Finite Group Databasebeta

← #44558382

Subgroups — #44558382

Order 32128.

478 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548696C211yesmaximal, characteristic
2C2×#22548692C211yesmaximal, characteristic, complemented
3#30631621C211yesmaximal, characteristic, complemented
4#30631621C211yesmaximal, characteristic, complemented
5#30631618C211yesmaximal, characteristic, complemented
6#30631615C211yesmaximal, characteristic, complemented
7#30631615C211yesmaximal, characteristic, complemented
8#30631612C211yesmaximal, characteristic, complemented
9#30631610C211yesmaximal, characteristic, complemented
10#30631610C211yesmaximal, characteristic, complemented
11#30631607C211yesmaximal, characteristic, complemented
12#30631601C211yesmaximal, characteristic
13#30631599C211yesmaximal, characteristic, complemented
14#30631545C211yesmaximal, characteristic, complemented
15#394×C251C211yesmaximal, characteristic
16C22×Dic502C2211yescharacteristic
17C23×Dic251C2211yescharacteristic
18#22548696—24no—
19#22548692—24no—
20C2×C4⋊Dic251C2211yescharacteristic
21C2×C4⋊Dic251C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×Dic251⋊C4C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×C22⋊Dic251C2211yescharacteristic
33C2×C22⋊Dic251C2211yescharacteristic
34C2×C22⋊Dic251C2211yescharacteristic
35C2×C22⋊Dic251C2211yescharacteristic
36C2×C22⋊Dic251C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
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
53C2×Dic502—22no—
54C2×Dic502—41no—
55C2×Dic502—41no—
56C22×Dic251—41no—
57C22×Dic251—41no—
58C22×Dic251D414yes—
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C4⋊Dic251—22no—
68C4⋊Dic251—41no—
69Dic251⋊C4—22no—
70Dic251⋊C4—22no—
71Dic251⋊C4—22no—
72Dic251⋊C4—22no—
73Dic251⋊C4—41no—
74Dic251⋊C4—41no—
75Dic251⋊C4—41no—
76Dic251⋊C4—41no—
77C22⋊Dic251—22no—
78C22⋊Dic251—22no—
79C22⋊Dic251—22no—
80C22⋊Dic251—22no—
81C22⋊Dic251—41no—
82C22⋊Dic251—41no—
83C22⋊Dic251—41no—
84C24×C251C2311yescharacteristic
85C24×C251C2311yescharacteristic
86C2×D4×C251—22no—
87C2×D4×C251—41no—
88C2×D4×C251—41no—
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C4⋊C4×C251—41no—
95C22⋊C4×C251—22no—
96C22⋊C4×C251—41no—
97C22⋊C4×C251—41no—
98C22⋊C4×C251—41no—
99Dic502—44no—
100C2×Dic251—21no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C2×Dic251—21no—
108C2×Dic251—21no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—44no—
123C2×Dic251C2×D412yes—
124C2×Dic251C2×D412yes—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251—21no—
137C23×C251—21no—
138C23×C251C2×D412yes—
139C23×C251C2411yescharacteristic
140D4×C251—44no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251—21no—
154C2×C4×C251—21no—
155C2×C4×C251—21no—
156C22×C251—22no—
157C22×C251—22no—
158C22×C251—22no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251—41no—
168C22×C251—41no—
169C22×C251—41no—
170C22×C251C22×D411yescharacteristic
171C22×C251H(5,𝔽2)11yescharacteristic
172C22×C251H(5,𝔽2)11yescharacteristic
173C22×C251H(5,𝔽2)11yescharacteristic
174C22×C2512−1+411yescharacteristic
175C22×C2512−1+411yescharacteristic
176C22×C2512−1+411yescharacteristic
177C4×C251—41no—
178C4×C251—41no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251—41no—
182Dic251—22no—
183Dic251—22no—
184Dic251—41no—
185Dic251—41no—
186Dic251—41no—
187Dic251—41no—
188Dic251—41no—
189Dic251—41no—
190C2×C251—42no—
191C2×C251—81no—
192C2×C251#53411yescharacteristic
193C2×C251#53611yescharacteristic
194C2×C251#53711yescharacteristic
195C2×C251#56211yescharacteristic
196C2×C251#56211yescharacteristic
197C2×C251#56311yescharacteristic
198C2×C251#56311yescharacteristic
199C251#281411yescharacteristic, complemented, Sylow
200#2814—2511nomaximal, Sylow
201C2×#123—2511no—
202C2×C22⋊Q8—2511no—
203#398—2511no—
204#398—2511no—
205#398—2511no—
206#397—2511no—
207#397—2511no—
208#397—2511no—
209#396—2511no—
210#396—2511no—
211#395—2511no—
212#394—2511no—
213#394D25111yescharacteristic
214#393—2511no—
215#380—2511no—
216C22×Q8—2511no—
217C22×D4C2×D25111yescharacteristic
218C23×C4—2511no—
219#123—5024no—
220C22⋊Q8—5024no—
221C2×C4⋊C4—2511no—
222C2×C4⋊C4—2511no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4—2511no—
225C2×C4⋊C4—2511no—
226C2×C4⋊C4—2511no—
227C2×C4⋊C4—2511no—
228C2×C4⋊C4—2511no—
229C2×C4⋊C4—2511no—
230C2×C4⋊C4—2511no—
231C2×C4⋊C4C2×D25111yescharacteristic
232C2×C22⋊C4—2511no—
233C2×C22⋊C4—2511no—
234C2×C22⋊C4—2511no—
235C2×C22⋊C4—2511no—
236C2×C22⋊C4—2511no—
237C2×C22⋊C4—2511no—
238C2×C22⋊C4—2511no—
239C2×C22⋊C4C2×D25111yescharacteristic
240C2×C22⋊C4C2×D25111yescharacteristic
241C2×C22⋊C4C2×D25111yescharacteristic
242C2×C22⋊C4C2×D25111yescharacteristic
243#95—2511no—
244#95—2511no—
245#95—2511no—
246#95—2511no—
247#95—2511no—
248#95—2511no—
249#95—2511no—
250#95—2511no—
251#95—2511no—
252#95C2×D25111yescharacteristic
253C24C22×D25111yescharacteristic
254C24C22×D25111yescharacteristic
255C2×Q8—10041no—
256C2×Q8—10041no—
257C2×Q8—5022no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C22×C4—10041no—
262C22×C4—10041no—
263C22×C4—2514no—
264C22×C4—2511no—
265C22×C4—2511no—
266C22×C4—2511no—
267C22×C4—2511no—
268C22×C4—2511no—
269C22×C4—2511no—
270C22×C4—2511no—
271C22×C4—2511no—
272C22×C4C22×D25111yescharacteristic
273C22×C4C22×D25111yescharacteristic
274C22×C4C22×D25111yescharacteristic
275C22×C4C22×D25111yescharacteristic
276C22×C4C22×D25111yescharacteristic
277C4⋊C4—41no—
278C4⋊C4—10041no—
279C4⋊C4—10041no—
280C4⋊C4—10041no—
281C4⋊C4—10041no—
282C4⋊C4—10041no—
283C4⋊C4—5022no—
284C4⋊C4—5022no—
285C4⋊C4—5022no—
286C4⋊C4—5022no—
287C4⋊C4—5022no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—10041no—
293C22⋊C4—10041no—
294C22⋊C4—10041no—
295C22⋊C4—5022no—
296C22⋊C4—5022no—
297C22⋊C4—5022no—
298C22⋊C4—5022no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23D4×D25112yes—
313C23C23×D25111yescharacteristic
314Q8—10044no—
315D4—44no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—10044no—
332C2×C4—5021no—
333C2×C4—5021no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—5021no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4—5021no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—5021no—
352C2×C4—5021no—
353C2×C4—5021no—
354C2×C4—2512no—
355C2×C4—2512no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C2×D4×D25111yescharacteristic
371C22#2254876011yescharacteristic
372C22#2254876011yescharacteristic
373C22#2254876011yescharacteristic
374C22#2254876211yescharacteristic
375C22#2254876211yescharacteristic
376C22#2254876211yescharacteristic
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—10041no—
383C4—10041no—
384C4—10041no—
385C4—10041no—
386C4—10041no—
387C4—10041no—
388C4—5022no—
389C4—5022no—
390C2—42no—
391C2—81no—
392C2#3063224711yescharacteristic
393C2#3063225911yescharacteristic
394C2#3063226411yescharacteristic
395C2#3063243211yescharacteristic
396C2#3063243211yescharacteristic
397C2#3063243711yescharacteristic
398C2#3063244011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
prime radicalC22×D4×C251C22
hypercenter#394D251
hypocenterC251#2814
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#22548692C2semidirect
3. #30631621C2semidirect
4. #30631621C2semidirect
5. #30631618C2semidirect
6. #30631615C2semidirect
7. #30631615C2semidirect
8. #30631612C2semidirect
9. #30631610C2semidirect
10. #30631610C2semidirect
11. #30631607C2semidirect
13. #30631599C2semidirect
14. #30631545C2semidirect
199. C251#2814semidirect

Maximal subgroups

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

subgroupindex
C2×#225486962
C2×#225486922
#306316212
#306316182
#306316152
#306316122
#306316102
#306316072
#306316012
#306315992
#306315452
#394×C2512
#2814251