Finite Group Databasebeta

← #37545381

Subgroups — #37545381

Order 23168.

482 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153848C211yesmaximal, characteristic, complemented
2#26041948C211yesmaximal, characteristic
3Q8×C22⋊D181C211yesmaximal, characteristic, complemented
4#26041650C211yesmaximal, characteristic, complemented
5#26041525C211yesmaximal, characteristic, complemented
6#26041522C211yesmaximal, characteristic, complemented
7#26041487C211yesmaximal, characteristic, complemented
8#26041477C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041440C211yesmaximal, characteristic, complemented
11#26041418C211yesmaximal, characteristic, complemented
12#26041416C211yesmaximal, characteristic, complemented
13Q16×Dic181C211yesmaximal, characteristic, complemented
14#26041355C211yesmaximal, characteristic, complemented
15C2×#137×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18#19153848—24no—
19C2×#15668439C2211yescharacteristic
20C2×#15668438C2211yescharacteristic
21C2×#15668437C2211yescharacteristic
22C2×#15668436C2211yescharacteristic
23C2×#15668436C2211yescharacteristic
24C2×#15668432C2211yescharacteristic
25#19153814—21no—
26#19153814C2211yescharacteristic
27#19153812—21no—
28#19153812C2211yescharacteristic
29#19153795—21no—
30#19153795C2211yescharacteristic
31#19153792—21no—
32#19153792C2211yescharacteristic
33#19153790—21no—
34#19153790C2211yescharacteristic
35#19153787—21no—
36#19153787C2211yescharacteristic
37Q8×Dic181C2211yescharacteristic
38Q8×Dic181C2211yescharacteristic
39C4×C22⋊D181—21no—
40C4×C22⋊D181—21no—
41C4×C22⋊D181C2211yescharacteristic
42#19153750C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153740C2211yescharacteristic
48#19153738C2211yescharacteristic
49#19153737C2211yescharacteristic
50C22⋊#13089678C2211yescharacteristic
51#19153728C2211yescharacteristic
52C8×Dic181C2211yescharacteristic
53C2×#41×C181C2211yescharacteristic
54C22×Q8×C181C2211yescharacteristic
55#137×C181—24no—
56C2×Q16×C181C2211yescharacteristic
57C2×Q16×C181C2211yescharacteristic
58C2×SD16×C181C2211yescharacteristic
59C2×SD16×C181C2211yescharacteristic
60C2×M16×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—41no—
63C2×Dic362—21no—
64C2×Dic362C2311yescharacteristic
65Q8×D181—22no—
66Q8×D181—41no—
67C2×D724—21no—
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181D411yescharacteristic
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76C2×C4×D181C2311yescharacteristic
77#15668439—22no—
78#15668438—22no—
79#15668437—41no—
80#15668436—22no—
81#15668436—41no—
82#15668432—22no—
83C2×#13089678C2311yescharacteristic
84C2×#13089678C2311yescharacteristic
85C4⋊Dic181—21no—
86C4⋊Dic181—21no—
87C4⋊Dic181C2311yescharacteristic
88C4⋊Dic181C2311yescharacteristic
89#15668428—21no—
90#15668428—21no—
91#15668428—21no—
92#15668428—21no—
93#15668428D411yescharacteristic
94C22⋊Dic181—21no—
95C22⋊Dic181—21no—
96C22⋊Dic181D411yescharacteristic
97#15668426—21no—
98#15668426—21no—
99#15668426D411yescharacteristic
100C4×Dic181—21no—
101C4×Dic181—21no—
102C4×Dic181C2311yescharacteristic
103#41×C181—22no—
104#41×C181—41no—
105C2×Q8×C181—21no—
106C2×Q8×C181—22no—
107C2×Q8×C181—41no—
108C2×Q8×C181C2311yescharacteristic
109C2×Q8×C181C2311yescharacteristic
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181—21no—
114C22×C4×C181—21no—
115C22×C4×C181C2311yescharacteristic
116Q16×C181—22no—
117Q16×C181D414yescomplemented
118SD16×C181—22no—
119SD16×C181—22no—
120M16×C181—22no—
121C2×C8×C181C2311yescharacteristic
122C2×C8×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181C2×D411yescharacteristic
125Dic362—21no—
126Dic362—21no—
127Dic362—42no—
128D724—21no—
129D724—21no—
130D724—42no—
131C22⋊D181—22no—
132C22⋊D181—42no—
133C22⋊D181—81no—
134C22⋊D181—81no—
135C22⋊D181—81no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181C2×D411yescharacteristic
140C2×Dic181C2×D411yescharacteristic
141C2×Dic181C2×D411yescharacteristic
142C4×D181—21no—
143C4×D181—21no—
144C4×D181—42no—
145C4×D181—42no—
146#13089678—21no—
147#13089678—21no—
148C23×C181—21no—
149C23×C181C2×D411yescharacteristic
150Q8×C181—21no—
151Q8×C181—21no—
152Q8×C181—21no—
153Q8×C181—21no—
154Q8×C181—42no—
155Q8×C181C2×D412yes—
156Q8×C181C2×D412yes—
157D4×C181—21no—
158D4×C181—21no—
159D4×C181—42no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—42no—
166C2×C4×C181—42no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2411yescharacteristic
169C8×C181—21no—
170C8×C181C2×D412yes—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181C22×D411yescharacteristic
178C2×D181—21no—
179C2×D181—21no—
180C2×D181—21no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C4×C181—21no—
185C4×C181—22no—
186C4×C181—22no—
187C4×C181—41no—
188C4×C181C22×D411yescharacteristic
189C4×C181H(5,𝔽2)11yescharacteristic
190Dic181—21no—
191Dic181—21no—
192Dic181—41no—
193Dic181—41no—
194Dic181—41no—
195Dic181#13712yescomplemented
196C2×C181—41no—
197C2×C181—81no—
198C2×C181D4211yescharacteristic
199C2×C181C2×#13711yescharacteristic
200C2×C181#57711yescharacteristic
201D181—41no—
202D181—81no—
203C181#327811yescharacteristic, complemented, Sylow
204#3278—1811nomaximal, Sylow
205C2×#137—1811no—
206C2×#137D18111yescharacteristic, complemented
207#550—1811no—
208D4×Q8—1811no—
209#492—1811no—
210#468—1811no—
211#468—1811no—
212#462—1811no—
213#460—1811no—
214#452—1811no—
215#452—1811no—
216#448—1811no—
217#448—1811no—
218C4×Q16—1811no—
219#436—1811no—
220C2×#41—1811no—
221C2×#41C2×D18111yescharacteristic
222C22×Q8—1811no—
223C22×Q8C2×D18111yescharacteristic
224#137—24no—
225#137—3624no—
226C2×Q16—1811no—
227C2×Q16C2×D18111yescharacteristic
228C2×Q16C2×D18111yescharacteristic
229C2×SD16—1811no—
230C2×SD16—1811no—
231C2×SD16—1811no—
232C2×SD16—1811no—
233C2×SD16C2×D18111yescharacteristic
234C2×SD16C2×D18111yescharacteristic
235C2×M16—1811no—
236C2×M16C2×D18111yescharacteristic
237C4⋊Q8—3621no—
238C4⋊Q8—1811no—
239#127—3621no—
240#127—1811no—
241C22⋊Q8—3621no—
242C22⋊Q8—3621no—
243C22⋊Q8—1811no—
244C22⋊Q8—1811no—
245#121—3621no—
246#121—3621no—
247#121—1811no—
248#121—1811no—
249C4×Q8—1811no—
250C4×Q8—1811no—
251C4×D4—3621no—
252C4×D4—3621no—
253C4×D4—1811no—
254#107—1811no—
255C4⋊C8—1811no—
256#103—1811no—
257#103—1811no—
258#103—1811no—
259#103—1811no—
260#102—1811no—
261#102—1811no—
262C22⋊C8—1811no—
263C22⋊C8—1811no—
264C4×C8—1811no—
265#41—22no—
266#41—41no—
267#41—3622no—
268#41—7241no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3621no—
273C2×Q8—3622no—
274C2×Q8—1811no—
275C2×Q8—7241no—
276C2×Q8C22×D18111yescharacteristic
277C2×Q8C22×D18111yescharacteristic
278C2×Q8C22×D18111yescharacteristic
279C2×D4—21no—
280C2×D4—3621no—
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—1811no—
286C2×D4—1811no—
287C2×D4C22×D18111yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—3621no—
291C22×C4—3621no—
292C22×C4—1811no—
293C22×C4C22×D18111yescharacteristic
294Q16—22no—
295Q16—3622no—
296Q16C22⋊D18114yescomplemented
297SD16—22no—
298SD16—22no—
299SD16—3622no—
300SD16—3622no—
301SD16—7241no—
302SD16—7241no—
303M16—22no—
304M16—3622no—
305C2×C8—1811no—
306C2×C8—1811no—
307C2×C8C22×D18111yescharacteristic
308C2×C8C22×D18111yescharacteristic
309C4⋊C4—3621no—
310C4⋊C4—3621no—
311C4⋊C4—3621no—
312C4⋊C4—3621no—
313C4⋊C4—3621no—
314C4⋊C4—3621no—
315C4⋊C4—1811no—
316C4⋊C4—1811no—
317C4⋊C4—1811no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—1811no—
323C22⋊C4—1811no—
324C42—3621no—
325C42—3621no—
326C42—1811no—
327C23—21no—
328C23—3621no—
329C23—1811no—
330C23D4×D18111yescharacteristic
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—42no—
336Q8—3621no—
337Q8—3621no—
338Q8—7242no—
339Q8C2×C22⋊D18112yes—
340Q8C2×C22⋊D18112yes—
341D4—21no—
342D4—21no—
343D4—42no—
344D4—3622no—
345D4—3621no—
346D4—3621no—
347D4—14481no—
348D4—14481no—
349D4—14481no—
350D4—7242no—
351D4—7242no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—3621no—
360C2×C4—3621no—
361C2×C4—3621no—
362C2×C4—3621no—
363C2×C4—3621no—
364C2×C4—1811no—
365C2×C4—1811no—
366C2×C4—1811no—
367C2×C4—7242no—
368C2×C4—7242no—
369C2×C4D4×D18111yescharacteristic
370C2×C4C23×D18111yescharacteristic
371C8—21no—
372C8—3621no—
373C8—3621no—
374C8C2×C22⋊D18112yes—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—3621no—
382C22—3621no—
383C22—3621no—
384C22—7241no—
385C22—7241no—
386C22—7241no—
387C22C2×D4×D18111yescharacteristic
388C4—21no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—3621no—
393C4—3621no—
394C4—1812no—
395C4—7241no—
396C4—7241no—
397C4—7241no—
398C4C22×C22⋊D18111yescharacteristic
399C4#1915386111yescharacteristic
400C2—41no—
401C2—81no—
402C2—14481no—
403C2—7241no—
404C2D4×C22⋊D18111yescharacteristic
405C2#137×D18111yescharacteristic
406C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#137×C181C2
socleC22×C181C22×D4
prime radical#26041948C2
hypercenterC2×#137D181
hypocenterC181#3278
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#19153848C2semidirect
3. Q8×C22⋊D181C2semidirect
4. #26041650C2semidirect
5. #26041525C2semidirect
6. #26041522C2semidirect
7. #26041487C2semidirect
8. #26041477C2semidirect
9. #26041441C2semidirect
10. #26041440C2semidirect
11. #26041418C2semidirect
12. #26041416C2semidirect
13. Q16×Dic181C2semidirect
14. #26041355C2semidirect
15. C2×#137×C181C2semidirect
117. Q16×C181D4semidirect
195. Dic181#137semidirect
203. C181#3278semidirect
206. C2×#137D181semidirect
296. Q16C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538482
#260419482
Q8×C22⋊D1812
#260416502
#260415252
#260415222
#260414872
#260414772
#260414412
#260414402
#260414182
#260414162
Q16×Dic1812
#260413552
C2×#137×C1812
#3278181