Finite Group Databasebeta

← #37536388

Subgroups — #37536388

Order 23168.

632 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153798C211yesmaximal, characteristic
2C2×C4×C22⋊D181C211yesmaximal, characteristic
3C2×C4⋊C4×D181C211yesmaximal, characteristic, complemented
4#26041176C211yesmaximal, characteristic, complemented
5#26041175C211yesmaximal, characteristic, complemented
6#26041172C211yesmaximal, characteristic, complemented
7#26041167C211yesmaximal, characteristic, complemented
8#26041164C211yesmaximal, characteristic, complemented
9#26041160C211yesmaximal, characteristic, complemented
10#26041155C211yesmaximal, characteristic
11#26041152C211yesmaximal, characteristic, complemented
12#26041150C211yesmaximal, characteristic, complemented
13#26041141C211yesmaximal, characteristic, complemented
14#26041126C211yesmaximal, characteristic, complemented
15#385×C181C211yesmaximal, characteristic, complemented
16C22×C22⋊D181C2211yescharacteristic
17C22×C4×D181C2211yescharacteristic
18C22×C4×D181C2211yescharacteristic
19C22×C4×D181C2211yescharacteristic
20#19153798C418yescomplemented
21C4×C22⋊D181—24no—
22C2×C4⋊Dic181C2211yescharacteristic
23C2×C4⋊Dic181C2211yescharacteristic
24C2×#15668428C2211yescharacteristic
25C2×#15668428C2211yescharacteristic
26C2×#15668428C2211yescharacteristic
27C2×#15668428C2211yescharacteristic
28C2×#15668428C2211yescharacteristic
29C4⋊C4×D181—24no—
30C2×C22⋊Dic181C2211yescharacteristic
31C2×C22⋊Dic181C2211yescharacteristic
32C2×C22⋊Dic181C2211yescharacteristic
33C2×#15668426C2211yescharacteristic
34C2×#15668426C2211yescharacteristic
35C2×#15668426C2211yescharacteristic
36C2×C4×Dic181C2211yescharacteristic
37C2×C4×Dic181C2211yescharacteristic
38C2×C4×Dic181C2211yescharacteristic
39#19153723C2211yescharacteristic
40#19153723C2211yescharacteristic
41#19153723C2211yescharacteristic
42#19153723C2211yescharacteristic
43#19153723C2211yescharacteristic
44#19153723C2211yescharacteristic
45#19153723C2211yescharacteristic
46#19153723C2211yescharacteristic
47C23×C4×C181C2211yescharacteristic
48C2×C4⋊C4×C181C2211yescharacteristic
49C2×C4⋊C4×C181C2211yescharacteristic
50C2×C22⋊C4×C181C2211yescharacteristic
51C2×C22⋊C4×C181C2211yescharacteristic
52#95×C181C2211yescharacteristic
53#95×C181C2211yescharacteristic
54C23×D181C2311yescharacteristic
55C2×C22⋊D181—22no—
56C2×C22⋊D181—22no—
57C2×C22⋊D181C2×C414yes—
58C22×Dic181C2311yescharacteristic
59C22×Dic181C2311yescharacteristic
60C22×Dic181C2311yescharacteristic
61C22×Dic181C2311yescharacteristic
62C22×Dic181C2311yescharacteristic
63C22×Dic181C2311yescharacteristic
64C22×Dic181C2311yescharacteristic
65C2×C4×D181—22no—
66C2×C4×D181—22no—
67C2×C4×D181—22no—
68C2×C4×D181—41no—
69C2×C4×D181—41no—
70C2×C4×D181—41no—
71C2×C4×D181—41no—
72C2×C4×D181C2×C414yescomplemented
73C2×C4×D181D414yes—
74C4⋊Dic181—22no—
75C4⋊Dic181C2×C414yescomplemented
76#15668428—22no—
77#15668428—22no—
78#15668428—22no—
79#15668428—41no—
80#15668428C2×C414yescomplemented
81C22⋊Dic181—22no—
82C22⋊Dic181—41no—
83C22⋊Dic181C2×C414yescomplemented
84#15668426—22no—
85#15668426—22no—
86#15668426C2×C414yescomplemented
87C4×Dic181—22no—
88C4×Dic181—41no—
89C4×Dic181—41no—
90C24×C181C2311yescharacteristic
91C22×C4×C181—22no—
92C22×C4×C181—41no—
93C22×C4×C181—41no—
94C22×C4×C181C2311yescharacteristic
95C22×C4×C181C2311yescharacteristic
96C22×C4×C181C2311yescharacteristic
97C22×C4×C181C2311yescharacteristic
98C22×C4×C181C2311yescharacteristic
99C22×C4×C181C2311yescharacteristic
100C4⋊C4×C181—22no—
101C4⋊C4×C181—41no—
102C22⋊C4×C181—41no—
103C22⋊C4×C181C2×C414yescomplemented
104C22×D181—21no—
105C22×D181—21no—
106C22×D181—21no—
107C22×D181—21no—
108C22×D181C22×C412yes—
109C22×D181C2×D412yes—
110C22×D181#4112yes—
111C22⋊D181—28no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—21no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C2×Dic181—21no—
128C2×Dic181—21no—
129C2×Dic181C22×C412yes—
130C2×Dic181C22×C412yes—
131C2×Dic181C22×C412yes—
132C2×Dic181C2×D412yes—
133C4×D181—28no—
134C4×D181—44no—
135C4×D181—44no—
136C23×C181—21no—
137C23×C181—21no—
138C23×C181—21no—
139C23×C181—21no—
140C23×C181—21no—
141C23×C181—21no—
142C23×C181C22×C412yes—
143C23×C181C2411yescharacteristic
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181—21no—
157C2×C4×C181—21no—
158C2×C4×C181—21no—
159C2×C4×C181—44no—
160C2×C4×C181C22×C412yes—
161C2×C4×C181C22×C412yes—
162C2×C4×C181C2×D412yes—
163C22×C181—22no—
164C22×C181—22no—
165C22×C181—22no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181—41no—
169C22×C181—41no—
170C22×C181C23×C411yescharacteristic
171C22×C181C22×D411yescharacteristic
172C22×C181C2×#4111yescharacteristic
173C22×C181H(5,𝔽2)11yescharacteristic
174C22×C181H(5,𝔽2)11yescharacteristic
175C22×C1812−1+411yescharacteristic
176C22×C1812−1+411yescharacteristic
177C2×D181—22no—
178C2×D181—22no—
179C2×D181—22no—
180C2×D181—22no—
181C2×D181—22no—
182C2×D181—22no—
183C2×D181C4×D414yes—
184C4×C181—22no—
185C4×C181—22no—
186C4×C181—22no—
187C4×C181—41no—
188C4×C181—41no—
189C4×C181—41no—
190Dic181—22no—
191Dic181—22no—
192Dic181—22no—
193Dic181—22no—
194Dic181—41no—
195Dic181—41no—
196Dic181—41no—
197C2×C181—42no—
198C2×C181C2×C4×D411yescharacteristic
199C2×C181#52011yescharacteristic
200C2×C181#52011yescharacteristic
201C2×C181#53411yescharacteristic
202C2×C181#53611yescharacteristic
203C2×C181#53911yescharacteristic
204C2×C181#53911yescharacteristic
205D181—24no—
206C181#232211yescharacteristic, complemented, Sylow
207#2322—1811nomaximal, Sylow
208C2×#123—1811no—
209C2×C4×D4—1811no—
210C22×C4⋊C4—1811no—
211#389—1811no—
212#389—1811no—
213#388—1811no—
214#388—1811no—
215#387—1811no—
216#387—1811no—
217#386—1811no—
218#385—1811no—
219#385—1811no—
220#385D18111yescharacteristic, complemented
221#381—1811no—
222C4×C22⋊C4—1811no—
223C22×D4—1811no—
224C23×C4—1811no—
225C23×C4—1811no—
226C23×C4—1811no—
227C23×C4C2×D18111yescharacteristic
228#123—1818no—
229C4×D4—3624no—
230C2×C4⋊C4—3624no—
231C2×C4⋊C4—1811no—
232C2×C4⋊C4—1811no—
233C2×C4⋊C4—1811no—
234C2×C4⋊C4—1811no—
235C2×C4⋊C4—1811no—
236C2×C4⋊C4—1811no—
237C2×C4⋊C4—1811no—
238C2×C4⋊C4C2×D18111yescharacteristic
239C2×C4⋊C4C2×D18111yescharacteristic
240C2×C22⋊C4—1811no—
241C2×C22⋊C4—1811no—
242C2×C22⋊C4—1811no—
243C2×C22⋊C4—1811no—
244C2×C22⋊C4—1811no—
245C2×C22⋊C4—1811no—
246C2×C22⋊C4C2×D18111yescharacteristic
247C2×C22⋊C4C2×D18111yescharacteristic
248C2×C42—1811no—
249C2×C42—1811no—
250C2×C42—1811no—
251#95—1811no—
252#95—1811no—
253#95—1811no—
254#95—1811no—
255#95—1811no—
256#95—1811no—
257#95—1811no—
258#95—1811no—
259#95C2×D18111yescharacteristic
260#95C2×D18111yescharacteristic
261C24—1811no—
262C24C22×D18111yescharacteristic
263C2×D4—3622no—
264C2×D4—3622no—
265C2×D4—1814no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—3622no—
270C22×C4—3622no—
271C22×C4—3622no—
272C22×C4—1811no—
273C22×C4—1811no—
274C22×C4—1811no—
275C22×C4—1811no—
276C22×C4—1811no—
277C22×C4—1811no—
278C22×C4—1811no—
279C22×C4—1814no—
280C22×C4—1814no—
281C22×C4—7241no—
282C22×C4—7241no—
283C22×C4—7241no—
284C22×C4—7241no—
285C22×C4C22×D18111yescharacteristic
286C22×C4C22×D18111yescharacteristic
287C22×C4C22×D18111yescharacteristic
288C22×C4C22×D18111yescharacteristic
289C22×C4C22×D18111yescharacteristic
290C22×C4C22×D18111yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—3622no—
294C4⋊C4—3622no—
295C4⋊C4—3622no—
296C4⋊C4—3622no—
297C4⋊C4—1814no—
298C4⋊C4—1814no—
299C4⋊C4—7241no—
300C22⋊C4—41no—
301C22⋊C4—3622no—
302C22⋊C4—3622no—
303C22⋊C4—3622no—
304C22⋊C4—1814no—
305C22⋊C4—1814no—
306C22⋊C4—7241no—
307C22⋊C4C4×D18114yescomplemented
308C42—3622no—
309C42—7241no—
310C42—7241no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—3621no—
318C23—3621no—
319C23—3621no—
320C23—3621no—
321C23—1812no—
322C23—1812no—
323C23—1812no—
324C23C2×C4×D18112yes—
325C23C23×D18111yescharacteristic
326D4—3628no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—3628no—
344C2×C4—3621no—
345C2×C4—3621no—
346C2×C4—3621no—
347C2×C4—3621no—
348C2×C4—3621no—
349C2×C4—3621no—
350C2×C4—3621no—
351C2×C4—3621no—
352C2×C4—3621no—
353C2×C4—3621no—
354C2×C4—3621no—
355C2×C4—3621no—
356C2×C4—3621no—
357C2×C4—3621no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4—3621no—
361C2×C4—1812no—
362C2×C4—1812no—
363C2×C4—1812no—
364C2×C4—1812no—
365C2×C4—7244no—
366C2×C4—7244no—
367C2×C4C2×C4×D18112yes—
368C2×C4C2×C4×D18112yes—
369C2×C4D4×D18112yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3622no—
378C22—3622no—
379C22—3622no—
380C22—3622no—
381C22—3622no—
382C22—3622no—
383C22—1814no—
384C22C22×C4×D18111yescharacteristic
385C22C2×D4×D18111yescharacteristic
386C22#41×D18111yescharacteristic
387C22#1915386111yescharacteristic
388C22#1915386111yescharacteristic
389C22#1915386311yescharacteristic
390C22#1915386411yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—3622no—
398C4—3622no—
399C4—3622no—
400C4—3622no—
401C4—7241no—
402C4—7241no—
403C4—7241no—
404C2—42no—
405C2—3624no—
406C2C4×D4×D18111yescharacteristic
407C2#2604175811yescharacteristic
408C2#2604176011yescharacteristic
409C2#2604183911yescharacteristic
410C2#2604185211yescharacteristic
411C2#2604187211yescharacteristic
412C2#2604187511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#385×C181C2
socle, cosocleC23×C181C24
prime radicalC22×C22⋊D181C22
hypercenter#385D181
hypocenterC181#2322
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D181C2semidirect
4. #26041176C2semidirect
5. #26041175C2semidirect
6. #26041172C2semidirect
7. #26041167C2semidirect
8. #26041164C2semidirect
9. #26041160C2semidirect
11. #26041152C2semidirect
12. #26041150C2semidirect
13. #26041141C2semidirect
14. #26041126C2semidirect
15. #385×C181C2semidirect
20. #19153798C4semidirect
72. C2×C4×D181C2×C4semidirect
75. C4⋊Dic181C2×C4semidirect
80. #15668428C2×C4semidirect
83. C22⋊Dic181C2×C4semidirect
86. #15668426C2×C4semidirect
103. C22⋊C4×C181C2×C4semidirect
206. C181#2322semidirect
220. #385D181semidirect
307. C22⋊C4C4×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537982
C2×C4×C22⋊D1812
C2×C4⋊C4×D1812
#260411762
#260411752
#260411722
#260411672
#260411642
#260411602
#260411552
#260411522
#260411502
#260411412
#260411262
#385×C1812
#2322181