Finite Group Databasebeta

← #36365421

Subgroups — #36365421

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D173C211yesmaximal, characteristic, complemented
2#26012141C211yesmaximal, characteristic
3#26011840C211yesmaximal, characteristic, complemented
4#26011822C211yesmaximal, characteristic, complemented
5#26011820C211yesmaximal, characteristic, complemented
6#26011774C211yesmaximal, characteristic, complemented
7#26011718C211yesmaximal, characteristic, complemented
8#26011717C211yesmaximal, characteristic, complemented
9#26011694C211yesmaximal, characteristic, complemented
10#26011628C211yesmaximal, characteristic, complemented
11SD16×Dic173C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13C2×#19149603C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#448×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×Q8×D173C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C22×#13063436C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149677—21no—
23#19149677C2211yescharacteristic
24#19149661—21no—
25#19149660—21no—
26#19149658—21no—
27#19149658C2211yescharacteristic
28#19149656—21no—
29#19149655—21no—
30#19149655C2211yescharacteristic
31#19149650C2211yescharacteristic
32Q8×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic
34C4×C22⋊D173—21no—
35C4×C22⋊D173C2211yescharacteristic
36C2×C4⋊Dic173C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149611C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149603C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583825—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—41no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173D411yescharacteristic
67C22×Dic173—21no—
68C22×Dic173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173C2311yescharacteristic
72#14583814—41no—
73C2×#13063436—41no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C8×D173—41no—
77C4⋊Dic173—21no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805D411yescharacteristic
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173D411yescharacteristic
93#14583803—21no—
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173—21no—
98C2×Q8×C173—22no—
99C2×Q8×C173—41no—
100C2×Q8×C173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173—21no—
105C22×C4×C173C2311yescharacteristic
106SD16×C173—22no—
107SD16×C173—41no—
108C2×C8×C173C2311yescharacteristic
109C2×C8×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—42no—
116Dic346—42no—
117C22⋊D173—42no—
118C22⋊D173SD1614yescomplemented
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—42no—
124C2×Dic173—42no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—21no—
129C4×D173—21no—
130C4×D173—42no—
131C4×D173—42no—
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135C23×C173C2×D411yescharacteristic
136Q8×C173—21no—
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173—42no—
140Q8×C173#4112yescomplemented
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—42no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—42no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173C2×SD1612yes—
160C22×C173C22×D411yescharacteristic
161C2×D173—21no—
162C2×D173—21no—
163C2×D173C2×SD1612yes—
164C4×C173—21no—
165C4×C173—22no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173C2×#4111yescharacteristic
169C4×C173H(5,𝔽2)11yescharacteristic
170Dic173—21no—
171Dic173—21no—
172Dic173—41no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176Dic173C2×SD1612yes—
177C2×C173—22no—
178C2×C173—81no—
179C2×C173#54611yescharacteristic
180C2×C173C22×SD1611yescharacteristic
181C2×C173#57811yescharacteristic
182D173—22no—
183C173#329811yescharacteristic, complemented, Sylow
184#3298—1731nomaximal, Sylow
185D4×Q8—1731no—
186#547—1731no—
187#487—1731no—
188#483—1731no—
189#483—1731no—
190#475—1731no—
191#465—1731no—
192#465—1731no—
193#461—1731no—
194#448—1731no—
195#448D17311yescharacteristic, complemented
196C4×SD16—1731no—
197C8×D4—1731no—
198C2×#103—1731no—
199C2×#103—1731no—
200C2×#41—3461no—
201C22×Q8—1731no—
202C22×Q8C2×D17311yescharacteristic
203C2×SD16—1731no—
204C2×SD16C2×D17311yescharacteristic
205C2×SD16C2×D17311yescharacteristic
206C22×C8—1731no—
207C22×C8—1731no—
208C4⋊Q8—3461no—
209C4⋊Q8—1731no—
210C4⋊Q8—1731no—
211#123—3461no—
212#123—3461no—
213C22⋊Q8—3461no—
214C22⋊Q8—3461no—
215C22⋊Q8—3461no—
216C22⋊Q8—1731no—
217C22⋊Q8—1731no—
218#121—1731no—
219#121C2×D17311yescharacteristic
220C4×Q8—1731no—
221C4×D4—3461no—
222C4×D4—1731no—
223C4×D4—1731no—
224C2×C4⋊C4—1731no—
225C2×C4⋊C4—1731no—
226#106—1731no—
227#106—1731no—
228#106—1731no—
229C4⋊C8—1731no—
230#103—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103C2×D17311yescharacteristic
238#103C2×D17311yescharacteristic
239#102—1731no—
240C22⋊C8—1731no—
241C22⋊C8C2×D17311yescharacteristic
242C4×C8—1731no—
243#41—6924no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—3461no—
248C2×Q8—3461no—
249C2×Q8—3462no—
250C2×Q8—1731no—
251C2×Q8—6921no—
252C2×Q8C22×D17311yescharacteristic
253C2×Q8C22×D17311yescharacteristic
254C2×D4—21no—
255C2×D4—3461no—
256C2×D4—1731no—
257C2×D4C22×D17311yescharacteristic
258C22×C4—21no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—3461no—
263C22×C4—1731no—
264C22×C4C22×D17311yescharacteristic
265SD16—22no—
266SD16—41no—
267SD16—6921no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—6921no—
271C2×C8—6921no—
272C2×C8C22×D17311yescharacteristic
273C2×C8C22×D17311yescharacteristic
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—1731no—
283C4⋊C4—1731no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4C22×D17311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—1731no—
293C22⋊C4—1731no—
294C42—3461no—
295C42—1731no—
296C23—21no—
297C23—1731no—
298C23D4×D17311yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—3461no—
304Q8—3461no—
305Q8—6922no—
306Q8—6922no—
307Q8#1458382512yescomplemented
308D4—21no—
309D4—21no—
310D4—42no—
311D4—1734no—
312D4—6922no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—1731no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—6922no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4—6922no—
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—3461no—
344C22—3461no—
345C22—1732no—
346C22SD16×D17312yes—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—3461no—
353C4—3461no—
354C4—1732no—
355C4—6921no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4C2×#1458382511yescharacteristic
360C4#1914972411yescharacteristic
361C2—22no—
362C2—81no—
363C2—3462no—
364C2#2601212811yescharacteristic
365C2C2×SD16×D17311yescharacteristic
366C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
prime radical#26012141C2
hypercenter#448D173
hypocenterC173#3298
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D173C2semidirect
3. #26011840C2semidirect
4. #26011822C2semidirect
5. #26011820C2semidirect
6. #26011774C2semidirect
7. #26011718C2semidirect
8. #26011717C2semidirect
9. #26011694C2semidirect
10. #26011628C2semidirect
11. SD16×Dic173C2semidirect
12. #26011554C2semidirect
13. C2×#19149603C2semidirect
14. #103×D173C2semidirect
15. #448×C173C2semidirect
118. C22⋊D173SD16semidirect
140. Q8×C173#41semidirect
183. C173#3298semidirect
195. #448D173semidirect
307. Q8#14583825semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1732
#260121412
#260118402
#260118222
#260118202
#260117742
#260117182
#260117172
#260116942
#260116282
SD16×Dic1732
#260115542
C2×#191496032
#103×D1732
#448×C1732
#3298173