Finite Group Databasebeta

← #37545308

Subgroups — #37545308

Order 23168.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153848C211yesmaximal, characteristic, complemented
2C2×#19153839C211yesmaximal, characteristic, complemented
3#26041951C211yesmaximal, characteristic, complemented
4#26041941C211yesmaximal, characteristic, complemented
5#26041668C211yesmaximal, characteristic, complemented
6#26041533C211yesmaximal, characteristic, complemented
7#26041509C211yesmaximal, characteristic, complemented
8#26041501C211yesmaximal, characteristic, complemented
9#26041479C211yesmaximal, characteristic, complemented
10#26041444C211yesmaximal, characteristic, complemented
11#26041427C211yesmaximal, characteristic, complemented
12#26041427C211yesmaximal, characteristic, complemented
13#26041419C211yesmaximal, characteristic, complemented
14#26041352C211yesmaximal, characteristic, complemented
15#441×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668450C2211yescharacteristic
18C2×#15668450C2211yescharacteristic
19C2×Q8×D181C2211yescharacteristic
20#19153848—24no—
21#19153839—24no—
22C2×#15668439C2211yescharacteristic
23C2×#15668438C2211yescharacteristic
24C2×#15668438C2211yescharacteristic
25C2×#15668436C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27C2×#15668436C2211yescharacteristic
28C2×D1448C2211yescharacteristic
29C2×D1448C2211yescharacteristic
30C2×#15668432C2211yescharacteristic
31C2×C8×D181C2211yescharacteristic
32#19153813—21no—
33#19153813C2211yescharacteristic
34#19153804—21no—
35#19153804C2211yescharacteristic
36#19153796—21no—
37#19153796C2211yescharacteristic
38#19153791—21no—
39#19153791—21no—
40#19153791—21no—
41#19153791C2211yescharacteristic
42#19153791C2211yescharacteristic
43#19153791C2211yescharacteristic
44C4×D724—21no—
45C4×D724—21no—
46C4×D724C2211yescharacteristic
47C4⋊#13089678C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153736C2211yescharacteristic
51#19153736C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153728C2211yescharacteristic
54C2×Q16×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57#106×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59#103×C181C2211yescharacteristic
60#97×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—22no—
63#15668450—22no—
64#15668450—41no—
65#15668450—41no—
66#15668450—41no—
67C2×Dic362—21no—
68C2×Dic362C2311yescharacteristic
69Q8×D181—22no—
70Q8×D181—41no—
71C2×D724—21no—
72C2×D724—21no—
73C2×D724—21no—
74C2×D724—21no—
75C2×D724—21no—
76C2×D724—21no—
77C2×D724—21no—
78C2×D724—21no—
79C2×D724D411yescharacteristic
80C2×D724C2311yescharacteristic
81C2×D724C2311yescharacteristic
82C2×D724C2311yescharacteristic
83C2×C4×D181—21no—
84C2×C4×D181—21no—
85C2×C4×D181—21no—
86C2×C4×D181—21no—
87C2×C4×D181C2311yescharacteristic
88C2×C4×D181C2311yescharacteristic
89#15668439—22no—
90#15668438—22no—
91#15668438—41no—
92#15668436—22no—
93#15668436—22no—
94#15668436—22no—
95D1448—22no—
96D1448—41no—
97#15668432—22no—
98C2×#13089678C2311yescharacteristic
99C2×#13089678C2311yescharacteristic
100C8×D181—22no—
101C4⋊Dic181—21no—
102C4⋊Dic181D411yescharacteristic
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426—21no—
109#15668426D411yescharacteristic
110#15668426D411yescharacteristic
111C2×Q8×C181C2311yescharacteristic
112C2×Q8×C181C2311yescharacteristic
113Q16×C181D414yescomplemented
114C2×C8×C181C2311yescharacteristic
115C2×C8×C181C2311yescharacteristic
116C4⋊C4×C181—21no—
117C4⋊C4×C181—21no—
118C4⋊C4×C181C2311yescharacteristic
119C4⋊C4×C181C2311yescharacteristic
120C42×C181—21no—
121C42×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—21no—
126C22×D181C2×D411yescharacteristic
127C22×D181C2×D411yescharacteristic
128Dic362—21no—
129Dic362—21no—
130Dic362—42no—
131D724—21no—
132D724—21no—
133D724—21no—
134D724—21no—
135D724—21no—
136D724—21no—
137D724—41no—
138D724—42no—
139D724—42no—
140D724—42no—
141D724—42no—
142D724—42no—
143D724—81no—
144D724—81no—
145D724—81no—
146C2×Dic181—21no—
147C2×Dic181C2×D411yescharacteristic
148C2×Dic181C2×D411yescharacteristic
149C4×D181—21no—
150C4×D181—21no—
151C4×D181—21no—
152C4×D181—21no—
153C4×D181—42no—
154C4×D181—42no—
155C4×D181—42no—
156C4×D181—42no—
157#13089678—21no—
158#13089678—21no—
159Q8×C181—21no—
160Q8×C181—21no—
161Q8×C181C2×D412yes—
162Q8×C181C2×D412yes—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181C2×D412yes—
172C22×C181C22×D411yescharacteristic
173C2×D181—21no—
174C2×D181—21no—
175C2×D181—21no—
176C2×D181—21no—
177C2×D181—21no—
178C2×D181—21no—
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C4×C181—22no—
189C4×C181—22no—
190C4×C181—21no—
191C4×C181—21no—
192C4×C181—41no—
193C4×C181—41no—
194C4×C181C22×D411yescharacteristic
195C4×C181H(5,𝔽2)11yescharacteristic
196Dic181—21no—
197Dic181—21no—
198Dic181—41no—
199C2×C181D4211yescharacteristic
200C2×C181#57511yescharacteristic
201C2×C181#57711yescharacteristic
202D181—41no—
203D181—41no—
204D181—81no—
205D181—81no—
206D181—81no—
207C181#327111yescharacteristic, complemented, Sylow
208#3271—1811nomaximal, Sylow
209C2×#137—1811no—
210C2×#135—1811no—
211#550—1811no—
212#548—1811no—
213#496—1811no—
214#469—1811no—
215#465—1811no—
216#464—1811no—
217#460—1811no—
218#452—1811no—
219#449—1811no—
220#449—1811no—
221#448—1811no—
222#441D18111yescharacteristic, complemented
223#435—1811no—
224C2×#41—1811no—
225C2×#41—1811no—
226C2×#41—1811no—
227C22×Q8—1811no—
228#137—3624no—
229#135—3624no—
230C2×Q16—1811no—
231C2×Q16C2×D18111yescharacteristic
232C2×SD16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×D8—1811no—
238C2×D8—1811no—
239C2×M16—1811no—
240C22×C8—1811no—
241#127—3621no—
242#127—1811no—
243#124—3621no—
244#124—1811no—
245C22⋊Q8—3621no—
246C22⋊Q8—1811no—
247#121—3621no—
248#121—3621no—
249#121—3621no—
250#121—1811no—
251#121—1811no—
252#121—1811no—
253C4×Q8C2×D18111yescharacteristic
254C4×Q8C2×D18111yescharacteristic
255C4×D4—3621no—
256C4×D4—3621no—
257C4×D4—1811no—
258#106C2×D18111yescharacteristic
259C4⋊C8—1811no—
260#103—1811no—
261#103C2×D18111yescharacteristic
262#103C2×D18111yescharacteristic
263#102—1811no—
264#102—1811no—
265#102—1811no—
266C22⋊C8—1811no—
267C22⋊C8—1811no—
268#97C2×D18111yescharacteristic
269#41—3622no—
270#41—3622no—
271#41—3622no—
272#41—7241no—
273#41—7241no—
274#41—7241no—
275C2×Q8—3621no—
276C2×Q8—3622no—
277C2×Q8—1811no—
278C2×Q8—7241no—
279C2×Q8C22×D18111yescharacteristic
280C2×Q8C22×D18111yescharacteristic
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—3621no—
288C2×D4—3621no—
289C2×D4—1811no—
290C2×D4—1811no—
291C2×D4—1811no—
292C2×D4—1811no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—3621no—
297C22×C4—1811no—
298C22×C4—1811no—
299Q16—3622no—
300Q16D72414yescomplemented
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—3622no—
305SD16—7241no—
306D8—3622no—
307D8—7241no—
308M16—3622no—
309C2×C8—3622no—
310C2×C8—1811no—
311C2×C8—1811no—
312C2×C8C22×D18111yescharacteristic
313C2×C8C22×D18111yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—3621no—
317C4⋊C4—1811no—
318C4⋊C4C22×D18111yescharacteristic
319C4⋊C4C22×D18111yescharacteristic
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—1811no—
327C22⋊C4—1811no—
328C42—21no—
329C42—21no—
330C42C22×D18111yescharacteristic
331C23—3621no—
332C23—3621no—
333C23—3621no—
334C23—1811no—
335C23—1811no—
336Q8—21no—
337Q8—21no—
338Q8—3621no—
339Q8—3621no—
340Q8—7242no—
341Q8C2×D72412yes—
342Q8C2×D72412yes—
343D4—3621no—
344D4—3621no—
345D4—3621no—
346D4—3621no—
347D4—3621no—
348D4—3621no—
349D4—14481no—
350D4—14481no—
351D4—14481no—
352D4—7241no—
353D4—7242no—
354D4—7242no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—3621no—
363C2×C4—3621no—
364C2×C4—3621no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—1811no—
368C2×C4—1811no—
369C2×C4—7242no—
370C2×C4—7242no—
371C2×C4—7242no—
372C2×C4—7242no—
373C2×C4D4×D18111yescharacteristic
374C2×C4D4×D18111yescharacteristic
375C2×C4C23×D18111yescharacteristic
376C8—21no—
377C8—3621no—
378C8—3621no—
379C8C2×D72412yes—
380C22—3621no—
381C22—3621no—
382C22—3621no—
383C22—3621no—
384C22—3621no—
385C22—3621no—
386C22—7241no—
387C22—7241no—
388C22—7241no—
389C22—7241no—
390C22—7241no—
391C22—7241no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22C2×D4×D18111yescharacteristic
396C4—22no—
397C4—22no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3621no—
403C4—3621no—
404C4—7241no—
405C4C22×D72411yescharacteristic
406C4#1915386211yescharacteristic
407C2—14481no—
408C2—14481no—
409C2—14481no—
410C2—7241no—
411C2—7241no—
412C2D4×D72411yescharacteristic
413C2#2604207811yescharacteristic
414C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#441×C181C2
socleC22×C181C22×D4
hypercenter#441D181
hypocenterC181#3271
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. C2×#19153848C2semidirect
2. C2×#19153839C2semidirect
3. #26041951C2semidirect
4. #26041941C2semidirect
5. #26041668C2semidirect
6. #26041533C2semidirect
7. #26041509C2semidirect
8. #26041501C2semidirect
9. #26041479C2semidirect
10. #26041444C2semidirect
11. #26041427C2semidirect
12. #26041427C2semidirect
13. #26041419C2semidirect
14. #26041352C2semidirect
15. #441×C181C2semidirect
113. Q16×C181D4semidirect
207. C181#3271semidirect
222. #441D181semidirect
300. Q16D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538482
C2×#191538392
#260419512
#260419412
#260416682
#260415332
#260415092
#260415012
#260414792
#260414442
#260414272
#260414192
#260413522
#441×C1812
#3271181