Finite Group Databasebeta

← #36365467

Subgroups — #36365467

Order 22144.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012187C211yesmaximal, characteristic, complemented
2#26012150C211yesmaximal, characteristic, complemented
3#26011846C211yesmaximal, characteristic, complemented
4#26011833C211yesmaximal, characteristic, complemented
5#26011810C211yesmaximal, characteristic, complemented
6#26011799C211yesmaximal, characteristic, complemented
7#26011727C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011707C211yesmaximal, characteristic, complemented
10#26011638C211yesmaximal, characteristic, complemented
11#26011632C211yesmaximal, characteristic, complemented
12C4×#14583815C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic, complemented
14C2×#19149604C211yesmaximal, characteristic, complemented
15C8×D4×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21#19149674—21no—
22#19149671C2211yescharacteristic
23#19149667C2211yescharacteristic
24#19149665—21no—
25#19149661—21no—
26#19149658C2211yescharacteristic
27#19149657—21no—
28#19149653C2211yescharacteristic
29#19149653C2211yescharacteristic
30#19149653C2211yescharacteristic
31#19149651—21no—
32#19149651—21no—
33C4×Dic346C2211yescharacteristic
34C4×D692C2211yescharacteristic
35#19149639—21no—
36#19149639—21no—
37#19149638C2211yescharacteristic
38C2×C4⋊Dic173C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic
52C22×C8×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—41no—
61C2×Dic346—22no—
62C2×Dic346—21no—
63C2×Dic346—41no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C22×Dic173—21no—
73C22×Dic173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76#14583815—22no—
77#14583815—41no—
78D1384—41no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C2×D4×C173D411yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C2×C8×C173—41no—
102C2×C8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117Dic346#4112yes—
118D692—21no—
119D692—21no—
120D692—21no—
121D692—21no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C22⋊D173—81no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—42no—
133C2×Dic173—81no—
134C4×D173—42no—
135C4×D173—81no—
136C23×C173C2×D411yescharacteristic
137C23×C173C2×D411yescharacteristic
138D4×C173—22no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C8×C173—21no—
149C8×C173—21no—
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173#13512yes—
158C22×C173C22×D411yescharacteristic
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—21no—
168C4×C173—21no—
169C4×C173C2×#4111yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—22no—
172Dic173—41no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—22no—
178C2×C173—41no—
179C2×C173#54611yescharacteristic
180C2×C173C2×#13511yescharacteristic
181C2×C173#57711yescharacteristic
182D173—81no—
183D173—81no—
184C173#330111yescharacteristic, complemented, Sylow
185#3301—1731nomaximal, Sylow
186#553—1731no—
187#548—1731no—
188#488—1731no—
189#485—1731no—
190#481—1731no—
191#479—1731no—
192#466—1731no—
193#465—1731no—
194#463—1731no—
195#449—1731no—
196#448—1731no—
197C4×SD16—1731no—
198C8×D4D17311yescharacteristic, complemented
199#416—1731no—
200C2×#103—1731no—
201C2×#41—1731no—
202C22×Q8—1731no—
203C2×SD16—1731no—
204C2×SD16—1731no—
205C2×D8—1731no—
206C22×C8C2×D17311yescharacteristic
207C22×C8C2×D17311yescharacteristic
208#126—3461no—
209#125—1731no—
210#124—3461no—
211#124—1731no—
212#123—3461no—
213C22⋊Q8—3461no—
214C22⋊Q8—1731no—
215#121—3461no—
216#121—3461no—
217#121—1731no—
218#121—1731no—
219#121—1731no—
220C4×Q8—1731no—
221C4×D4—3461no—
222C4×D4—3461no—
223C4×D4—1731no—
224C4×D4C2×D17311yescharacteristic
225#117—1731no—
226C2×C4⋊C4—1731no—
227#107—1731no—
228#106—1731no—
229#106—1731no—
230C4⋊C8C2×D17311yescharacteristic
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241C22⋊C8C2×D17311yescharacteristic
242C22⋊C8C2×D17311yescharacteristic
243C4×C8C2×D17311yescharacteristic
244#41—3462no—
245#41—6921no—
246C2×Q8—3462no—
247C2×Q8—3461no—
248C2×Q8—1731no—
249C2×Q8—1731no—
250C2×Q8—6921no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4—1731no—
257C2×D4D69211yescharacteristic
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4C22×D17311yescharacteristic
263C22×C4C22×D17311yescharacteristic
264SD16—3462no—
265SD16—6921no—
266D8—6921no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8C22×D17311yescharacteristic
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—1731no—
279C4⋊C4—1731no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4D69211yescharacteristic
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4D69211yescharacteristic
292C22⋊C4D69211yescharacteristic
293C42—3461no—
294C42—3461no—
295C42C22×D17311yescharacteristic
296C23—3461no—
297C23—3461no—
298C23C2×D69211yescharacteristic
299C23C2×D69211yescharacteristic
300Q8—3461no—
301Q8—3461no—
302Q8—3461no—
303Q8—1732no—
304Q8—6922no—
305D4—22no—
306D4—3461no—
307D4—3461no—
308D4—3461no—
309D4—3461no—
310D4—13841no—
311D4—6922no—
312D4—6922no—
313D4—6922no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—13841no—
325C2×C4—13841no—
326C2×C4—6922no—
327C2×C4—6922no—
328C2×C4C2×D69211yescharacteristic
329C2×C4C2×D69211yescharacteristic
330C2×C4C2×D69211yescharacteristic
331C2×C4C2×D69211yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—6921no—
343C22—6921no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22—6921no—
348C22#1914970112yes—
349C22C22×D69211yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—3462no—
355C4—6921no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4#41×D17311yescharacteristic
361C4#1914972411yescharacteristic
362C2—22no—
363C2—41no—
364C2—13841no—
365C2—13841no—
366C2#2601213611yescharacteristic
367C2C2×#1914970111yescharacteristic
368C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC8×D4×C173C2
socleC22×C173C22×D4
hypercenterC8×D4D173
hypocenterC173#3301
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #26012187C2semidirect
2. #26012150C2semidirect
3. #26011846C2semidirect
4. #26011833C2semidirect
5. #26011810C2semidirect
6. #26011799C2semidirect
7. #26011727C2semidirect
8. #26011720C2semidirect
9. #26011707C2semidirect
10. #26011638C2semidirect
11. #26011632C2semidirect
12. C4×#14583815C2semidirect
13. #26011488C2semidirect
14. C2×#19149604C2semidirect
15. C8×D4×C173C2semidirect
184. C173#3301semidirect
198. C8×D4D173semidirect

Maximal subgroups

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

subgroupindex
#260121872
#260121502
#260118462
#260118332
#260118102
#260117992
#260117272
#260117202
#260117072
#260116382
#260116322
C4×#145838152
#260114882
C2×#191496042
C8×D4×C1732
#3301173