Finite Group Databasebeta

← #36363091

Subgroups — #36363091

Order 22144.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149708C211yesmaximal, characteristic
2C2×Q16×D173C211yesmaximal, characteristic, complemented
3#26012061C211yesmaximal, characteristic
4#26011967C211yesmaximal, characteristic, complemented
5#26011712C211yesmaximal, characteristic, complemented
6#26011711C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011702C211yesmaximal, characteristic, complemented
9#26011701C211yesmaximal, characteristic, complemented
10#26011700C211yesmaximal, characteristic, complemented
11#26011699C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011498C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic
15#464×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19#19149708—24no—
20C2×Dic692C2211yescharacteristic
21C2×Dic692C2211yescharacteristic
22C2×#14583816C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583816C2211yescharacteristic
25Q16×D173—24no—
26C2×#14583814C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149679C2211yescharacteristic
30#19149677C2211yescharacteristic
31#19149666C2211yescharacteristic
32#19149661—21no—
33#19149661—21no—
34#19149656—21no—
35#19149656—21no—
36Q8×Dic173C2211yescharacteristic
37C4×Dic346—21no—
38C4×Dic346C2211yescharacteristic
39#19149644—21no—
40#19149644C2211yescharacteristic
41#19149637—21no—
42#19149637C2211yescharacteristic
43#19149636—21no—
44#19149636C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic
46#19149608C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149604C2211yescharacteristic
50#19149603C2211yescharacteristic
51#19149602C2211yescharacteristic
52#19149602C2211yescharacteristic
53#19149600C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57C4×Q8×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—22no—
63C2×Dic346—21no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68Q8×D173—22no—
69Q8×D173—22no—
70Q8×D173—22no—
71Q8×D173D414yes—
72C2×C22⋊D173—21no—
73C22×Dic173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77Dic692—22no—
78Dic692—22no—
79#14583816—22no—
80#14583816—22no—
81#14583816—22no—
82#14583814—22no—
83#14583809—22no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C8×D173—22no—
87C4⋊Dic173—21no—
88C4⋊Dic173—21no—
89C4⋊Dic173—21no—
90C4⋊Dic173C2311yescharacteristic
91#14583805—21no—
92#14583805—21no—
93#14583805—21no—
94#14583805—21no—
95#14583805D411yescharacteristic
96#14583805D411yescharacteristic
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99#14583803—21no—
100#14583803D411yescharacteristic
101#14583803D411yescharacteristic
102C4×Dic173—21no—
103C4×Dic173—21no—
104C4×Dic173—21no—
105C4×Dic173—21no—
106C4×Dic173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110Q16×C173—22no—
111SD16×C173—22no—
112C2×C8×C173C2311yescharacteristic
113C2×C8×C173C2311yescharacteristic
114C4⋊C4×C173—21no—
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C42×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173C2×D411yescharacteristic
121Dic346—21no—
122Dic346—24no—
123Dic346—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—42no—
128Dic346C2×D412yes—
129C22⋊D173—42no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—42no—
135C2×Dic173C2×D411yescharacteristic
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C4×D173—24no—
139C4×D173—42no—
140C4×D173C2×D412yes—
141C4×D173#4112yes—
142#13063436—21no—
143#13063436—21no—
144C23×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148Q8×C173C2×D412yes—
149D4×C173—21no—
150D4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2411yescharacteristic
157C8×C173—21no—
158C8×C173—21no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173C22×D411yescharacteristic
163C2×D173—21no—
164C2×D173—21no—
165C2×D173#12112yes—
166C4×C173—22no—
167C4×C173—21no—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173C22×D411yescharacteristic
172C4×C173C2×#4111yescharacteristic
173Dic173—22no—
174Dic173—21no—
175Dic173—21no—
176Dic173—41no—
177Dic173—41no—
178Dic173—41no—
179Dic173#12112yes—
180C2×C173—81no—
181C2×C173C2×#12111yescharacteristic
182C2×C173#57811yescharacteristic
183C2×C173#57811yescharacteristic
184D173—22no—
185C173#304411yescharacteristic, complemented, Sylow
186#3044—1731nomaximal, Sylow
187C2×#137—1731no—
188C22×Q16—1731no—
189#536—1731no—
190#519—1731no—
191#464—1731no—
192#464—1731no—
193#464—1731no—
194#464D17311yescharacteristic, complemented
195#462—1731no—
196#462—1731no—
197#462—1731no—
198#462—1731no—
199#424—1731no—
200#419—1731no—
201#416—1731no—
202C2×#41—1731no—
203C22×Q8—1731no—
204C22×Q8—1731no—
205#137—3464no—
206C2×Q16—3464no—
207C2×Q16—1731no—
208C2×Q16—1731no—
209C2×Q16—1731no—
210C2×Q16—1731no—
211C2×Q16—1731no—
212C2×Q16C2×D17311yescharacteristic
213C2×SD16—1731no—
214C2×SD16C2×D17311yescharacteristic
215C2×M16—1731no—
216C22×C8—1731no—
217C4⋊Q8—1731no—
218C4⋊Q8—1731no—
219#124—1731no—
220#124C2×D17311yescharacteristic
221#123—3461no—
222#123—3461no—
223C22⋊Q8—3461no—
224C22⋊Q8—3461no—
225C4×Q8—3461no—
226C4×Q8—3461no—
227C4×Q8—1731no—
228C4×Q8—1731no—
229C4×Q8—1731no—
230C4×Q8C2×D17311yescharacteristic
231#117—3461no—
232#117—3461no—
233#117—1731no—
234#117—1731no—
235C4⋊C8—1731no—
236C4⋊C8—1731no—
237C4⋊C8—1731no—
238C4⋊C8C2×D17311yescharacteristic
239#103—1731no—
240#103—1731no—
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#103C2×D17311yescharacteristic
245#102—1731no—
246#102C2×D17311yescharacteristic
247#41—3462no—
248#41—3462no—
249C2×Q8—3461no—
250C2×Q8—3461no—
251C2×Q8—3462no—
252C2×Q8—3462no—
253C2×Q8—3462no—
254C2×Q8—1734no—
255C2×Q8—1731no—
256C2×Q8—1731no—
257C2×Q8—1731no—
258C2×Q8C22×D17311yescharacteristic
259C2×Q8C22×D17311yescharacteristic
260C2×D4—3461no—
261C2×D4C22×D17311yescharacteristic
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—1731no—
266Q16—22no—
267Q16—3462no—
268Q16—3462no—
269Q16—3462no—
270Q16—3462no—
271Q16—3462no—
272SD16—22no—
273SD16—3462no—
274M16—3462no—
275C2×C8—3462no—
276C2×C8—1731no—
277C2×C8—1731no—
278C2×C8C22×D17311yescharacteristic
279C2×C8C22×D17311yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—1731no—
289C4⋊C4—1731no—
290C4⋊C4—1731no—
291C4⋊C4C22×D17311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—1731no—
298C22⋊C4—1731no—
299C42—21no—
300C42—3461no—
301C42—3461no—
302C42—3461no—
303C42—3461no—
304C42—1731no—
305C42C22×D17311yescharacteristic
306C23—21no—
307C23—1731no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—3464no—
312Q8—3461no—
313Q8—3461no—
314Q8—3461no—
315Q8—3461no—
316Q8—3461no—
317Q8—1732no—
318Q8—6922no—
319Q8D4×D17312yes—
320D4—21no—
321D4—21no—
322D4—6922no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—3464no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—1731no—
332C2×C4—1731no—
333C2×C4—1731no—
334C2×C4—1732no—
335C2×C4—1732no—
336C2×C4—6922no—
337C2×C4—6922no—
338C2×C4D4×D17311yescharacteristic
339C2×C4D4×D17311yescharacteristic
340C2×C4C23×D17311yescharacteristic
341C8—21no—
342C8—21no—
343C8—3461no—
344C8—3461no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—3461no—
349C22—3461no—
350C22—1732no—
351C22C2×D4×D17311yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—3462no—
358C4—3461no—
359C4—3461no—
360C4—1732no—
361C4—6921no—
362C4—6921no—
363C4—6921no—
364C4C2×D4×D17311yescharacteristic
365C4#41×D17311yescharacteristic
366C2—81no—
367C2—3462no—
368C2#121×D17311yescharacteristic
369C2#2601230511yescharacteristic
370C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583825C22
hypercenter#464D173
hypocenterC173#3044
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D173C2semidirect
4. #26011967C2semidirect
5. #26011712C2semidirect
6. #26011711C2semidirect
7. #26011710C2semidirect
8. #26011702C2semidirect
9. #26011701C2semidirect
10. #26011700C2semidirect
11. #26011699C2semidirect
12. #26011516C2semidirect
13. #26011498C2semidirect
15. #464×C173C2semidirect
185. C173#3044semidirect
194. #464D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C2×Q16×D1732
#260120612
#260119672
#260117122
#260117112
#260117102
#260117022
#260117012
#260117002
#260116992
#260115162
#260114982
#260114842
#464×C1732
#3044173