Finite Group Databasebeta

← #44558662

Subgroups — #44558662

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×#22548692C211yesmaximal, characteristic, complemented
3C2×#22548687C211yesmaximal, characteristic
4C22×C4×Dic251C211yesmaximal, characteristic, complemented
5#30631591C211yesmaximal, characteristic, complemented
6#30631587C211yesmaximal, characteristic, complemented
7#30631580C211yesmaximal, characteristic, complemented
8#30631579C211yesmaximal, characteristic, complemented
9#30631576C211yesmaximal, characteristic, complemented
10#30631575C211yesmaximal, characteristic, complemented
11#30631567C211yesmaximal, characteristic, complemented
12#30631566C211yesmaximal, characteristic, complemented
13#30631564C211yesmaximal, characteristic, complemented
14#30631551C211yesmaximal, characteristic, complemented
15#386×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C22×C22⋊D251C2211yescharacteristic
19C23×Dic251C2211yescharacteristic
20C23×Dic251C2211yescharacteristic
21#22548701—24no—
22#22548692—24no—
23#22548687—24no—
24C2×C4⋊Dic251C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×(C2×D251)⋊C4C2211yescharacteristic
37C2×C4×Dic251—24no—
38C2×C4×Dic251C2211yescharacteristic
39C2×C4×Dic251C2211yescharacteristic
40C2×C4×Dic251C2211yescharacteristic
41C2×C4×Dic251C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C251⋊#95C2211yescharacteristic
47C251⋊#95C2211yescharacteristic
48C22×D4×C251C2211yescharacteristic
49C23×C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C22⋊C4×C251C2211yescharacteristic
52C2×C22⋊C4×C251C2211yescharacteristic
53#95×C251C2211yescharacteristic
54#95×C251C2211yescharacteristic
55C23×D251C2311yescharacteristic
56C2×Dic502—22no—
57C2×Dic502—22no—
58C2×Dic502—22no—
59C2×C22⋊D251—22no—
60C2×C22⋊D251—22no—
61C2×C22⋊D251—22no—
62C2×C22⋊D251—22no—
63C2×C22⋊D251—22no—
64C2×C22⋊D251—22no—
65C22×Dic251—22no—
66C22×Dic251—22no—
67C22×Dic251—22no—
68C22×Dic251—22no—
69C22×Dic251—22no—
70C22×Dic251D414yes—
71C22×Dic251C2311yescharacteristic
72C22×Dic251C2311yescharacteristic
73C22×Dic251C2311yescharacteristic
74C22×Dic251C2311yescharacteristic
75C22×Dic251C2311yescharacteristic
76C22×Dic251C2311yescharacteristic
77C22×Dic251C2311yescharacteristic
78C4⋊Dic251—22no—
79Dic251⋊C4—22no—
80Dic251⋊C4—22no—
81Dic251⋊C4—22no—
82C22⋊Dic251—22no—
83C22⋊Dic251—22no—
84C22⋊Dic251—22no—
85C22⋊Dic251—22no—
86(C2×D251)⋊C4—22no—
87(C2×D251)⋊C4—22no—
88(C2×D251)⋊C4—22no—
89(C2×D251)⋊C4—22no—
90(C2×D251)⋊C4—22no—
91C4×Dic251—22no—
92C4×Dic251—22no—
93C4×Dic251—22no—
94C4×Dic251D414yescomplemented
95C24×C251C2311yescharacteristic
96C24×C251C2311yescharacteristic
97C2×D4×C251—22no—
98C2×D4×C251—22no—
99C2×D4×C251—22no—
100C22×C4×C251—22no—
101C22×C4×C251—22no—
102C22×C4×C251—22no—
103C22×C4×C251C2311yescharacteristic
104C22×C4×C251C2311yescharacteristic
105C22×C4×C251C2311yescharacteristic
106C22×C4×C251C2311yescharacteristic
107C22×C4×C251C2311yescharacteristic
108C22⋊C4×C251—22no—
109C22⋊C4×C251—22no—
110C22⋊C4×C251—22no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—21no—
116C22×D251—21no—
117C22×D251—21no—
118Dic502—44no—
119C22⋊D251—44no—
120C22⋊D251—44no—
121C2×Dic251—28no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—28no—
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×Dic251C2×D412yes—
139C2×Dic251C2×D412yes—
140C2×Dic251C2×D412yes—
141C2×Dic251C2×D412yes—
142C2×Dic251#4112yes—
143C2×Dic251#4112yes—
144C23×C251—21no—
145C23×C251—21no—
146C23×C251—21no—
147C23×C251—21no—
148C23×C251—21no—
149C23×C251—21no—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251—21no—
153C23×C251—21no—
154C23×C251—21no—
155C23×C251C2×D412yes—
156C23×C251#4112yes—
157C23×C251#4112yes—
158C23×C251C2411yescharacteristic
159D4×C251—44no—
160C2×C4×C251—28no—
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×C251—21no—
174C2×C4×C251C2×D412yes—
175C2×C4×C251#4112yes—
176C22×C251—22no—
177C22×C251—22no—
178C22×C251—22no—
179C22×C251—22no—
180C22×C251—22no—
181C22×C251—22no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251—41no—
188C22×C251—41no—
189C22×C251#12414yescomplemented
190C22×C251C22×D411yescharacteristic
191C22×C251C22×D411yescharacteristic
192C22×C251C2×#4111yescharacteristic
193C22×C251C2×#4111yescharacteristic
194C22×C251C2×#4111yescharacteristic
195C22×C251C2×#4111yescharacteristic
196C22×C251C2×#4111yescharacteristic
197C2×D251—41no—
198C2×D251—41no—
199C2×D251—41no—
200C2×D251—41no—
201C2×D251—41no—
202C2×D251—41no—
203C2×D251—41no—
204C4×C251—22no—
205C4×C251—22no—
206C4×C251—41no—
207C4×C251—41no—
208C4×C251—41no—
209Dic251—22no—
210Dic251—22no—
211Dic251—22no—
212Dic251—41no—
213Dic251—41no—
214Dic251—41no—
215Dic251#12114yes—
216C2×C251—24no—
217C2×C251—81no—
218C2×C251C2×#12111yescharacteristic
219C2×C251#52511yescharacteristic
220C2×C251#52511yescharacteristic
221C2×C251C2×#12411yescharacteristic
222C2×C251#52911yescharacteristic
223C2×C251#52911yescharacteristic
224C2×C251#53211yescharacteristic
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#388—2511no—
238#386—2511no—
239#386—2511no—
240#386—2511no—
241#386D25111yescharacteristic, complemented
242#382—2511no—
243C22×Q8—2511no—
244C22×D4—2511no—
245C22×D4—2511no—
246C22×D4C2×D25111yescharacteristic
247C23×C4—2511no—
248C23×C4—2511no—
249C23×C4C2×D25111yescharacteristic
250#124—5024no—
251C22⋊Q8—5024no—
252#121—5024no—
253C2×C4⋊C4—2511no—
254C2×C4⋊C4—2511no—
255C2×C4⋊C4—2511no—
256C2×C4⋊C4—2511no—
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×C42—2511no—
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—
283C24C22×D25111yescharacteristic
284C24C22×D25111yescharacteristic
285C2×Q8—5022no—
286C2×Q8—5022no—
287C2×Q8—5022no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—5022no—
292C2×D4—5022no—
293C2×D4—5022no—
294C2×D4—5022no—
295C2×D4—5022no—
296C2×D4—5022no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
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—2514no—
309C22×C4—2511no—
310C22×C4—2511no—
311C22×C4—2511no—
312C22×C4—2511no—
313C22×C4C22×D25111yescharacteristic
314C22×C4C22×D25111yescharacteristic
315C22×C4C22×D25111yescharacteristic
316C22×C4C22×D25111yescharacteristic
317C22×C4C22×D25111yescharacteristic
318C4⋊C4—5022no—
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—5022no—
335C42—5022no—
336C42—5022no—
337C42—2514no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—5021no—
350C23—5021no—
351C23—5021no—
352C23—5021no—
353C23—5021no—
354C23—5021no—
355C23—5021no—
356C23D4×D25112yes—
357C23#1686080912yes—
358C23#1686081012yes—
359C23C23×D25111yescharacteristic
360Q8—10044no—
361D4—44no—
362D4—10044no—
363D4—10044no—
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—5021no—
379C2×C4—5021no—
380C2×C4—5021no—
381C2×C4—5021no—
382C2×C4—5021no—
383C2×C4—5021no—
384C2×C4—5021no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5028no—
388C2×C4—5021no—
389C2×C4—5021no—
390C2×C4—5021no—
391C2×C4—5021no—
392C2×C4—5021no—
393C2×C4—5021no—
394C2×C4—5028no—
395C2×C4—2512no—
396C2×C4—2512no—
397C2×C4—2512no—
398C2×C4—2512no—
399C2×C4—2512no—
400C2×C4—2512no—
401C2×C4D4×D25112yes—
402C2×C4#1686080912yes—
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—10041no—
417C22—10041no—
418C22—10041no—
419C22—10041no—
420C22—10041no—
421C22—10041no—
422C22—10041no—
423C22#2254870114yescomplemented
424C22C2×D4×D25111yescharacteristic
425C22C2×D4×D25111yescharacteristic
426C22#41×D25111yescharacteristic
427C22#41×D25111yescharacteristic
428C22C2×#1686080911yescharacteristic
429C22C2×#1686080911yescharacteristic
430C22C2×#1686081011yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—10041no—
437C4—10041no—
438C4—10041no—
439C4—5022no—
440C4—5022no—
441C4—5022no—
442C4—2514no—
443C2—24no—
444C2—81no—
445C2—20081no—
446C2#121×D25111yescharacteristic
447C2#3063219311yescharacteristic
448C2#3063219611yescharacteristic
449C2C2×#2254870111yescharacteristic
450C2#3063221411yescharacteristic
451C2#3063221711yescharacteristic
452C2#3063223211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#386×C251C2
prime radicalC2×#22548687C2
hypercenter#386D251
hypocenterC251#2850
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#22548701C2semidirect
2. C2×#22548692C2semidirect
4. C22×C4×Dic251C2semidirect
5. #30631591C2semidirect
6. #30631587C2semidirect
7. #30631580C2semidirect
8. #30631579C2semidirect
9. #30631576C2semidirect
10. #30631575C2semidirect
11. #30631567C2semidirect
12. #30631566C2semidirect
13. #30631564C2semidirect
14. #30631551C2semidirect
15. #386×C251C2semidirect
94. C4×Dic251D4semidirect
189. C22×C251#124semidirect
226. C251#2850semidirect
241. #386D251semidirect
423. C22#22548701semidirect

Maximal subgroups

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

subgroupindex
C2×#225487012
C2×#225486922
C2×#225486872
C22×C4×Dic2512
#306315912
#306315872
#306315802
#306315792
#306315762
#306315752
#306315672
#306315662
#306315642
#306315512
#386×C2512
#2850251