Finite Group Databasebeta

← #27174374

Subgroups — #27174374

Order 12928.

696 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D101C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D101C211yesmaximal, characteristic, complemented
3C2×#15692214C211yesmaximal, characteristic, complemented
4#20265110C211yesmaximal, characteristic, complemented
5#20265105C211yesmaximal, characteristic, complemented
6#20265100C211yesmaximal, characteristic, complemented
7#20265099C211yesmaximal, characteristic, complemented
8#20265093C211yesmaximal, characteristic, complemented
9#20265092C211yesmaximal, characteristic, complemented
10#20265091C211yesmaximal, characteristic, complemented
11#20265085C211yesmaximal, characteristic
12C4×#13108053C211yesmaximal, characteristic, complemented
13C4×#13108051C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic101C211yesmaximal, characteristic, complemented
15C2×C4×D4×C101C211yesmaximal, characteristic, complemented
16C22×C22⋊D101C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19C4×C22⋊D101—24no—
20C4×C22⋊D101—24no—
21#15692214—24no—
22C2×C4⋊Dic101C2211yescharacteristic
23C2×C4⋊Dic101C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×C22⋊Dic101C2211yescharacteristic
32C2×C22⋊Dic101C2211yescharacteristic
33C2×C22⋊Dic101C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47C22×D4×C101C2211yescharacteristic
48C23×C4×C101C2211yescharacteristic
49C23×C4×C101C2211yescharacteristic
50C4×D4×C101—24no—
51C2×C4⋊C4×C101C2211yescharacteristic
52C2×C22⋊C4×C101C2211yescharacteristic
53C2×C22⋊C4×C101C2211yescharacteristic
54C2×C42×C101C2211yescharacteristic
55C23×D101C2311yescharacteristic
56C2×C22⋊D101—22no—
57C2×C22⋊D101—22no—
58C2×C22⋊D101—22no—
59C2×C22⋊D101—22no—
60C2×C22⋊D101—22no—
61C2×C22⋊D101—22no—
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C2×C4×D101—22no—
70C2×C4×D101—22no—
71C2×C4×D101—22no—
72C4⋊Dic101—22no—
73C4⋊Dic101—22no—
74#13108053—22no—
75#13108053—22no—
76#13108053—22no—
77#13108053—22no—
78C22⋊Dic101—22no—
79C22⋊Dic101—22no—
80C22⋊Dic101—22no—
81C22⋊Dic101—22no—
82C22⋊Dic101—22no—
83C22⋊Dic101—22no—
84#13108051—22no—
85#13108051—22no—
86#13108051—22no—
87#13108051—22no—
88C4×Dic101—22no—
89C4×Dic101—22no—
90C4×Dic101D414yescomplemented
91C24×C101C2311yescharacteristic
92C24×C101C2311yescharacteristic
93C2×D4×C101—22no—
94C2×D4×C101—22no—
95C2×D4×C101—22no—
96C22×C4×C101—22no—
97C22×C4×C101—22no—
98C22×C4×C101—22no—
99C22×C4×C101—22no—
100C22×C4×C101—22no—
101C22×C4×C101D414yescomplemented
102C22×C4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104C22×C4×C101C2311yescharacteristic
105C22×C4×C101C2311yescharacteristic
106C22×C4×C101C2311yescharacteristic
107C4⋊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×C101—21no—
119C23×C101—21no—
120C23×C101—21no—
121C23×C101—21no—
122C23×C101—21no—
123C23×C101C2×D412yes—
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#4112yes—
133D4×C101—28no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101—28no—
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—44no—
146C2×C4×C101C2×D412yes—
147C2×C4×C101C2×D412yes—
148C2×C4×C101C2×D412yes—
149C2×C4×C101#4112yes—
150C2×C4×C101#4112yes—
151C22⋊D101—28no—
152C22⋊D101—28no—
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×Dic101—21no—
167C2×Dic101—21no—
168C2×Dic101—21no—
169C2×Dic101—21no—
170C2×Dic101—21no—
171C2×Dic101C2×D412yes—
172C2×Dic101C2×D412yes—
173C2×Dic101#4112yes—
174C4×D101—44no—
175C22×C101—22no—
176C22×C101—22no—
177C22×C101—22no—
178C22×C101—22no—
179C22×C101—22no—
180C22×C101—22no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101C22×D411yescharacteristic
190C22×C101C22×D411yescharacteristic
191C22×C101C2×#4111yescharacteristic
192C22×C101C2×#4111yescharacteristic
193C22×C101C2×#4111yescharacteristic
194C22×C101C2×#4111yescharacteristic
195C22×C101C2×#4111yescharacteristic
196C2×D101—22no—
197C2×D101—22no—
198C2×D101—22no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C4×C101—22no—
204C4×C101—22no—
205C4×C101—41no—
206C4×C101—41no—
207C4×C101#12114yes—
208Dic101—22no—
209Dic101—22no—
210Dic101—22no—
211Dic101—41no—
212Dic101—41no—
213Dic101—41no—
214Dic101—41no—
215C2×C101—42no—
216C2×C101—42no—
217C2×C101C2×#12111yescharacteristic
218C2×C101#52511yescharacteristic
219C2×C101#52511yescharacteristic
220C2×C101#52511yescharacteristic
221C2×C101#52911yescharacteristic
222C2×C101#52911yescharacteristic
223C2×C101#53211yescharacteristic
224D101—42no—
225#2392—1011nomaximal, Sylow
226C101#239211yescharacteristic, complemented, Sylow
227C2×C4×D4—1011no—
228C2×C4×D4—1011no—
229C2×C4×D4D10111yescharacteristic, complemented
230C2×#117—1011no—
231#398—1011no—
232#397—1011no—
233#396—1011no—
234#396—1011no—
235#394—1011no—
236#394—1011no—
237#394—1011no—
238#392—1011no—
239C4×C4⋊C4—1011no—
240C4×C22⋊C4—1011no—
241C4×C22⋊C4—1011no—
242C22×D4—1011no—
243C22×D4—1011no—
244C22×D4C2×D10111yescharacteristic
245C23×C4—1011no—
246C23×C4C2×D10111yescharacteristic
247C23×C4C2×D10111yescharacteristic
248C4×D4—24no—
249C4×D4—2024no—
250C4×D4—2024no—
251#117—2024no—
252C2×C4⋊C4—1011no—
253C2×C4⋊C4—1011no—
254C2×C4⋊C4—1011no—
255C2×C4⋊C4—1011no—
256C2×C4⋊C4—1011no—
257C2×C4⋊C4—1011no—
258C2×C4⋊C4C2×D10111yescharacteristic
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⋊C4—1011no—
267C2×C22⋊C4—1011no—
268C2×C22⋊C4—1011no—
269C2×C22⋊C4C2×D10111yescharacteristic
270C2×C22⋊C4C2×D10111yescharacteristic
271C2×C42—1011no—
272C2×C42—1011no—
273C2×C42—1011no—
274C2×C42C2×D10111yescharacteristic
275#95—1011no—
276#95—1011no—
277#95—1011no—
278#95—1011no—
279#95—1011no—
280#95—1011no—
281C24—1011no—
282C24C22×D10111yescharacteristic
283C24C22×D10111yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—2022no—
288C2×D4—2022no—
289C2×D4—2022no—
290C2×D4—2022no—
291C2×D4—2022no—
292C2×D4—2022no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—1011no—
299C22×C4—1011no—
300C22×C4—1011no—
301C22×C4—1011no—
302C22×C4—1011no—
303C22×C4—1011no—
304C22×C4—1011no—
305C22×C4—2022no—
306C22×C4—2022no—
307C22×C4—2022no—
308C22×C4C22⋊D10114yescomplemented
309C22×C4C22×D10111yescharacteristic
310C22×C4C22×D10111yescharacteristic
311C22×C4C22×D10111yescharacteristic
312C22×C4C22×D10111yescharacteristic
313C22×C4C22×D10111yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—2022no—
316C4⋊C4—2022no—
317C4⋊C4—2022no—
318C4⋊C4—2022no—
319C4⋊C4—2022no—
320C4⋊C4—2022no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—2022no—
324C22⋊C4—2022no—
325C22⋊C4—2022no—
326C22⋊C4—2022no—
327C22⋊C4—2022no—
328C22⋊C4—2022no—
329C22⋊C4—2022no—
330C22⋊C4—2022no—
331C22⋊C4—2022no—
332C22⋊C4—2022no—
333C42—22no—
334C42—1014no—
335C42—2022no—
336C42—2022no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—1012no—
350C23—2021no—
351C23—2021no—
352C23—2021no—
353C23—2021no—
354C23—2021no—
355C23—2021no—
356C23C2×C22⋊D10112yes—
357C23#1310807412yes—
358C23C23×D10111yescharacteristic
359D4—28no—
360D4—2028no—
361D4—2028no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—28no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—1012no—
375C2×C4—1012no—
376C2×C4—1012no—
377C2×C4—4044no—
378C2×C4—2021no—
379C2×C4—2021no—
380C2×C4—2021no—
381C2×C4—2021no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4—2021no—
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—2021no—
395C2×C4—2021no—
396C2×C4D4×D10112yes—
397C2×C4C2×C22⋊D10112yes—
398C2×C4C2×C22⋊D10112yes—
399C2×C4#1310807312yes—
400C2×C4#1310807412yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—4041no—
416C22—4041no—
417C22—4041no—
418C22—4041no—
419C22—2022no—
420C22—2022no—
421C22—2022no—
422C22C2×D4×D10111yescharacteristic
423C22C22×C22⋊D10111yescharacteristic
424C22#41×D10111yescharacteristic
425C22#41×D10111yescharacteristic
426C22C2×#1310807311yescharacteristic
427C22C2×#1310807411yescharacteristic
428C22C2×#1310807411yescharacteristic
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—4041no—
434C4—4041no—
435C4—4041no—
436C4—4041no—
437C4—2022no—
438C4—2022no—
439C4—2022no—
440C4#1569222714yes—
441C2—42no—
442C2—42no—
443C2—4042no—
444C2C2×#1569222711yescharacteristic
445C2#2026568411yescharacteristic
446C2#2026568611yescharacteristic
447C2#2026568811yescharacteristic
448C2#2026570711yescharacteristic
449C2#2026570911yescharacteristic
450C2#2026572411yescharacteristic
subgroupG/H
centerC22×C4C22×D101
commutator, socle, cosocleC23×C101C24
FrattiniC23C23×D101
FittingC2×C4×D4×C101C2
prime radical#20265085C2
hypercenterC2×C4×D4D101
hypocenterC101#2392
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D101C2semidirect
2. C2×C4×C22⋊D101C2semidirect
3. C2×#15692214C2semidirect
4. #20265110C2semidirect
5. #20265105C2semidirect
6. #20265100C2semidirect
7. #20265099C2semidirect
8. #20265093C2semidirect
9. #20265092C2semidirect
10. #20265091C2semidirect
12. C4×#13108053C2semidirect
13. C4×#13108051C2semidirect
14. C4×C22⋊Dic101C2semidirect
15. C2×C4×D4×C101C2semidirect
90. C4×Dic101D4semidirect
101. C22×C4×C101D4semidirect
226. C101#2392semidirect
229. C2×C4×D4D101semidirect
308. C22×C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1012
C2×#156922142
#202651102
#202651052
#202651002
#202650992
#202650932
#202650922
#202650912
#202650852
C4×#131080532
C4×#131080512
C4×C22⋊Dic1012
C2×C4×D4×C1012
#2392101