Finite Group Databasebeta

← #44558397

Subgroups — #44558397

Order 32128.

528 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×#22548692C211yesmaximal, characteristic, complemented
2C2×#22548687C211yesmaximal, characteristic
3#30631623C211yesmaximal, characteristic, complemented
4#30631622C211yesmaximal, characteristic, complemented
5#30631621C211yesmaximal, characteristic, complemented
6#30631620C211yesmaximal, characteristic, complemented
7#30631603C211yesmaximal, characteristic, complemented
8#30631603C211yesmaximal, characteristic, complemented
9#30631602C211yesmaximal, characteristic, complemented
10#30631602C211yesmaximal, characteristic, complemented
11#30631601C211yesmaximal, characteristic, complemented
12#30631600C211yesmaximal, characteristic, complemented
13#30631599C211yesmaximal, characteristic, complemented
14#30631548C211yesmaximal, characteristic, complemented
15#394×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C22×C22⋊D251C2211yescharacteristic
19C23×Dic251C2211yescharacteristic
20#22548692—24no—
21#22548687—24no—
22C2×C4⋊Dic251C2211yescharacteristic
23C2×C4⋊Dic251C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×C22⋊Dic251C2211yescharacteristic
33C2×C22⋊Dic251C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×(C2×D251)⋊C4C2211yescharacteristic
37C2×(C2×D251)⋊C4C2211yescharacteristic
38C2×(C2×D251)⋊C4C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C22×D4×C251C2211yescharacteristic
47C2×C4⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C22⋊C4×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53C23×D251C2311yescharacteristic
54C2×Dic502—22no—
55C2×Dic502—41no—
56C2×Dic502—41no—
57C2×C22⋊D251—22no—
58C2×C22⋊D251—22no—
59C2×C22⋊D251—41no—
60C2×C22⋊D251—41no—
61C2×C22⋊D251—41no—
62C2×C22⋊D251—41no—
63C22×Dic251—41no—
64C22×Dic251—41no—
65C22×Dic251D414yes—
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C22×Dic251C2311yescharacteristic
70C22×Dic251C2311yescharacteristic
71C22×Dic251C2311yescharacteristic
72C22×Dic251C2311yescharacteristic
73C4⋊Dic251—22no—
74C4⋊Dic251—41no—
75Dic251⋊C4—22no—
76Dic251⋊C4—22no—
77Dic251⋊C4—22no—
78Dic251⋊C4—41no—
79Dic251⋊C4—41no—
80C22⋊Dic251—22no—
81C22⋊Dic251—22no—
82C22⋊Dic251—41no—
83C22⋊Dic251—41no—
84C22⋊Dic251—41no—
85(C2×D251)⋊C4—22no—
86(C2×D251)⋊C4—41no—
87(C2×D251)⋊C4—41no—
88(C2×D251)⋊C4—41no—
89(C2×D251)⋊C4—41no—
90C24×C251C2311yescharacteristic
91C24×C251C2311yescharacteristic
92C2×D4×C251—22no—
93C2×D4×C251—41no—
94C2×D4×C251—41no—
95C22×C4×C251C2311yescharacteristic
96C22×C4×C251C2311yescharacteristic
97C22×C4×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C4⋊C4×C251—41no—
101C22⋊C4×C251—22no—
102C22⋊C4×C251—41no—
103C22⋊C4×C251—41no—
104C22⋊C4×C251—41no—
105C22×D251—21no—
106C22×D251—21no—
107C22×D251—21no—
108C22×D251—21no—
109C22×D251—21no—
110C22×D251—21no—
111C22×D251—21no—
112Dic502—44no—
113C22⋊D251—44no—
114C22⋊D251—44no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—44no—
135C2×Dic251C2×D412yes—
136C2×Dic251C2×D412yes—
137C23×C251—21no—
138C23×C251—21no—
139C23×C251—21no—
140C23×C251—21no—
141C23×C251—21no—
142C23×C251—21no—
143C23×C251—21no—
144C23×C251—21no—
145C23×C251—21no—
146C23×C251—21no—
147C23×C251—21no—
148C23×C251—21no—
149C23×C251—21no—
150C23×C251C2×D412yes—
151C23×C251C2411yescharacteristic
152D4×C251—44no—
153C2×C4×C251—21no—
154C2×C4×C251—21no—
155C2×C4×C251—21no—
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—
168C22×C251—22no—
169C22×C251—22no—
170C22×C251—22no—
171C22×C251—41no—
172C22×C251—41no—
173C22×C251—41no—
174C22×C251—41no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251C22×D411yescharacteristic
183C22×C251H(5,𝔽2)11yescharacteristic
184C22×C251H(5,𝔽2)11yescharacteristic
185C22×C251H(5,𝔽2)11yescharacteristic
186C22×C251H(5,𝔽2)11yescharacteristic
187C22×C2512−1+411yescharacteristic
188C22×C2512−1+411yescharacteristic
189C2×D251—41no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C2×D251—41no—
196C4×C251—41no—
197C4×C251—41no—
198C4×C251—41no—
199C4×C251—41no—
200C4×C251—41no—
201Dic251—22no—
202Dic251—22no—
203Dic251—41no—
204Dic251—41no—
205Dic251—41no—
206Dic251—41no—
207Dic251—41no—
208C2×C251—42no—
209C2×C251—81no—
210C2×C251#53411yescharacteristic
211C2×C251#53511yescharacteristic
212C2×C251#53611yescharacteristic
213C2×C251#56211yescharacteristic
214C2×C251#56211yescharacteristic
215C2×C251#56211yescharacteristic
216C2×C251#56211yescharacteristic
217D251—81no—
218C251#281611yescharacteristic, complemented, Sylow
219#2816—2511nomaximal, Sylow
220C2×C22⋊Q8—2511no—
221C2×#121—2511no—
222#399—2511no—
223#399—2511no—
224#398—2511no—
225#398—2511no—
226#394—2511no—
227#394—2511no—
228#394—2511no—
229#394—2511no—
230#394—2511no—
231#394D25111yescharacteristic, complemented
232#393—2511no—
233#393—2511no—
234#381—2511no—
235C22×Q8—2511no—
236C22×D4—2511no—
237C22×D4—2511no—
238C22×D4C2×D25111yescharacteristic
239C23×C4—2511no—
240C22⋊Q8—5024no—
241#121—5024no—
242C2×C4⋊C4—2511no—
243C2×C4⋊C4—2511no—
244C2×C4⋊C4—2511no—
245C2×C4⋊C4—2511no—
246C2×C4⋊C4—2511no—
247C2×C4⋊C4—2511no—
248C2×C4⋊C4—2511no—
249C2×C4⋊C4C2×D25111yescharacteristic
250C2×C22⋊C4—2511no—
251C2×C22⋊C4—2511no—
252C2×C22⋊C4—2511no—
253C2×C22⋊C4—2511no—
254C2×C22⋊C4—2511no—
255C2×C22⋊C4—2511no—
256C2×C22⋊C4—2511no—
257C2×C22⋊C4—2511no—
258C2×C22⋊C4—2511no—
259C2×C22⋊C4—2511no—
260C2×C22⋊C4C2×D25111yescharacteristic
261C2×C22⋊C4C2×D25111yescharacteristic
262C2×C22⋊C4C2×D25111yescharacteristic
263C2×C22⋊C4C2×D25111yescharacteristic
264#95—2511no—
265#95—2511no—
266#95—2511no—
267#95—2511no—
268#95—2511no—
269#95—2511no—
270#95—2511no—
271#95C2×D25111yescharacteristic
272C24—2511no—
273C24C22×D25111yescharacteristic
274C24C22×D25111yescharacteristic
275C2×Q8—10041no—
276C2×Q8—10041no—
277C2×Q8—5022no—
278C2×D4—22no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—10041no—
282C2×D4—10041no—
283C2×D4—10041no—
284C2×D4—10041no—
285C2×D4—5022no—
286C2×D4—5022no—
287C22×C4—10041no—
288C22×C4—10041no—
289C22×C4—2514no—
290C22×C4—2511no—
291C22×C4—2511no—
292C22×C4—2511no—
293C22×C4—2511no—
294C22×C4—2511no—
295C22×C4—2511no—
296C22×C4—2511no—
297C22×C4C22×D25111yescharacteristic
298C22×C4C22×D25111yescharacteristic
299C22×C4C22×D25111yescharacteristic
300C22×C4C22×D25111yescharacteristic
301C22×C4C22×D25111yescharacteristic
302C4⋊C4—41no—
303C4⋊C4—10041no—
304C4⋊C4—10041no—
305C4⋊C4—10041no—
306C4⋊C4—5022no—
307C4⋊C4—5022no—
308C4⋊C4—5022no—
309C4⋊C4—5022no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—10041no—
315C22⋊C4—10041no—
316C22⋊C4—10041no—
317C22⋊C4—10041no—
318C22⋊C4—10041no—
319C22⋊C4—10041no—
320C22⋊C4—10041no—
321C22⋊C4—5022no—
322C22⋊C4—5022no—
323C22⋊C4—5022no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—5021no—
338C23—5021no—
339C23—5021no—
340C23—5021no—
341C23—5021no—
342C23—5021no—
343C23—5021no—
344C23D4×D25112yes—
345C23C23×D25111yescharacteristic
346Q8—10044no—
347D4—44no—
348D4—10044no—
349D4—10044no—
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—21no—
365C2×C4—10044no—
366C2×C4—5021no—
367C2×C4—5021no—
368C2×C4—5021no—
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—5021no—
383C2×C4—5021no—
384C2×C4—5021no—
385C2×C4—2512no—
386C2×C4—2512no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—10041no—
402C22—10041no—
403C22—10041no—
404C22—10041no—
405C22—10041no—
406C22—10041no—
407C22—10041no—
408C22C2×D4×D25111yescharacteristic
409C22#2254876011yescharacteristic
410C22#2254876011yescharacteristic
411C22#2254876011yescharacteristic
412C22#2254876111yescharacteristic
413C22#2254876211yescharacteristic
414C22#2254876211yescharacteristic
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—10041no—
421C4—10041no—
422C4—10041no—
423C4—10041no—
424C4—10041no—
425C4—5022no—
426C4—5022no—
427C2—42no—
428C2—81no—
429C2—20081no—
430C2#3063224511yescharacteristic
431C2#3063225211yescharacteristic
432C2#3063225911yescharacteristic
433C2#3063243111yescharacteristic
434C2#3063243211yescharacteristic
435C2#3063243211yescharacteristic
436C2#3063243411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
prime radicalC2×#22548687C2
hypercenter#394D251
hypocenterC251#2816
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22548692C2semidirect
3. #30631623C2semidirect
4. #30631622C2semidirect
5. #30631621C2semidirect
6. #30631620C2semidirect
7. #30631603C2semidirect
8. #30631603C2semidirect
9. #30631602C2semidirect
10. #30631602C2semidirect
11. #30631601C2semidirect
12. #30631600C2semidirect
13. #30631599C2semidirect
14. #30631548C2semidirect
15. #394×C251C2semidirect
218. C251#2816semidirect
231. #394D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486922
C2×#225486872
#306316232
#306316222
#306316212
#306316202
#306316032
#306316022
#306316012
#306316002
#306315992
#306315482
#394×C2512
#2816251