22⋊Q8—1011no—
244C22⋊Q8—1011no—
245C22⋊Q8—2021no—
246C22⋊Q8—2021no—
247#121—1011no—
248#121—2021no—
249#120—2021no—
250C4×Q8—1011no—
251C4×D4—1011no—
252C4×D4—1011no—
253C4×D4—2021no—
254C2×C22⋊C4—2021no—
255#107—1011no—
256C4⋊C8—1011no—
257#103—1011no—
258#103—1011no—
259#103—1011no—
260#103—1011no—
261#102—1011no—
262#102—1011no—
263C22⋊C8—1011no—
264C22⋊C8—1011no—
265#97—1011no—
266C24—21no—
267#41—4041no—
268#41—2022no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1011no—
273C2×Q8—1011no—
274C2×Q8—4041no—
275C2×Q8—2021no—
276C2×Q8—2022no—
277C2×Q8C22×D10111yescharacteristic
278C2×Q8C22×D10111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—1011no—
283C2×D4—2021no—
284C2×D4—2021no—
285C2×D4C22×D10111yescharacteristic
286C2×D4C22×D10111yescharacteristic
287C22×C4—21no—
288C22×C4—1011no—
289C22×C4—2021no—
290C22×C4—2021no—
291C22×C4—2021no—
292C22×C4C22×D10111yescharacteristic
293Q16—4041no—
294Q16—2022no—
295Q16—2022no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—4041no—
300SD16—2022no—
301SD16C22⋊D10114yescomplemented
302M16—2022no—
303C2×C8—22no—
304C2×C8—1011no—
305C2×C8—1011no—
306C2×C8C22×D10111yescharacteristic
307C2×C8C22×D10111yescharacteristic
308C4⋊C4—1011no—
309C4⋊C4—1011no—
310C4⋊C4—1011no—
311C4⋊C4—2021no—
312C4⋊C4—2021no—
313C4⋊C4—2021no—
314C4⋊C4—2021no—
315C4⋊C4—2021no—
316C4⋊C4—2021no—
317C22⋊C4—1011no—
318C22⋊C4—1011no—
319C22⋊C4—2021no—
320C22⋊C4—2021no—
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C42—1011no—
327C42—2021no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1011no—
334C23D4×D10111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—4042no—
340Q8—2021no—
341Q8—2021no—
342Q8—2021no—
343Q8—2021no—
344Q8C2×C22⋊D10112yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—8081no—
350D4—4042no—
351D4—2022no—
352D4C2×C22⋊D10112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—8081no—
359C2×C4—1011no—
360C2×C4—1011no—
361C2×C4—1011no—
362C2×C4—4042no—
363C2×C4—4042no—
364C2×C4—2021no—
365C2×C4—2021no—
366C2×C4—2021no—
367C2×C4—2021no—
368C2×C4—2021no—
369C2×C4—2021no—
370C2×C4D4×D10111yescharacteristic
371C2×C4C23×D10111yescharacteristic
372C8—21no—
373C8—2021no—
374C8—2021no—
375C8C2×C22⋊D10112yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2021no—
386C22—2021no—
387C22—2021no—
388C22#1569228412yescomplemented
389C22C2×D4×D10111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—4041no—
394C4—4041no—
395C4—4041no—
396C4—4041no—
397C4—2021no—
398C4—2021no—
399C4—2021no—
400C4C22×C22⋊D10111yescharacteristic
401C4#1569230011yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—4041no—
406C2D4×C22⋊D10111yescharacteristic
407C2C2×#1569228411yescharacteristic
408C2#2026598311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×SD16×C101C2
socleC22×C101C22×D4
prime radical#20265821C2
hypercenterC22×SD16D101
hypocenterC101#3266
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15692284C2semidirect
2. Q8×C22⋊D101C2semidirect
4. #20265568C2semidirect
5. #20265420C2semidirect
6. #20265418C2semidirect
7. #20265397C2semidirect
8. #20265380C2semidirect
9. #20265336C2semidirect
10. #20265333C2semidirect
11. #20265328C2semidirect
12. #20265314C2semidirect
13. #20265278C2semidirect
14. #20265248C2semidirect
15. C22×SD16×C101C2semidirect
17. C2×Q8×D101C22semidirect
19. C2×Dic404C22semidirect
20. C2×#13108064C22semidirect
21. C2×#13108064C22semidirect
24. C2×#13108057C22semidirect
26. #15692253C22semidirect
33. #15692231C22semidirect
37. Q8×Dic101C22semidirect
42. #15692189C22semidirect
43. #15692184C22semidirect
44. #15692180C22semidirect
45. #15692180C22semidirect
46. #15692179C22semidirect
47. #15692179C22semidirect
51. #15692167C22semidirect
52. #15692165C22semidirect
119. SD16×C101D4semidirect
180. C22×C101#137semidirect
205. C101#3266semidirect
207. C22×SD16D101semidirect
301. SD16C22⋊D101semidirect
388. C22#15692284semidirect

Maximal subgroups

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

subgroupindex
C2×#156922842
Q8×C22⋊D1012
#202658212
#202655682
#202654202
#202654182
#202653972
#202653802
#202653362
#202653332
#202653282
#202653142
#202652782
#202652482
C22×SD16×C1012
#3266101