Finite Group Databasebeta

← #37545305

Subgroups — #37545305

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×#19153845C211yesmaximal, characteristic, complemented
2C2×#19153835C211yesmaximal, characteristic, complemented
3#26041949C211yesmaximal, characteristic, complemented
4#26041937C211yesmaximal, characteristic, complemented
5#26041529C211yesmaximal, characteristic, complemented
6#26041505C211yesmaximal, characteristic, complemented
7#26041498C211yesmaximal, characteristic, complemented
8#26041476C211yesmaximal, characteristic, complemented
9#26041438C211yesmaximal, characteristic, complemented
10#26041421C211yesmaximal, characteristic, complemented
11#26041421C211yesmaximal, characteristic, complemented
12#26041416C211yesmaximal, characteristic, complemented
13#26041387C211yesmaximal, characteristic, complemented
14#26041352C211yesmaximal, characteristic, complemented
15#496×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×#15668448C2211yescharacteristic
18C2×#15668448C2211yescharacteristic
19C2×Q8×D181C2211yescharacteristic
20#19153845—24no—
21#19153835—24no—
22C2×Dic724C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668437C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668432C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic
29#19153812—21no—
30#19153802—21no—
31#19153793—21no—
32#19153792C2211yescharacteristic
33#19153789—21no—
34#19153788—21no—
35#19153788—21no—
36#19153787C2211yescharacteristic
37#19153787C2211yescharacteristic
38#19153787C2211yescharacteristic
39C4×Dic362C2211yescharacteristic
40C4×Dic362C2211yescharacteristic
41C4×D724C2211yescharacteristic
42D4×Dic181—21no—
43#19153776—21no—
44#19153748C2211yescharacteristic
45C4⋊#13089678C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153741C2211yescharacteristic
48#19153740C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153737C2211yescharacteristic
51#19153737C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153728C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic
56C2×D8×C181C2211yescharacteristic
57C2×D8×C181C2211yescharacteristic
58#127×C181C2211yescharacteristic
59#124×C181C2211yescharacteristic
60#97×C181C2211yescharacteristic
61#15668448—22no—
62#15668448—22no—
63#15668448—22no—
64#15668448—41no—
65#15668448—41no—
66#15668448—41no—
67C2×Dic362—21no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70Q8×D181—22no—
71Q8×D181—41no—
72C2×D724D411yescharacteristic
73C2×C22⋊D181—21no—
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C2×C22⋊D181—21no—
77C2×C22⋊D181—21no—
78C2×C22⋊D181—21no—
79C22×Dic181—21no—
80C22×Dic181—21no—
81C22×Dic181—21no—
82C2×C4×D181—21no—
83C2×C4×D181C2311yescharacteristic
84C2×C4×D181C2311yescharacteristic
85Dic724D414yescomplemented
86#15668439—22no—
87#15668437—22no—
88#15668437—22no—
89#15668437—22no—
90#15668432—22no—
91C2×#13089678C2311yescharacteristic
92C2×#13089678C2311yescharacteristic
93C8×D181—22no—
94C4⋊Dic181D411yescharacteristic
95C4⋊Dic181C2311yescharacteristic
96C4⋊Dic181C2311yescharacteristic
97#15668428—21no—
98#15668428—21no—
99#15668428—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181—21no—
102C22⋊Dic181—21no—
103#15668426—21no—
104#15668426D411yescharacteristic
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181—21no—
108C2×Q8×C181C2311yescharacteristic
109C2×D4×C181—21no—
110C2×D4×C181—21no—
111C2×D4×C181C2311yescharacteristic
112C2×D4×C181C2311yescharacteristic
113C2×D4×C181C2311yescharacteristic
114SD16×C181—22no—
115SD16×C181—41no—
116D8×C181—22no—
117D8×C181—41no—
118C2×C8×C181C2311yescharacteristic
119C2×C8×C181C2311yescharacteristic
120C22⋊C4×C181—21no—
121C22⋊C4×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181C2×D411yescharacteristic
124C22×D181C2×D411yescharacteristic
125Dic362—21no—
126Dic362—21no—
127Dic362—42no—
128Dic362C2×D412yes—
129Dic362C2×D412yes—
130D724—41no—
131C22⋊D181—42no—
132C22⋊D181—42no—
133C22⋊D181—42no—
134C22⋊D181—42no—
135C22⋊D181—81no—
136C22⋊D181—81no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—21no—
140C2×Dic181—21no—
141C2×Dic181—42no—
142C2×Dic181—42no—
143C2×Dic181—42no—
144C2×Dic181C2×D411yescharacteristic
145C2×Dic181C2×D411yescharacteristic
146C4×D181—21no—
147C4×D181—21no—
148C4×D181—21no—
149C4×D181—21no—
150C4×D181—42no—
151#13089678—21no—
152#13089678—21no—
153C23×C181—21no—
154C23×C181—21no—
155C23×C181—21no—
156Q8×C181—21no—
157Q8×C181—21no—
158D4×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—21no—
162D4×C181—21no—
163D4×C181—21no—
164D4×C181—42no—
165D4×C181—81no—
166C2×C4×C181—21no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181C2×D412yes—
172C22×C181—41no—
173C22×C181—41no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181—41no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181C22×D411yescharacteristic
182C2×D181—21no—
183C2×D181—21no—
184C2×D181—21no—
185C2×D181—21no—
186C2×D181—21no—
187C2×D181—21no—
188C4×C181—21no—
189C4×C181—21no—
190C4×C181—41no—
191C4×C181C22×D411yescharacteristic
192C4×C181H(5,𝔽2)11yescharacteristic
193Dic181—21no—
194Dic181—21no—
195Dic181—22no—
196Dic181—22no—
197Dic181—41no—
198Dic181—41no—
199C2×C181—81no—
200C2×C181—81no—
201C2×C181—81no—
202C2×C181D4211yescharacteristic
203C2×C181#57511yescharacteristic
204C2×C181#57711yescharacteristic
205D181—41no—
206D181—41no—
207C181#327111yescharacteristic, complemented, Sylow
208#3271—1811nomaximal, Sylow
209C2×#137—1811no—
210C2×#135—1811no—
211#550—1811no—
212#548—1811no—
213#496D18111yescharacteristic, complemented
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#441—1811no—
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×Q16—1811no—
232C2×SD16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16C2×D18111yescharacteristic
236C2×SD16C2×D18111yescharacteristic
237C2×D8C2×D18111yescharacteristic
238C2×D8C2×D18111yescharacteristic
239C2×M16—1811no—
240C22×C8—1811no—
241#127—3621no—
242#127C2×D18111yescharacteristic
243#124—3621no—
244#124C2×D18111yescharacteristic
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×Q8—1811no—
254C4×Q8—1811no—
255C4×D4—3621no—
256C4×D4—3621no—
257C4×D4—1811no—
258#106—1811no—
259C4⋊C8—1811no—
260#103—1811no—
261#103—1811no—
262#103—1811no—
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—1811no—
279C2×Q8—7241no—
280C2×Q8C22×D18111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—3621no—
288C2×D4—3621no—
289C2×D4—1811no—
290C2×D4C22×D18111yescharacteristic
291C2×D4C22×D18111yescharacteristic
292C2×D4C22×D18111yescharacteristic
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—3621no—
297C22×C4—1811no—
298C22×C4—1811no—
299Q16—3622no—
300Q16—1814no—
301SD16—22no—
302SD16—41no—
303SD16—3622no—
304SD16—3622no—
305SD16—3622no—
306D8—22no—
307D8—41no—
308M16—3622no—
309C2×C8—3622no—
310C2×C8—1811no—
311C2×C8—1811no—
312C2×C8C22×D18111yescharacteristic
313C2×C8C22×D18111yescharacteristic
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—3621no—
317C4⋊C4—1811no—
318C4⋊C4—1811no—
319C4⋊C4—1811no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—1811no—
327C22⋊C4—1811no—
328C42—3621no—
329C42—3621no—
330C42C22×D18111yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—1811no—
335C23—1811no—
336Q8—21no—
337Q8—21no—
338Q8—3621no—
339Q8—3621no—
340Q8—1812no—
341Q8—1812no—
342Q8—7242no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—81no—
351D4—14481no—
352D4—14481no—
353D4—7241no—
354D4—7242no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358C2×C4—21no—
359C2×C4—3621no—
360C2×C4—3621no—
361C2×C4—3621no—
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—
379C8D4×D18112yes—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3621no—
390C22—3621no—
391C22—3621no—
392C22—3621no—
393C22—3621no—
394C22—3621no—
395C22C2×D4×D18111yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—3621no—
400C4—3621no—
401C4—3622no—
402C4—3622no—
403C4—7241no—
404C4—7241no—
405C4C2×D4×D18111yescharacteristic
406C4#1915386111yescharacteristic
407C2—81no—
408C2—81no—
409C2—81no—
410C2—7241no—
411C2—7241no—
412C2#2604191411yescharacteristic
413C2#2604207511yescharacteristic
414C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#496×C181C2
socleC22×C181C22×D4
hypercenter#496D181
hypocenterC181#3271
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19153845C2semidirect
2. C2×#19153835C2semidirect
3. #26041949C2semidirect
4. #26041937C2semidirect
5. #26041529C2semidirect
6. #26041505C2semidirect
7. #26041498C2semidirect
8. #26041476C2semidirect
9. #26041438C2semidirect
10. #26041421C2semidirect
11. #26041421C2semidirect
12. #26041416C2semidirect
13. #26041387C2semidirect
14. #26041352C2semidirect
15. #496×C181C2semidirect
85. Dic724D4semidirect
207. C181#3271semidirect
213. #496D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538452
C2×#191538352
#260419492
#260419372
#260415292
#260415052
#260414982
#260414762
#260414382
#260414212
#260414162
#260413872
#260413522
#496×C1812
#3271181