Finite Group Databasebeta

← #44553263

Subgroups — #44553263

Order 32128.

776 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D251C211yesmaximal, characteristic
2C2×(C22⋊D251)⋊C4C211yesmaximal, characteristic
3C2×(C22⋊D251)⋊C4C211yesmaximal, characteristic
4C22×Dic251⋊C4C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D251C211yesmaximal, characteristic, complemented
6#30631584C211yesmaximal, characteristic, complemented
7#30631582C211yesmaximal, characteristic, complemented
8#30631562C211yesmaximal, characteristic, complemented
9#30631561C211yesmaximal, characteristic, complemented
10#30631558C211yesmaximal, characteristic, complemented
11#30631556C211yesmaximal, characteristic, complemented
12#30631546C211yesmaximal, characteristic, complemented
13#30631545C211yesmaximal, characteristic, complemented
14#30631538C211yesmaximal, characteristic, complemented
15#385×C251C211yesmaximal, characteristic, complemented
16C22×C22⋊D251C2211yescharacteristic
17C23×Dic251C2211yescharacteristic
18C23×Dic251C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21C22×C4×D251C2211yescharacteristic
22C4×C22⋊D251—24no—
23(C22⋊D251)⋊C4—24no—
24(C22⋊D251)⋊C4C418yescomplemented
25C2×C4⋊Dic251C2211yescharacteristic
26C2×C4⋊Dic251C2211yescharacteristic
27C2×Dic251⋊C4—24no—
28C2×Dic251⋊C4C2211yescharacteristic
29C2×Dic251⋊C4C2211yescharacteristic
30C2×Dic251⋊C4C2211yescharacteristic
31C2×Dic251⋊C4C2211yescharacteristic
32C2×Dic251⋊C4C2211yescharacteristic
33C2×Dic251⋊C4C2211yescharacteristic
34C2×Dic251⋊C4C2211yescharacteristic
35C2×Dic251⋊C4C2211yescharacteristic
36C2×Dic251⋊C4C2211yescharacteristic
37C4⋊C4×D251—24no—
38C2×C22⋊Dic251C2211yescharacteristic
39C2×(C2×D251)⋊C4C2211yescharacteristic
40C2×(C2×D251)⋊C4C2211yescharacteristic
41C2×(C2×D251)⋊C4C2211yescharacteristic
42C2×C4×Dic251C2211yescharacteristic
43C2×C4×Dic251C2211yescharacteristic
44C2×C4×Dic251C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C251⋊#95C2211yescharacteristic
47C251⋊#95C2211yescharacteristic
48C251⋊#95C2211yescharacteristic
49C23×C4×C251C2211yescharacteristic
50C2×C4⋊C4×C251C2211yescharacteristic
51C2×C4⋊C4×C251C2211yescharacteristic
52C2×C22⋊C4×C251C2211yescharacteristic
53C2×C22⋊C4×C251C2211yescharacteristic
54#95×C251C2211yescharacteristic
55#95×C251C2211yescharacteristic
56C23×D251C2311yescharacteristic
57C2×C22⋊D251C2×C414yes—
58C2×C22⋊D251D414yes—
59C2×C22⋊D251Q814yes—
60C22×Dic251—22no—
61C22×Dic251—22no—
62C22×Dic251—22no—
63C22×Dic251—22no—
64C22×Dic251—41no—
65C22×Dic251C2×C414yescomplemented
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C22×Dic251C2311yescharacteristic
70C22×Dic251C2311yescharacteristic
71C22×Dic251C2311yescharacteristic
72C22×Dic251C2311yescharacteristic
73C2×C4×D251—22no—
74C2×C4×D251—22no—
75C2×C4×D251—22no—
76C2×C4×D251—22no—
77C2×C4×D251—22no—
78C2×C4×D251—22no—
79C2×C4×D251—41no—
80C2×C4×D251—41no—
81C2×C4×D251C2×C414yescomplemented
82C4⋊Dic251—22no—
83C4⋊Dic251—41no—
84Dic251⋊C4—22no—
85Dic251⋊C4—22no—
86Dic251⋊C4—22no—
87Dic251⋊C4—22no—
88Dic251⋊C4—22no—
89Dic251⋊C4—22no—
90Dic251⋊C4—22no—
91Dic251⋊C4C2×C414yescomplemented
92Dic251⋊C4D414yescomplemented
93C22⋊Dic251—22no—
94(C2×D251)⋊C4—22no—
95(C2×D251)⋊C4—22no—
96(C2×D251)⋊C4C2×C414yescomplemented
97C4×Dic251—22no—
98C4×Dic251—22no—
99C4×Dic251C2×C414yescomplemented
100C24×C251C2311yescharacteristic
101C22×C4×C251—22no—
102C22×C4×C251—22no—
103C22×C4×C251—41no—
104C22×C4×C251C2311yescharacteristic
105C22×C4×C251C2311yescharacteristic
106C22×C4×C251C2311yescharacteristic
107C22×C4×C251C2311yescharacteristic
108C22×C4×C251C2311yescharacteristic
109C22×C4×C251C2311yescharacteristic
110C4⋊C4×C251—22no—
111C4⋊C4×C251—41no—
112C22⋊C4×C251—22no—
113C22⋊C4×C251C2×C414yescomplemented
114C22×D251—21no—
115C22×D251—21no—
116C22×D251—21no—
117C22×D251—21no—
118C22×D251C22×C412yes—
119C22×D251C2×D412yes—
120C22×D251C2×Q812yes—
121C22⋊D251C4⋊C4116yescomplemented
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—28no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—44no—
137C2×Dic251C22×C412yes—
138C2×Dic251C22×C412yes—
139C2×Dic251C22×C412yes—
140C2×Dic251C2×D412yes—
141C2×Dic251C2×D412yes—
142C2×Dic251C2×D412yes—
143C2×Dic251C2×Q812yes—
144C2×Dic251#4112yes—
145C4×D251—28no—
146C4×D251—44no—
147C4×D251—44no—
148C23×C251—21no—
149C23×C251—21no—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251C22×C412yes—
153C23×C251C2×D412yes—
154C23×C251C2×Q812yes—
155C23×C251C2411yescharacteristic
156C2×C4×C251—21no—
157C2×C4×C251—21no—
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—44no—
172C2×C4×C251C22×C412yes—
173C2×C4×C251C22×C412yes—
174C2×C4×C251C2×D412yes—
175C22×C251—22no—
176C22×C251—22no—
177C22×C251—22no—
178C22×C251—22no—
179C22×C251—22no—
180C22×C251—22no—
181C22×C251C2×C4⋊C414yescomplemented
182C22×C251C23×C411yescharacteristic
183C22×C251C22×D411yescharacteristic
184C22×C251C22×D411yescharacteristic
185C22×C251C22×Q811yescharacteristic
186C22×C251C2×#4111yescharacteristic
187C22×C251H(5,𝔽2)11yescharacteristic
188C22×C2512−1+411yescharacteristic
189C2×D251—22no—
190C2×D251—22no—
191C2×D251—22no—
192C2×D251—22no—
193C2×D251—22no—
194C2×D251—22no—
195C2×D251C2×C4⋊C414yescomplemented
196C4×C251—22no—
197C4×C251—22no—
198C4×C251—22no—
199C4×C251—41no—
200C4×C251—41no—
201C4×C251—41no—
202Dic251—22no—
203Dic251—22no—
204Dic251—41no—
205Dic251—41no—
206Dic251—41no—
207Dic251C2×C4⋊C414yescomplemented
208Dic251C4×D414yescomplemented
209C2×C251—24no—
210C2×C251C22×C4⋊C411yescharacteristic
211C2×C251C2×C4×D411yescharacteristic
212C2×C251#52011yescharacteristic
213C2×C251D4211yescharacteristic
214C2×C251#54711yescharacteristic
215C2×C251D4×Q811yescharacteristic
216C2×C251#55411yescharacteristic
217D251—24no—
218C251D4×C4⋊C411yescharacteristic, complemented, Sylow
219D4×C4⋊C4—2511nomaximal, Sylow
220C2×C4×D4—2511no—
221C2×C4×D4—2511no—
222C2×C4×D4—2511no—
223C22×C4⋊C4—2511no—
224C22×C4⋊C4—2511no—
225#389—2511no—
226#389—2511no—
227#385—2511no—
228#385—2511no—
229#385—2511no—
230#385D25111yescharacteristic, complemented
231#384—2511no—
232#380—2511no—
233#380—2511no—
234C4×C4⋊C4—2511no—
235C22×D4—2511no—
236C23×C4—2511no—
237C23×C4—2511no—
238C23×C4—2511no—
239C23×C4—2511no—
240C23×C4—2511no—
241C23×C4C2×D25111yescharacteristic
242C4×D4—5024no—
243C4×D4—5024no—
244C4×D4—2518no—
245C2×C4⋊C4—5024no—
246C2×C4⋊C4—5024no—
247C2×C4⋊C4—2511no—
248C2×C4⋊C4—2511no—
249C2×C4⋊C4—2511no—
250C2×C4⋊C4—2511no—
251C2×C4⋊C4—2511no—
252C2×C4⋊C4—2511no—
253C2×C4⋊C4—2511no—
254C2×C4⋊C4—2511no—
255C2×C4⋊C4—2511no—
256C2×C4⋊C4—2511no—
257C2×C4⋊C4—2511no—
258C2×C4⋊C4C2×D25111yescharacteristic
259C2×C4⋊C4C2×D25111yescharacteristic
260C2×C22⋊C4—2511no—
261C2×C22⋊C4—2511no—
262C2×C22⋊C4—2511no—
263C2×C22⋊C4—2511no—
264C2×C22⋊C4C2×D25111yescharacteristic
265C2×C22⋊C4C2×D25111yescharacteristic
266C2×C42—2511no—
267C2×C42—2511no—
268C2×C42—2511no—
269#95—2511no—
270#95—2511no—
271#95—2511no—
272#95—2511no—
273#95C2×D25111yescharacteristic
274#95C2×D25111yescharacteristic
275C24—2511no—
276C24C22×D25111yescharacteristic
277C2×D4—2514no—
278C2×D4—2514no—
279C2×D4—2514no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—10041no—
284C22×C4—10041no—
285C22×C4—10041no—
286C22×C4—5022no—
287C22×C4—5022no—
288C22×C4—5022no—
289C22×C4—5022no—
290C22×C4—5022no—
291C22×C4—5022no—
292C22×C4—5022no—
293C22×C4—5022no—
294C22×C4—5022no—
295C22×C4—5022no—
296C22×C4—2511no—
297C22×C4—2511no—
298C22×C4—2511no—
299C22×C4—2514no—
300C22×C4—2511no—
301C22×C4—2511no—
302C22×C4—2514no—
303C22×C4—2511no—
304C22×C4—2511no—
305C22×C4C22×D25111yescharacteristic
306C22×C4C22×D25111yescharacteristic
307C22×C4C22×D25111yescharacteristic
308C22×C4C22×D25111yescharacteristic
309C22×C4C22×D25111yescharacteristic
310C22×C4C22×D25111yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—10041no—
314C4⋊C4—5022no—
315C4⋊C4—5022no—
316C4⋊C4—5022no—
317C4⋊C4—5022no—
318C4⋊C4—5022no—
319C4⋊C4—5022no—
320C4⋊C4—5022no—
321C4⋊C4—5022no—
322C4⋊C4—2514no—
323C4⋊C4—2514no—
324C22⋊C4—22no—
325C22⋊C4—5022no—
326C22⋊C4—5022no—
327C22⋊C4—5022no—
328C22⋊C4—2514no—
329C22⋊C4C4×D25114yescomplemented
330C42—5022no—
331C42—5022no—
332C42—2514no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—5021no—
338C23—5021no—
339C23—5021no—
340C23—5021no—
341C23—2512no—
342C23—2512no—
343C23—2512no—
344C23C2×C4×D25112yes—
345C23D4×D25112yes—
346C23Q8×D25112yes—
347C23C23×D25111yescharacteristic
348D4—25116no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—44no—
365C2×C4—10044no—
366C2×C4—10044no—
367C2×C4—10044no—
368C2×C4—5028no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4—5021no—
373C2×C4—5021no—
374C2×C4—5021no—
375C2×C4—5021no—
376C2×C4—5021no—
377C2×C4—5021no—
378C2×C4—5021no—
379C2×C4—5021no—
380C2×C4—5021no—
381C2×C4—5021no—
382C2×C4—5028no—
383C2×C4—2512no—
384C2×C4—2512no—
385C2×C4—2512no—
386C2×C4—2512no—
387C2×C4—2512no—
388C2×C4—2512no—
389C2×C4—2512no—
390C2×C4—2512no—
391C2×C4C2×C4×D25112yes—
392C2×C4C2×C4×D25112yes—
393C2×C4D4×D25112yes—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—5022no—
401C22—5022no—
402C22—5022no—
403C22—5022no—
404C22—5022no—
405C22—5022no—
406C22—2514no—
407C22C4⋊C4×D25114yescomplemented
408C22C22×C4×D25111yescharacteristic
409C22C2×D4×D25111yescharacteristic
410C22C2×D4×D25111yescharacteristic
411C22C2×Q8×D25111yescharacteristic
412C22#41×D25111yescharacteristic
413C22#2254876011yescharacteristic
414C22#2254876211yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—10041no—
422C4—10041no—
423C4—10041no—
424C4—5022no—
425C4—5022no—
426C4—2514no—
427C4—2514no—
428C2—24no—
429C2—5024no—
430C2C2×C4⋊C4×D25111yescharacteristic
431C2C4×D4×D25111yescharacteristic
432C2#3063216611yescharacteristic
433C2#3063231911yescharacteristic
434C2#3063233511yescharacteristic
435C2#3063235011yescharacteristic
436C2#3063239111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#385×C251C2
socle, cosocleC23×C251C24
prime radicalC22×C22⋊D251C22
hypercenter#385D251
hypocenterC251D4×C4⋊C4
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. C22×Dic251⋊C4C2semidirect
5. C2×C4⋊C4×D251C2semidirect
6. #30631584C2semidirect
7. #30631582C2semidirect
8. #30631562C2semidirect
9. #30631561C2semidirect
10. #30631558C2semidirect
11. #30631556C2semidirect
12. #30631546C2semidirect
13. #30631545C2semidirect
14. #30631538C2semidirect
15. #385×C251C2semidirect
24. (C22⋊D251)⋊C4C4semidirect
65. C22×Dic251C2×C4semidirect
81. C2×C4×D251C2×C4semidirect
91. Dic251⋊C4C2×C4semidirect
92. Dic251⋊C4D4semidirect
96. (C2×D251)⋊C4C2×C4semidirect
99. C4×Dic251C2×C4semidirect
113. C22⋊C4×C251C2×C4semidirect
121. C22⋊D251C4⋊C4semidirect
181. C22×C251C2×C4⋊C4semidirect
195. C2×D251C2×C4⋊C4semidirect
207. Dic251C2×C4⋊C4semidirect
208. Dic251C4×D4semidirect
218. C251D4×C4⋊C4semidirect
230. #385D251semidirect
329. C22⋊C4C4×D251semidirect
407. C22C4⋊C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D2512
C2×(C22⋊D251)⋊C42
C22×Dic251⋊C42
C2×C4⋊C4×D2512
#306315842
#306315822
#306315622
#306315612
#306315582
#306315562
#306315462
#306315452
#306315382
#385×C2512
D4×C4⋊C4251