Finite Group Databasebeta

← #37545273

Subgroups — #37545273

Order 23168.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153848C211yesmaximal, characteristic, complemented
2C2×#19153838C211yesmaximal, characteristic, complemented
3#26041939C211yesmaximal, characteristic, complemented
4#26041931C211yesmaximal, characteristic, complemented
5#26041672C211yesmaximal, characteristic, complemented
6#26041526C211yesmaximal, characteristic, complemented
7#26041497C211yesmaximal, characteristic, complemented
8#26041488C211yesmaximal, characteristic, complemented
9#26041439C211yesmaximal, characteristic, complemented
10#26041439C211yesmaximal, characteristic, complemented
11#26041436C211yesmaximal, characteristic, complemented
12#26041427C211yesmaximal, characteristic, complemented
13#26041378C211yesmaximal, characteristic, complemented
14#26041350C211yesmaximal, characteristic, complemented
15#467×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668449C2211yescharacteristic
18C2×#15668449C2211yescharacteristic
19C2×Q8×D181C2211yescharacteristic
20#19153848—24no—
21#19153838—24no—
22C2×Dic724C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668439C2211yescharacteristic
25C2×#15668438C2211yescharacteristic
26C2×#15668438C2211yescharacteristic
27C2×#15668436C2211yescharacteristic
28C2×#15668436C2211yescharacteristic
29C2×D1448C2211yescharacteristic
30C2×#15668432C2211yescharacteristic
31#19153814C2211yescharacteristic
32#19153805C2211yescharacteristic
33#19153802—21no—
34#19153801—21no—
35#19153798—21no—
36#19153796C2211yescharacteristic
37#19153794—21no—
38#19153791C2211yescharacteristic
39#19153789—21no—
40#19153789—21no—
41#19153781C2211yescharacteristic
42#19153778—21no—
43#19153778C2211yescharacteristic
44C4×C22⋊D181C2211yescharacteristic
45C4⋊C4×D181—21no—
46#19153745C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153736C2211yescharacteristic
51C22⋊#13089678C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153726C2211yescharacteristic
54C2×Q16×C181C2211yescharacteristic
55C22×C8×C181C2211yescharacteristic
56C22⋊Q8×C181C2211yescharacteristic
57C22⋊Q8×C181C2211yescharacteristic
58#107×C181C2211yescharacteristic
59#103×C181C2211yescharacteristic
60#103×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—41no—
63#15668449—22no—
64#15668449—22no—
65#15668449—41no—
66#15668449—41no—
67C2×Dic362—21no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70Q8×D181—22no—
71Q8×D181—41no—
72C2×D724—21no—
73C2×D724—21no—
74C2×D724C2311yescharacteristic
75C2×D724C2311yescharacteristic
76C2×C22⋊D181—21no—
77C2×C22⋊D181—21no—
78C2×C22⋊D181D411yescharacteristic
79C2×C4×D181—21no—
80C2×C4×D181—21no—
81C2×C4×D181—21no—
82C2×C4×D181—21no—
83C2×C4×D181—21no—
84C2×C4×D181C2311yescharacteristic
85Dic724—22no—
86#15668439—22no—
87#15668439—41no—
88#15668438—22no—
89#15668438D414yescomplemented
90#15668436—22no—
91#15668436—41no—
92D1448—22no—
93#15668432—22no—
94C2×#13089678C2311yescharacteristic
95C2×#13089678C2311yescharacteristic
96C4⋊Dic181—21no—
97#15668428—21no—
98#15668428—21no—
99#15668428—21no—
100#15668428D411yescharacteristic
101C22⋊Dic181D411yescharacteristic
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426D411yescharacteristic
108C4×Dic181—21no—
109C4×Dic181C2311yescharacteristic
110C2×Q8×C181C2311yescharacteristic
111C2×Q8×C181C2311yescharacteristic
112C22×C4×C181C2311yescharacteristic
113Q16×C181—22no—
114C2×C8×C181—22no—
115C2×C8×C181C2311yescharacteristic
116C2×C8×C181C2311yescharacteristic
117C4⋊C4×C181—21no—
118C4⋊C4×C181—21no—
119C4⋊C4×C181C2311yescharacteristic
120C4⋊C4×C181C2311yescharacteristic
121C22⋊C4×C181—21no—
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—21no—
125C22×D181C2×D411yescharacteristic
126Dic362—21no—
127Dic362—21no—
128Dic362—21no—
129Dic362—42no—
130Dic362C2×D412yes—
131D724—21no—
132D724—21no—
133D724—21no—
134D724—42no—
135D724—42no—
136D724C2×D412yes—
137C22⋊D181—41no—
138C22⋊D181—42no—
139C22⋊D181—42no—
140C2×Dic181—21no—
141C2×Dic181—21no—
142C2×Dic181C2×D411yescharacteristic
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C4×D181—21no—
146C4×D181—21no—
147C4×D181—42no—
148C4×D181—42no—
149C4×D181—42no—
150C4×D181—42no—
151C4×D181—42no—
152#13089678—21no—
153#13089678—21no—
154C23×C181C2×D411yescharacteristic
155Q8×C181—21no—
156Q8×C181—21no—
157Q8×C181—21no—
158Q8×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181C2×D411yescharacteristic
166C2×C4×C181C2411yescharacteristic
167C8×C181—21no—
168C8×C181C2×D412yes—
169C22×C181—21no—
170C22×C181—21no—
171C22×C181—21no—
172C22×C181C22×D411yescharacteristic
173C2×D181—21no—
174C2×D181—22no—
175C2×D181—21no—
176C2×D181—21no—
177C2×D181—41no—
178C2×D181—41no—
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C4×C181—21no—
183C4×C181—41no—
184C4×C181—41no—
185C4×C181—41no—
186C4×C181—41no—
187C4×C181C22×D411yescharacteristic
188C4×C181H(5,𝔽2)11yescharacteristic
189Dic181—21no—
190Dic181—22no—
191Dic181—21no—
192Dic181—21no—
193Dic181—41no—
194C2×C181—41no—
195C2×C181D4211yescharacteristic
196C2×C181#57511yescharacteristic
197C2×C181#57811yescharacteristic
198D181—41no—
199D181—42no—
200D181—81no—
201C181#326911yescharacteristic, complemented, Sylow
202#3269—1811nomaximal, Sylow
203C2×#137—1811no—
204C2×#135—1811no—
205#548—1811no—
206#547—1811no—
207#497—1811no—
208#468—1811no—
209#467D18111yescharacteristic, complemented
210#463—1811no—
211#462—1811no—
212#452—1811no—
213#452—1811no—
214#451—1811no—
215#449—1811no—
216#440—1811no—
217#435—1811no—
218C2×#41—1811no—
219C2×#41—1811no—
220C2×#41—1811no—
221C22×Q8—1811no—
222#137—3624no—
223#135—3624no—
224C2×Q16—1811no—
225C2×Q16—1811no—
226C2×Q16—1811no—
227C2×Q16C2×D18111yescharacteristic
228C2×SD16—1811no—
229C2×SD16—1811no—
230C2×SD16—1811no—
231C2×SD16—1811no—
232C2×D8—1811no—
233C2×M16—1811no—
234C22×C8C2×D18111yescharacteristic
235C4⋊Q8—1811no—
236#124—3621no—
237#124—1811no—
238#123—3621no—
239#123—3621no—
240C22⋊Q8—3621no—
241C22⋊Q8—1811no—
242C22⋊Q8C2×D18111yescharacteristic
243C22⋊Q8C2×D18111yescharacteristic
244#121—3621no—
245#121—3621no—
246#121—1811no—
247C4×Q8—1811no—
248C4×D4—3621no—
249C4×D4—1811no—
250C4×D4—1811no—
251C2×C4⋊C4—3621no—
252#107C2×D18111yescharacteristic
253C4⋊C8—1811no—
254#103—1811no—
255#103—1811no—
256#103C2×D18111yescharacteristic
257#103C2×D18111yescharacteristic
258#102—1811no—
259#102—1811no—
260C22⋊C8—1811no—
261C22⋊C8—1811no—
262#97—1811no—
263#41—3622no—
264#41—3622no—
265#41—3622no—
266#41—7241no—
267#41—7241no—
268#41—7241no—
269C2×Q8—3621no—
270C2×Q8—3622no—
271C2×Q8—1811no—
272C2×Q8—1811no—
273C2×Q8—7241no—
274C2×Q8C22×D18111yescharacteristic
275C2×Q8C22×D18111yescharacteristic
276C2×D4—3621no—
277C2×D4—3621no—
278C2×D4—3621no—
279C2×D4—3621no—
280C2×D4—1811no—
281C2×D4—1811no—
282C2×D4—1811no—
283C22×C4—3621no—
284C22×C4—3621no—
285C22×C4—3621no—
286C22×C4—3621no—
287C22×C4—3621no—
288C22×C4—1811no—
289C22×C4C22×D18111yescharacteristic
290Q16—22no—
291Q16—3622no—
292Q16—3622no—
293Q16—7241no—
294SD16—3622no—
295SD16—3622no—
296SD16—1814no—
297SD16—7241no—
298D8—3622no—
299M16—3622no—
300C2×C8—22no—
301C2×C8—1811no—
302C2×C8—1811no—
303C2×C8C22×D18111yescharacteristic
304C2×C8C22×D18111yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—3621no—
308C4⋊C4—3621no—
309C4⋊C4—3621no—
310C4⋊C4—3621no—
311C4⋊C4—1811no—
312C4⋊C4C22×D18111yescharacteristic
313C4⋊C4C22×D18111yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—3621no—
317C22⋊C4—3621no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—1811no—
322C22⋊C4—1811no—
323C42—3621no—
324C42—1811no—
325C23—3621no—
326C23—3621no—
327C23—1811no—
328C23D4×D18111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—3621no—
334Q8—3621no—
335Q8—3621no—
336Q8—1812no—
337Q8—7242no—
338D4—3621no—
339D4—3621no—
340D4—3621no—
341D4—1812no—
342D4—7242no—
343D4—7242no—
344D4—7241no—
345D4—7242no—
346D4—7242no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—3621no—
354C2×C4—3621no—
355C2×C4—3621no—
356C2×C4—3621no—
357C2×C4—1811no—
358C2×C4—1811no—
359C2×C4—1811no—
360C2×C4—7242no—
361C2×C4—7242no—
362C2×C4—7242no—
363C2×C4—7242no—
364C2×C4—7242no—
365C2×C4D4×D18111yescharacteristic
366C2×C4C23×D18111yescharacteristic
367C8—21no—
368C8—3621no—
369C8—3621no—
370C8D4×D18112yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—3621no—
375C22—3622no—
376C22—3621no—
377C22—3621no—
378C22—7241no—
379C22—7241no—
380C22—7241no—
381C22—7241no—
382C22—7241no—
383C22C2×D4×D18111yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—3621no—
390C4—3622no—
391C4—3621no—
392C4—3621no—
393C4—7241no—
394C4C2×D4×D18111yescharacteristic
395C4#1915386211yescharacteristic
396C2—41no—
397C2—14481no—
398C2—7241no—
399C2—7242no—
400C2#2604191311yescharacteristic
401C2#2604207611yescharacteristic
402C2#2604209211yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#467×C181C2
socleC22×C181C22×D4
hypercenter#467D181
hypocenterC181#3269
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#19153848C2semidirect
2. C2×#19153838C2semidirect
3. #26041939C2semidirect
4. #26041931C2semidirect
5. #26041672C2semidirect
6. #26041526C2semidirect
7. #26041497C2semidirect
8. #26041488C2semidirect
9. #26041439C2semidirect
10. #26041439C2semidirect
11. #26041436C2semidirect
12. #26041427C2semidirect
13. #26041378C2semidirect
14. #26041350C2semidirect
15. #467×C181C2semidirect
89. #15668438D4semidirect
201. C181#3269semidirect
209. #467D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538482
C2×#191538382
#260419392
#260419312
#260416722
#260415262
#260414972
#260414882
#260414392
#260414362
#260414272
#260413782
#260413502
#467×C1812
#3269181