Finite Group Databasebeta

← #44558663

Subgroups — #44558663

Order 32128.

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×#22548701C211yesmaximal, characteristic, complemented
2C2×#22548693C211yesmaximal, characteristic, complemented
3C2×#22548688C211yesmaximal, characteristic
4C2×C42×D251C211yesmaximal, characteristic, complemented
5#30631591C211yesmaximal, characteristic, complemented
6#30631587C211yesmaximal, characteristic, complemented
7#30631581C211yesmaximal, characteristic, complemented
8#30631578C211yesmaximal, characteristic, complemented
9#30631577C211yesmaximal, characteristic, complemented
10#30631565C211yesmaximal, characteristic, complemented
11#30631565C211yesmaximal, characteristic, complemented
12#30631563C211yesmaximal, characteristic, complemented
13#30631563C211yesmaximal, characteristic, complemented
14#30631551C211yesmaximal, characteristic, complemented
15#388×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C22×C22⋊D251C2211yescharacteristic
19C22×C22⋊D251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21C22×C4×D251C2211yescharacteristic
22C22×C4×D251C2211yescharacteristic
23#22548701—24no—
24#22548693—24no—
25#22548688—24no—
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×(C2×D251)⋊C4C2211yescharacteristic
37C2×(C2×D251)⋊C4C2211yescharacteristic
38C2×C4×Dic251C2211yescharacteristic
39C2×C4×Dic251C2211yescharacteristic
40C2×C4×Dic251C2211yescharacteristic
41C42×D251—24no—
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C251⋊#95C2211yescharacteristic
47C251⋊#95C2211yescharacteristic
48C2×C4⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C22⋊C4×C251C2211yescharacteristic
52C2×C42×C251C2211yescharacteristic
53#95×C251C2211yescharacteristic
54#95×C251C2211yescharacteristic
55C23×D251C2311yescharacteristic
56C23×D251C2311yescharacteristic
57C2×Dic502—22no—
58C2×Dic502—22no—
59C2×Dic502—22no—
60C2×D1004—22no—
61C2×D1004—22no—
62C2×D1004—22no—
63C2×C22⋊D251—22no—
64C2×C22⋊D251—22no—
65C2×C22⋊D251—22no—
66C2×C22⋊D251—22no—
67C2×C22⋊D251—22no—
68C2×C22⋊D251—22no—
69C22×Dic251C2311yescharacteristic
70C22×Dic251C2311yescharacteristic
71C22×Dic251C2311yescharacteristic
72C22×Dic251C2311yescharacteristic
73C22×Dic251C2311yescharacteristic
74C22×Dic251C2311yescharacteristic
75C2×C4×D251—22no—
76C2×C4×D251—22no—
77C2×C4×D251—22no—
78C2×C4×D251—22no—
79C2×C4×D251—22no—
80C2×C4×D251—22no—
81C2×C4×D251—22no—
82C2×C4×D251—22no—
83C2×C4×D251D414yes—
84Dic251⋊C4—22no—
85Dic251⋊C4—22no—
86Dic251⋊C4—22no—
87C22⋊Dic251—22no—
88C22⋊Dic251—22no—
89(C2×D251)⋊C4—22no—
90(C2×D251)⋊C4—22no—
91(C2×D251)⋊C4—22no—
92(C2×D251)⋊C4—22no—
93(C2×D251)⋊C4—22no—
94(C2×D251)⋊C4—22no—
95(C2×D251)⋊C4—22no—
96C4×Dic251—22no—
97C4×Dic251—22no—
98C4×Dic251D414yescomplemented
99C24×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102C22×C4×C251C2311yescharacteristic
103C22×C4×C251C2311yescharacteristic
104C22×C4×C251C2311yescharacteristic
105C22×C4×C251C2311yescharacteristic
106C4⋊C4×C251—22no—
107C22⋊C4×C251—22no—
108C22⋊C4×C251—22no—
109C22⋊C4×C251—22no—
110C42×C251—22no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—21no—
116C22×D251—21no—
117C22×D251—21no—
118C22×D251—21no—
119C22×D251—21no—
120C22×D251—21no—
121C22×D251—21no—
122C22×D251C2×D412yes—
123C22×D251#4112yes—
124C22×D251#4112yes—
125Dic502—44no—
126D1004—44no—
127C22⋊D251—44no—
128C22⋊D251—44no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—21no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251C2×D412yes—
143C2×Dic251C2×D412yes—
144C2×Dic251C2×D412yes—
145C2×Dic251#4112yes—
146C2×Dic251#4112yes—
147C4×D251—28no—
148C4×D251—28no—
149C4×D251—28no—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251—21no—
153C23×C251—21no—
154C23×C251—21no—
155C23×C251—21no—
156C23×C251—21no—
157C23×C251C2411yescharacteristic
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251C2×D412yes—
174C2×C4×C251C2×D412yes—
175C2×C4×C251#4112yes—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251C22×D411yescharacteristic
184C22×C251C22×D411yescharacteristic
185C22×C251C2×#4111yescharacteristic
186C22×C251C2×#4111yescharacteristic
187C22×C251C2×#4111yescharacteristic
188C22×C251C2×#4111yescharacteristic
189C22×C251C2×#4111yescharacteristic
190C2×D251—22no—
191C2×D251—22no—
192C2×D251—22no—
193C2×D251—22no—
194C2×D251—22no—
195C2×D251—22no—
196C2×D251—41no—
197C2×D251—41no—
198C2×D251—41no—
199C2×D251—41no—
200C2×D251—41no—
201C2×D251—41no—
202C2×D251—41no—
203C2×D251#12414yescomplemented
204C4×C251—22no—
205C4×C251—22no—
206C4×C251—22no—
207C4×C251—41no—
208C4×C251—41no—
209C4×C251—41no—
210Dic251—22no—
211Dic251—22no—
212Dic251—41no—
213Dic251—41no—
214Dic251—41no—
215Dic251#12114yes—
216C2×C251—81no—
217C2×C251C2×#12111yescharacteristic
218C2×C251#52511yescharacteristic
219C2×C251#52511yescharacteristic
220C2×C251C2×#12411yescharacteristic
221C2×C251#52911yescharacteristic
222C2×C251#52911yescharacteristic
223C2×C251#53211yescharacteristic
224D251—24no—
225D251—81no—
226C251#285011yescharacteristic, complemented, Sylow
227#2850—2511nomaximal, Sylow
228C2×#124—2511no—
229C2×C22⋊Q8—2511no—
230C2×#121—2511no—
231C22×C42—2511no—
232#391—2511no—
233#390—2511no—
234#388—2511no—
235#388—2511no—
236#388—2511no—
237#388D25111yescharacteristic, complemented
238#386—2511no—
239#386—2511no—
240#386—2511no—
241#386—2511no—
242#382—2511no—
243C22×Q8—2511no—
244C22×D4—2511no—
245C22×D4—2511no—
246C22×D4—2511no—
247C23×C4—2511no—
248C23×C4—2511no—
249C23×C4—2511no—
250#124—5024no—
251C22⋊Q8—5024no—
252#121—5024no—
253C2×C4⋊C4—2511no—
254C2×C4⋊C4—2511no—
255C2×C4⋊C4—2511no—
256C2×C4⋊C4C2×D25111yescharacteristic
257C2×C22⋊C4—2511no—
258C2×C22⋊C4—2511no—
259C2×C22⋊C4—2511no—
260C2×C22⋊C4—2511no—
261C2×C22⋊C4—2511no—
262C2×C22⋊C4—2511no—
263C2×C22⋊C4—2511no—
264C2×C22⋊C4—2511no—
265C2×C22⋊C4—2511no—
266C2×C22⋊C4C2×D25111yescharacteristic
267C2×C22⋊C4C2×D25111yescharacteristic
268C2×C22⋊C4C2×D25111yescharacteristic
269C2×C42—5024no—
270C2×C42—2511no—
271C2×C42—2511no—
272C2×C42—2511no—
273C2×C42C2×D25111yescharacteristic
274#95—2511no—
275#95—2511no—
276#95—2511no—
277#95—2511no—
278#95—2511no—
279#95—2511no—
280#95C2×D25111yescharacteristic
281#95C2×D25111yescharacteristic
282C24—2511no—
283C24—2511no—
284C24C22×D25111yescharacteristic
285C2×Q8—5022no—
286C2×Q8—5022no—
287C2×Q8—5022no—
288C2×D4—5022no—
289C2×D4—5022no—
290C2×D4—5022no—
291C2×D4—5022no—
292C2×D4—5022no—
293C2×D4—5022no—
294C2×D4—5022no—
295C2×D4—5022no—
296C2×D4—5022no—
297C22×C4—5022no—
298C22×C4—5022no—
299C22×C4—5022no—
300C22×C4—5022no—
301C22×C4—5022no—
302C22×C4—5022no—
303C22×C4—5022no—
304C22×C4—5022no—
305C22×C4—2511no—
306C22×C4—2511no—
307C22×C4—2511no—
308C22×C4—2511no—
309C22×C4—2511no—
310C22×C4—2514no—
311C22×C4—2511no—
312C22×C4C22×D25111yescharacteristic
313C22×C4C22×D25111yescharacteristic
314C22×C4C22×D25111yescharacteristic
315C22×C4C22×D25111yescharacteristic
316C22×C4C22×D25111yescharacteristic
317C22×C4C22×D25111yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—5022no—
320C4⋊C4—5022no—
321C4⋊C4—5022no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—5022no—
326C22⋊C4—5022no—
327C22⋊C4—5022no—
328C22⋊C4—5022no—
329C22⋊C4—5022no—
330C22⋊C4—5022no—
331C22⋊C4—5022no—
332C22⋊C4—5022no—
333C22⋊C4—5022no—
334C42—22no—
335C42—5022no—
336C42—5022no—
337C42—2514no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—5021no—
346C23—5021no—
347C23—5021no—
348C23—5021no—
349C23—5021no—
350C23—5021no—
351C23—5021no—
352C23—5021no—
353C23—5021no—
354C23—5021no—
355C23—5021no—
356C23—2512no—
357C23—2512no—
358C23—2512no—
359C23C23×D25111yescharacteristic
360Q8—10044no—
361D4—10044no—
362D4—10044no—
363D4—10044no—
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—5028no—
380C2×C4—5028no—
381C2×C4—5028no—
382C2×C4—5021no—
383C2×C4—5021no—
384C2×C4—5021no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4—5021no—
389C2×C4—5021no—
390C2×C4—5021no—
391C2×C4—5021no—
392C2×C4—5021no—
393C2×C4—5021no—
394C2×C4—5021no—
395C2×C4—2512no—
396C2×C4—2512no—
397C2×C4—2512no—
398C2×C4—2512no—
399C2×C4—2512no—
400C2×C4D4×D25112yes—
401C2×C4D4×D25112yes—
402C2×C4#1686081012yes—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—10041no—
411C22—10041no—
412C22—10041no—
413C22—10041no—
414C22—10041no—
415C22—10041no—
416C22—10041no—
417C22—5022no—
418C22—5022no—
419C22—5022no—
420C22—5022no—
421C22—5022no—
422C22—5022no—
423C22—2514no—
424C22C2×D4×D25111yescharacteristic
425C22C2×D4×D25111yescharacteristic
426C22#41×D25111yescharacteristic
427C22#41×D25111yescharacteristic
428C22#41×D25111yescharacteristic
429C22#41×D25111yescharacteristic
430C22C2×#1686081011yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—10041no—
438C4—10041no—
439C4—10041no—
440C4—5022no—
441C4—5022no—
442C4—2514no—
443C2—81no—
444C2—20081no—
445C2—5024no—
446C2#121×D25111yescharacteristic
447C2#3063219511yescharacteristic
448C2#3063219611yescharacteristic
449C2#124×D25111yescharacteristic
450C2#3063221511yescharacteristic
451C2#3063222011yescharacteristic
452C2#3063223411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#388×C251C2
prime radicalC2×#22548688C2
hypercenter#388D251
hypocenterC251#2850
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#22548701C2semidirect
2. C2×#22548693C2semidirect
4. C2×C42×D251C2semidirect
5. #30631591C2semidirect
6. #30631587C2semidirect
7. #30631581C2semidirect
8. #30631578C2semidirect
9. #30631577C2semidirect
10. #30631565C2semidirect
11. #30631565C2semidirect
12. #30631563C2semidirect
13. #30631563C2semidirect
14. #30631551C2semidirect
15. #388×C251C2semidirect
98. C4×Dic251D4semidirect
203. C2×D251#124semidirect
226. C251#2850semidirect
237. #388D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487012
C2×#225486932
C2×#225486882
C2×C42×D2512
#306315912
#306315872
#306315812
#306315782
#306315772
#306315652
#306315632
#306315512
#388×C2512
#2850251