Finite Group Databasebeta

← #36367209

Subgroups — #36367209

Order 22144.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149726C211yesmaximal, characteristic, complemented
2C22×Q8×D173C211yesmaximal, characteristic, complemented
3Q8×D692C211yesmaximal, characteristic, complemented
4D4×Dic346C213yesmaximal, complemented
5Q8×C22⋊D173C211yesmaximal, characteristic, complemented
6#26012154C213yesmaximal, complemented
7#26012149C213yesmaximal, complemented
8#26012148C211yesmaximal, characteristic, complemented
9#26012147C213yesmaximal, complemented
10#26012064C213yesmaximal, complemented
11#26012061C213yesmaximal, complemented
12#26012001C213yesmaximal, complemented
13#26012000C213yesmaximal, complemented
14#26011967C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16#19149726—28no—
17C2×#14583826C2213yes—
18C2×#14583825C2213yes—
19C2×#14583825C2213yescomplemented
20C22×Dic346C2213yes—
21C22×Dic346C2213yescomplemented
22C2×Q8×D173—28no—
23C2×Q8×D173C2213yes—
24C2×Q8×D173C2211yescharacteristic
25C2×Q8×D173C2213yes—
26C2×Q8×D173C2211yescharacteristic
27C2×Q8×D173C2211yescharacteristic, complemented
28C22×C4×D173C2213yes—
29#19149679C2213yescomplemented
30#19149678C2216yescomplemented
31#19149677C2213yescomplemented
32#19149666C2213yescomplemented
33#19149665C2216yescomplemented
34#19149664C2213yes—
35#19149661C2213yes—
36#19149661C2216yes—
37#19149659C2213yes—
38#19149659C2213yescomplemented
39#19149658C2213yes—
40#19149658C2213yescomplemented
41#19149657C2213yes—
42#19149656C2213yes—
43#19149656C2216yescomplemented
44#19149656C2216yescomplemented
45#19149655C2213yescomplemented
46#19149652C2213yes—
47#19149651C2213yes—
48#19149650C2213yes—
49#19149649C2211yescharacteristic
50#19149647C2213yes—
51Q8×Dic173C2211yescharacteristic, complemented
52C4×Dic346C2213yescomplemented
53#19149644C2213yescomplemented
54C4×D692C2213yes—
55D4×Dic173C2216yes—
56C4×C22⋊D173C2213yes—
57C4×C22⋊D173C2213yes—
58#19149639C2216yes—
59#19149637C2213yes—
60#19149636C2213yes—
61C2×#41×C173C2211yescharacteristic, complemented
62C22×Q8×C173C2211yescharacteristic
63#124×C173C2213yescomplemented
64C22⋊Q8×C173C2213yescomplemented
65#121×C173C2213yes—
66C4×Q8×C173C2211yescharacteristic, complemented
67C4×D4×C173C2213yes—
68C23×D173C2311yescharacteristic
69#14583826—212no—
70#14583825—212no—
71#14583825—212no—
72C2×Dic346—212no—
73C2×Dic346—212no—
74C2×Dic346C2313yes—
75C2×Dic346C2316yes—
76C2×Dic346C2313yes—
77C2×Dic346C2313yes—
78C2×Dic346C2313yes—
79Q8×D173—212no—
80Q8×D173—212no—
81Q8×D173—24no—
82Q8×D173—24no—
83Q8×D173D418yescomplemented
84C2×D692C2311yescharacteristic
85C2×D692C2311yescharacteristic
86C2×C22⋊D173C2313yes—
87C2×C22⋊D173C2311yescharacteristic
88C2×C22⋊D173C2313yes—
89C2×C22⋊D173C2313yes—
90C22×Dic173C2316yes—
91C22×Dic173C2313yes—
92C2×C4×D173—212no—
93C2×C4×D173C2313yes—
94C2×C4×D173C2313yes—
95C2×C4×D173C2313yes—
96C2×C4×D173C2313yes—
97C2×C4×D173C2313yes—
98C4⋊Dic173C2313yes—
99C4⋊Dic173C2316yes—
100C4⋊Dic173C2313yes—
101#14583805C2313yes—
102#14583805C2313yes—
103#14583805C2316yes—
104#14583805C2313yes—
105#14583805C2313yes—
106C22⋊Dic173C2313yes—
107C22⋊Dic173C2313yes—
108C22⋊Dic173C2316yes—
109#14583803C2311yescharacteristic
110#14583803C2313yes—
111#14583803C2313yes—
112#14583803C2311yescharacteristic
113#14583803C2313yes—
114#14583803C2313yes—
115C4×Dic173C2313yes—
116C4×Dic173C2316yes—
117#41×C173—24no—
118C2×Q8×C173—24no—
119C2×Q8×C173C2313yes—
120C2×Q8×C173C2311yescharacteristic
121C2×D4×C173C2313yes—
122C2×D4×C173C2313yes—
123C22×C4×C173C2313yes—
124C22×C4×C173C2313yes—
125C4⋊C4×C173C2313yes—
126C4⋊C4×C173C2313yes—
127C22⋊C4×C173C2316yes—
128C22⋊C4×C173C2311yescharacteristic
129C22⋊C4×C173C2313yes—
130C42×C173C2313yes—
131C22×D173—22no—
132C22×D173—22no—
133C22×D173—22no—
134C22×D173C2411yescharacteristic
135C22×D173C2411yescharacteristic
136C22×D173C2411yescharacteristic
137Dic346—26no—
138Dic346—26no—
139Dic346—212no—
140Dic346—26no—
141Dic346C2×D4112yes—
142D692—22no—
143D692—22no—
144C22⋊D173—22no—
145C22⋊D173—26no—
146C22⋊D173—26no—
147C22⋊D173—26no—
148C2×Dic173—26no—
149C2×Dic173—212no—
150C2×Dic173C2413yes—
151C2×Dic173C2413yes—
152C2×Dic173C2411yescharacteristic
153C2×Dic173C2413yes—
154C2×Dic173C2413yes—
155C4×D173—26no—
156C4×D173—26no—
157C4×D173—26no—
158C4×D173—26no—
159C4×D173C2×D4112yes—
160C23×C173C2413yes—
161C23×C173C2411yescharacteristic
162Q8×C173—26no—
163Q8×C173C2×D414yes—
164D4×C173—26no—
165D4×C173—26no—
166C2×C4×C173—26no—
167C2×C4×C173—26no—
168C2×C4×C173C2413yes—
169C2×C4×C173C2411yescharacteristic
170C2×C4×C173C2411yescharacteristic
171C2×C4×C173C2413yes—
172C2×C4×C173C2413yes—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—23no—
176C22×C173—23no—
177C22×C173—23no—
178C22×C1732−1+412yescomplemented
179C22×C173C2511yescharacteristic
180C2×D173—28no—
181C2×D173—21no—
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—21no—
188C2×D173C22×D412yes—
189C2×D1732−1+412yescomplemented
190C4×C173—21no—
191C4×C173—23no—
192C4×C173—23no—
193C4×C173—21no—
194C4×C173C22×D416yes—
195Dic173—23no—
196Dic173—23no—
197Dic173—23no—
198Dic173—21no—
199Dic173C22×D416yes—
200C2×C173—22no—
201C2×C173—43no—
202C2×C173C23×D411yescharacteristic
203C2×C173C2×2−1+411yescharacteristic
204C2×C173C2×2−1+411yescharacteristic
205D173—22no—
206D173—22no—
207D173—41no—
208C173#348611yescharacteristic, complemented, Sylow
209#3486—1731nomaximal, Sylow
210C2×2−1+4—1731no—
211C23×Q8—1731no—
212D4×Q8—1731no—
213D4×Q8—1733no—
214D4×Q8—1731no—
215D4×Q8—1733no—
216#548—1731no—
217#548—1733no—
218#548—1733no—
219#548D17311yescharacteristic, complemented
220#536—1733no—
221#536—1733no—
222#525—1733no—
223#525—1733no—
224#519—1731no—
2252−1+4—3468no—
226C2×#41—1733no—
227C2×#41—1733no—
228C2×#41—1733no—
229C2×#41C2×D17311yescharacteristic, complemented
230C22×Q8—3468no—
231C22×Q8—1733no—
232C22×Q8—1733no—
233C22×Q8—1731no—
234C22×Q8—1731no—
235C22×Q8—1731no—
236C22×Q8—1733no—
237C22×Q8—1733no—
238C22×Q8C2×D17311yescharacteristic
239C23×C4—1733no—
240C4⋊Q8—1733no—
241C4⋊Q8—1736no—
242C4⋊Q8—1733no—
243#124—1733no—
244#124—1736no—
245#124C2×D17313yescomplemented
246#123—1733no—
247#123—1736no—
248#123—1733no—
249C22⋊Q8—1733no—
250C22⋊Q8—1733no—
251C22⋊Q8—1733no—
252C22⋊Q8—1733no—
253C22⋊Q8—1733no—
254C22⋊Q8—1733no—
255C22⋊Q8—1733no—
256C22⋊Q8—1736no—
257C22⋊Q8—1736no—
258C22⋊Q8C2×D17313yescomplemented
259#121—1733no—
260#121—1733no—
261#121—1733no—
262#121C2×D17313yes—
263#120—1733no—
264#120—1731no—
265C4×Q8—1731no—
266C4×Q8—1733no—
267C4×Q8—1733no—
268C4×Q8C2×D17311yescharacteristic, complemented
269C4×D4—1736no—
270C4×D4—1733no—
271C4×D4—1733no—
272C4×D4—1736no—
273C4×D4—1733no—
274C4×D4C2×D17313yes—
275#117—1733no—
276#117—1733no—
277C24—1731no—
278#41—24no—
279#41—34612no—
280#41—34612no—
281#41—34612no—
282C2×Q8—24no—
283C2×Q8—34612no—
284C2×Q8—34612no—
285C2×Q8—34612no—
286C2×Q8—3464no—
287C2×Q8—3464no—
288C2×Q8—34612no—
289C2×Q8—1738no—
290C2×Q8—1733no—
291C2×Q8—1736no—
292C2×Q8—1733no—
293C2×Q8—1733no—
294C2×Q8—1733no—
295C2×Q8C22×D17313yes—
296C2×Q8C22×D17311yescharacteristic
297C2×D4—1733no—
298C2×D4—1731no—
299C2×D4—1733no—
300C2×D4—1733no—
301C2×D4—1731no—
302C2×D4—1731no—
303C2×D4C22×D17313yes—
304C2×D4C22×D17313yes—
305C22×C4—34612no—
306C22×C4—1733no—
307C22×C4—1736no—
308C22×C4—1733no—
309C22×C4—1733no—
310C22×C4—1733no—
311C22×C4—1733no—
312C22×C4—1733no—
313C22×C4C22×D17313yes—
314C22×C4C22×D17313yes—
315C4⋊C4—1733no—
316C4⋊C4—1733no—
317C4⋊C4—1733no—
318C4⋊C4—1733no—
319C4⋊C4—1733no—
320C4⋊C4—1736no—
321C4⋊C4—1736no—
322C4⋊C4—1733no—
323C4⋊C4C22×D17313yes—
324C4⋊C4C22×D17313yes—
325C22⋊C4—1733no—
326C22⋊C4—1736no—
327C22⋊C4—1733no—
328C22⋊C4—1733no—
329C22⋊C4—1731no—
330C22⋊C4—1731no—
331C22⋊C4—1733no—
332C22⋊C4—1733no—
333C22⋊C4—1733no—
334C22⋊C4C22×D17316yes—
335C22⋊C4C22×D17311yescharacteristic
336C22⋊C4C22×D17313yes—
337C42—1733no—
338C42—1736no—
339C42C22×D17313yes—
340C23—3462no—
341C23—3462no—
342C23—3462no—
343C23—1731no—
344C23—1731no—
345C23—1731no—
346C23C23×D17313yes—
347C23C23×D17311yescharacteristic
348Q8—26no—
349Q8—34612no—
350Q8—3466no—
351Q8—3466no—
352Q8—3466no—
353Q8—17312no—
354Q8D4×D17314yes—
355D4—26no—
356D4—26no—
357D4—3462no—
358D4—3462no—
359D4—3466no—
360D4—3466no—
361D4—3462no—
362D4—3466no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—3466no—
366C2×C4—3466no—
367C2×C4—3466no—
368C2×C4—34612no—
369C2×C4—3466no—
370C2×C4—3466no—
371C2×C4—1733no—
372C2×C4—1731no—
373C2×C4—1733no—
374C2×C4—1733no—
375C2×C4—1733no—
376C2×C4—17312no—
377C2×C4C23×D17313yes—
378C2×C4C23×D17311yescharacteristic
379C2×C4C23×D17313yes—
380C2×C4C23×D17311yescharacteristic
381C2×C4C23×D17313yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—3468no—
388C22—3461no—
389C22—3461no—
390C22—3461no—
391C22—3461no—
392C22—3461no—
393C22—3461no—
394C22—3461no—
395C22—1732no—
396C22—1732no—
397C22#1914972612yescomplemented
398C22C24×D17311yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—3463no—
404C4—3463no—
405C4—3463no—
406C4—3461no—
407C4—1736no—
408C4C2×D4×D17316yes—
409C2—22no—
410C2—43no—
411C2—3462no—
412C2—3462no—
413C2—6921no—
414C2C22×D4×D17311yescharacteristic
415C22−1+4×D17311yescharacteristic
416C2C2×#1914972611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#548×C173C2
hypercenter#548D173
hypocenterC173#3486
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19149726C2semidirect
2. C22×Q8×D173C2semidirect
3. Q8×D692C2semidirect
4. D4×Dic346C2semidirect
5. Q8×C22⋊D173C2semidirect
6. #26012154C2semidirect
7. #26012149C2semidirect
8. #26012148C2semidirect
9. #26012147C2semidirect
10. #26012064C2semidirect
11. #26012061C2semidirect
12. #26012001C2semidirect
13. #26012000C2semidirect
14. #26011967C2semidirect
15. #548×C173C2semidirect
19. C2×#14583825C22semidirect
21. C22×Dic346C22semidirect
27. C2×Q8×D173C22semidirect
29. #19149679C22semidirect
30. #19149678C22semidirect
31. #19149677C22semidirect
32. #19149666C22semidirect
33. #19149665C22semidirect
38. #19149659C22semidirect
40. #19149658C22semidirect
43. #19149656C22semidirect
44. #19149656C22semidirect
45. #19149655C22semidirect
51. Q8×Dic173C22semidirect
52. C4×Dic346C22semidirect
53. #19149644C22semidirect
61. C2×#41×C173C22semidirect
63. #124×C173C22semidirect
64. C22⋊Q8×C173C22semidirect
66. C4×Q8×C173C22semidirect
83. Q8×D173D4semidirect
178. C22×C1732−1+4semidirect
189. C2×D1732−1+4semidirect
208. C173#3486semidirect
219. #548D173semidirect
229. C2×#41C2×D173semidirect
245. #124C2×D173semidirect
258. C22⋊Q8C2×D173semidirect
268. C4×Q8C2×D173semidirect
397. C22#19149726semidirect

Maximal subgroups

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

subgroupindex
C2×#191497262
C22×Q8×D1732
Q8×D6922
D4×Dic3462
Q8×C22⋊D1732
#260121542
#260121492
#260121482
#260121472
#260120642
#260120612
#260120012
#260120002
#260119672
#548×C1732
#3486173