Finite Group Databasebeta

← #36363017

Subgroups — #36363017

Order 22144.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149702C211yesmaximal, characteristic, complemented
2C2×#19149700C211yesmaximal, characteristic, complemented
3#26012040C211yesmaximal, characteristic, complemented
4C4×#14583827C211yesmaximal, characteristic, complemented
5#26011713C211yesmaximal, characteristic, complemented
6#26011707C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011699C211yesmaximal, characteristic, complemented
9#26011695C211yesmaximal, characteristic, complemented
10#26011689C211yesmaximal, characteristic, complemented
11#26011685C211yesmaximal, characteristic, complemented
12C4⋊C8×D173C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#464×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583827C2211yescharacteristic
18C2×#14583825C2211yescharacteristic
19#19149702—24no—
20#19149700—24no—
21C2×#14583816C2211yescharacteristic
22C2×#14583815C2211yescharacteristic
23C2×#14583813C2211yescharacteristic
24C2×#14583813C2211yescharacteristic
25C2×D1384C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×C8×D173C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149677C2211yescharacteristic
30#19149675C2211yescharacteristic
31#19149667C2211yescharacteristic
32#19149652—21no—
33#19149652—21no—
34Q8×Dic173C2211yescharacteristic
35C4×D692—21no—
36C4×D692C2211yescharacteristic
37#19149641—21no—
38#19149641C2211yescharacteristic
39#19149639—21no—
40#19149639—21no—
41#19149637—21no—
42#19149637C2211yescharacteristic
43C42×D173—21no—
44C42×D173C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic
46#19149608C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149603C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149601C2211yescharacteristic
52#19149600C2211yescharacteristic
53#19149599C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57C4×Q8×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—22no—
63#14583827—22no—
64#14583827D414yescomplemented
65#14583825—22no—
66#14583825—22no—
67C2×Dic346C2311yescharacteristic
68C2×D692—21no—
69C2×D692—21no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C22×Dic173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173D411yescharacteristic
81C2×C4×D173D411yescharacteristic
82C2×C4×D173C2311yescharacteristic
83#14583816—22no—
84#14583815—22no—
85#14583813—22no—
86#14583813—22no—
87D1384—22no—
88#14583811—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C8×D173—22no—
92C8×D173—22no—
93C4⋊Dic173—21no—
94C4⋊Dic173C2311yescharacteristic
95#14583805—21no—
96#14583805—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102C4×Dic173—21no—
103C4×Dic173—21no—
104C4×Dic173D411yescharacteristic
105C4×Dic173D411yescharacteristic
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×D173—21no—
121C22×D173—21no—
122C22×D173C2×D411yescharacteristic
123Dic346—21no—
124Dic346—21no—
125D692—24no—
126D692—21no—
127D692—21no—
128D692—21no—
129D692—42no—
130D692C2×D412yes—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—42no—
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—24no—
141C4×D173—22no—
142C4×D173—41no—
143C4×D173—42no—
144C4×D173—42no—
145C4×D173—42no—
146C4×D173C2×D412yes—
147C4×D173#4112yes—
148#13063436—21no—
149#13063436—21no—
150C23×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173C2×D412yes—
155D4×C173—21no—
156D4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2×D411yescharacteristic
162C2×C4×C173C2411yescharacteristic
163C8×C173—21no—
164C8×C173—21no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173C22×D411yescharacteristic
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—22no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173#12112yes—
178C4×C173—22no—
179C4×C173—21no—
180C4×C173—21no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173C22×D411yescharacteristic
184C4×C173C2×#4111yescharacteristic
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188Dic173#12112yes—
189Dic173#13512yescomplemented
190C2×C173—81no—
191C2×C173C2×#12111yescharacteristic
192C2×C173C2×#13511yescharacteristic
193C2×C173#57511yescharacteristic
194D173—22no—
195D173—42no—
196D173—81no—
197C173#303511yescharacteristic, complemented, Sylow
198#3035—1731nomaximal, Sylow
199C2×#135—1731no—
200C2×#135—1731no—
201#532—1731no—
202C4×#41—1731no—
203#464—1731no—
204#464D17311yescharacteristic, complemented
205#463—1731no—
206#463—1731no—
207#462—1731no—
208#461—1731no—
209#460—1731no—
210#459—1731no—
211C2×C4⋊C8—1731no—
212#416—1731no—
213#416—1731no—
214C2×#41—1731no—
215C2×#41—1731no—
216C2×#41—1731no—
217#135—3464no—
218#135—3464no—
219C2×Q16—1731no—
220C2×Q16C2×D17311yescharacteristic
221C2×SD16—1731no—
222C2×SD16—1731no—
223C2×SD16—1731no—
224C2×SD16C2×D17311yescharacteristic
225C2×D8—1731no—
226C2×D8—1731no—
227C22×C8—1731no—
228C22×C8—1731no—
229C4⋊Q8—1731no—
230#127—1731no—
231#124—1731no—
232#124C2×D17311yescharacteristic
233#121—3461no—
234#121—3461no—
235C4×Q8—1731no—
236C4×Q8C2×D17311yescharacteristic
237C4×D4—3461no—
238C4×D4—3461no—
239C4×D4—3461no—
240C4×D4—3461no—
241C4×D4—1731no—
242C4×D4—1731no—
243#117—3461no—
244#117—1731no—
245C2×C42—3461no—
246C2×C42—1731no—
247C4⋊C8—1731no—
248C4⋊C8—1731no—
249C4⋊C8—1731no—
250C4⋊C8C2×D17311yescharacteristic
251#103—1731no—
252#103—1731no—
253#103—1731no—
254#103C2×D17311yescharacteristic
255#102—1731no—
256#102—1731no—
257#102—1731no—
258#102C2×D17311yescharacteristic
259#41—3462no—
260#41—3462no—
261#41—3462no—
262#41—3462no—
263#41—3462no—
264#41—1734no—
265C2×Q8—1731no—
266C2×Q8C22×D17311yescharacteristic
267C2×Q8C22×D17311yescharacteristic
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4C22×D17311yescharacteristic
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—1731no—
282C22×C4—1731no—
283C22×C4—1731no—
284Q16—22no—
285Q16—3462no—
286SD16—22no—
287SD16—3462no—
288SD16—3462no—
289SD16—3462no—
290D8—3462no—
291D8—3462no—
292C2×C8—3462no—
293C2×C8—3462no—
294C2×C8—1731no—
295C2×C8—1731no—
296C2×C8C22×D17311yescharacteristic
297C2×C8C22×D17311yescharacteristic
298C4⋊C4—21no—
299C4⋊C4—3461no—
300C4⋊C4—3461no—
301C4⋊C4—3461no—
302C4⋊C4—1731no—
303C4⋊C4C22×D17311yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C42—21no—
312C42—3461no—
313C42—3461no—
314C42—1731no—
315C42—1731no—
316C42—1731no—
317C42C22×D17311yescharacteristic
318C23—21no—
319C23—3461no—
320C23—3461no—
321C23—1731no—
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—3461no—
326Q8—3461no—
327Q8D4×D17312yes—
328D4—21no—
329D4—21no—
330D4—3464no—
331D4—3461no—
332D4—3461no—
333D4—3461no—
334D4—1732no—
335D4—6922no—
336D4—6922no—
337D4—6922no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3464no—
343C2×C4—3462no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—1731no—
347C2×C4—1731no—
348C2×C4—1731no—
349C2×C4—1732no—
350C2×C4—1732no—
351C2×C4—6921no—
352C2×C4—6922no—
353C2×C4—6922no—
354C2×C4—6922no—
355C2×C4—6922no—
356C2×C4D4×D17311yescharacteristic
357C2×C4D4×D17311yescharacteristic
358C2×C4C23×D17311yescharacteristic
359C8—21no—
360C8—21no—
361C8—3461no—
362C8—3461no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—3461no—
367C22—3461no—
368C22—3462no—
369C22—1732no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22C2×D4×D17311yescharacteristic
376C4—22no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—3461no—
382C4—1732no—
383C4—1732no—
384C4—6921no—
385C4—6921no—
386C4C2×D4×D17311yescharacteristic
387C4#41×D17311yescharacteristic
388C2—81no—
389C2—3462no—
390C2—13841no—
391C2—6922no—
392C2#121×D17311yescharacteristic
393C2#135×D17311yescharacteristic
394C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
hypercenter#464D173
hypocenterC173#3035
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
1. C2×#19149702C2semidirect
2. C2×#19149700C2semidirect
3. #26012040C2semidirect
4. C4×#14583827C2semidirect
5. #26011713C2semidirect
6. #26011707C2semidirect
7. #26011703C2semidirect
8. #26011699C2semidirect
9. #26011695C2semidirect
10. #26011689C2semidirect
11. #26011685C2semidirect
12. C4⋊C8×D173C2semidirect
13. #26011486C2semidirect
14. #26011484C2semidirect
15. #464×C173C2semidirect
64. #14583827D4semidirect
189. Dic173#135semidirect
197. C173#3035semidirect
204. #464D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497022
C2×#191497002
#260120402
C4×#145838272
#260117132
#260117072
#260117032
#260116992
#260116952
#260116892
#260116852
C4⋊C8×D1732
#260114862
#260114842
#464×C1732
#3035173