Finite Group Databasebeta

← #36365424

Subgroups — #36365424

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#26012142C211yesmaximal, characteristic
3#26011840C211yesmaximal, characteristic, complemented
4#26011821C211yesmaximal, characteristic, complemented
5#26011820C211yesmaximal, characteristic, complemented
6#26011774C211yesmaximal, characteristic, complemented
7#26011723C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011695C211yesmaximal, characteristic, complemented
10#26011631C211yesmaximal, characteristic, complemented
11#26011630C211yesmaximal, characteristic, complemented
12#26011576C211yesmaximal, characteristic, complemented
13C8×C22⋊D173C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15C2×#103×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×Q8×D173C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149678C2211yescharacteristic
23#19149677—21no—
24#19149677C2211yescharacteristic
25#19149664—21no—
26#19149662—21no—
27#19149658—21no—
28#19149658C2211yescharacteristic
29#19149657—21no—
30#19149655—21no—
31#19149655C2211yescharacteristic
32#19149654C2211yescharacteristic
33#19149653C2211yescharacteristic
34Q8×Dic173C2211yescharacteristic
35#19149641C2211yescharacteristic
36C4×C22⋊D173—21no—
37C4×C22⋊D173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149601C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C2×C4⋊C4×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#14583826—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—41no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173D411yescharacteristic
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73#14583815—41no—
74#14583813—22no—
75#14583813—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C8×D173—41no—
79C4⋊Dic173—21no—
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⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×Q8×C173—21no—
99C2×Q8×C173—22no—
100C2×Q8×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C22×C4×C173—21no—
104C22×C4×C173—21no—
105C22×C4×C173C2311yescharacteristic
106C2×C8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22×D173—21no—
113C22×D173C2×D411yescharacteristic
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117Dic346—42no—
118D692—21no—
119D692—21no—
120D692—42no—
121C22⋊D173—42no—
122C22⋊D173SD1614yescomplemented
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—42no—
132C4×D173—42no—
133C4×D173—42no—
134#13063436—21no—
135#13063436—21no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173—21no—
140Q8×C173—42no—
141Q8×C173#4112yescomplemented
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—42no—
149C2×C4×C173—42no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173C2×SD1612yes—
157C22×C173C22×D411yescharacteristic
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173C2×SD1612yes—
164C4×C173—21no—
165C4×C173—22no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176Dic173C2×SD1612yes—
177C2×C173—22no—
178C2×C173#54611yescharacteristic
179C2×C173C22×SD1611yescharacteristic
180C2×C173#57811yescharacteristic
181D173—22no—
182D173—81no—
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#448—1731no—
196C4×SD16—1731no—
197C8×D4—1731no—
198C2×#103—1731no—
199C2×#103D17311yescharacteristic, complemented
200C2×#41—3461no—
201C22×Q8—1731no—
202C22×Q8C2×D17311yescharacteristic
203C2×SD16—1731no—
204C2×SD16—1731no—
205C2×SD16—1731no—
206C22×C8—1731no—
207C22×C8C2×D17311yescharacteristic
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#121—1731no—
220C4×Q8—1731no—
221C4×D4—3461no—
222C4×D4—1731no—
223C4×D4—1731no—
224C2×C4⋊C4—1731no—
225C2×C4⋊C4C2×D17311yescharacteristic
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#103C2×D17311yescharacteristic
236#103C2×D17311yescharacteristic
237#103C2×D17311yescharacteristic
238#103C2×D17311yescharacteristic
239#102—1731no—
240C22⋊C8—1731no—
241C22⋊C8—1731no—
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—3461no—
255C2×D4—3461no—
256C2×D4—1731no—
257C2×D4—1731no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—3461no—
263C22×C4—1731no—
264C22×C4C22×D17311yescharacteristic
265SD16—3462no—
266SD16—6921no—
267SD16—6921no—
268C2×C8—41no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8—6921no—
272C2×C8C22×D17311yescharacteristic
273C2×C8C22×D17311yescharacteristic
274C4⋊C4—21no—
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⋊C4C22×D17311yescharacteristic
286C4⋊C4C22×D17311yescharacteristic
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—1731no—
293C22⋊C4—1731no—
294C42—3461no—
295C42—1731no—
296C23—3461no—
297C23—1731no—
298C23D4×D17311yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—3461no—
304Q8—3461no—
305Q8—6922no—
306Q8—6922no—
307Q8#1458382612yescomplemented
308D4—3461no—
309D4—3461no—
310D4—1734no—
311D4—6922no—
312D4—6922no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—1731no—
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—3461no—
341C22—3461no—
342C22—1732no—
343C22—6921no—
344C22—6921no—
345C22—6921no—
346C22SD16×D17312yes—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—3461no—
354C4—3461no—
355C4—1732no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4C2×#1458382611yescharacteristic
360C4#1914972411yescharacteristic
361C2—22no—
362C2—3462no—
363C2—13841no—
364C2#2601213111yescharacteristic
365C2C2×SD16×D17311yescharacteristic
366C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#103×C173C2
socleC22×C173C22×D4
prime radical#26012142C2
hypercenterC2×#103D173
hypocenterC173#3298
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D173C2semidirect
3. #26011840C2semidirect
4. #26011821C2semidirect
5. #26011820C2semidirect
6. #26011774C2semidirect
7. #26011723C2semidirect
8. #26011720C2semidirect
9. #26011695C2semidirect
10. #26011631C2semidirect
11. #26011630C2semidirect
12. #26011576C2semidirect
13. C8×C22⋊D173C2semidirect
14. #103×D173C2semidirect
15. C2×#103×C173C2semidirect
122. C22⋊D173SD16semidirect
141. Q8×C173#41semidirect
183. C173#3298semidirect
199. C2×#103D173semidirect
307. Q8#14583826semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1732
#260121422
#260118402
#260118212
#260118202
#260117742
#260117232
#260117202
#260116952
#260116312
#260116302
#260115762
C8×C22⋊D1732
#103×D1732
C2×#103×C1732
#3298173