Finite Group Databasebeta

← #36363033

Subgroups — #36363033

Order 22144.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C2×#19149708C211yesmaximal, characteristic, complemented
3#26012040C211yesmaximal, characteristic
4C4×Q8×D173C211yesmaximal, characteristic, complemented
5#26011714C211yesmaximal, characteristic, complemented
6#26011713C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011700C211yesmaximal, characteristic, complemented
9#26011699C211yesmaximal, characteristic, complemented
10#26011690C211yesmaximal, characteristic, complemented
11#26011689C211yesmaximal, characteristic, complemented
12#26011513C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14#26011489C211yesmaximal, characteristic, complemented
15#464×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19#19149711—24no—
20#19149708—24no—
21C2×Dic692C2211yescharacteristic
22C2×#14583816C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583815C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583813C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×#14583809C2211yescharacteristic
29#19149677C2211yescharacteristic
30#19149675C2211yescharacteristic
31#19149667C2211yescharacteristic
32#19149652—21no—
33#19149652—21no—
34Q8×Dic173C2211yescharacteristic
35C4×Dic346—21no—
36C4×Dic346C2211yescharacteristic
37#19149644—21no—
38#19149644C2211yescharacteristic
39#19149639—21no—
40#19149639—21no—
41C4⋊C4×D173—21no—
42C4⋊C4×D173C2211yescharacteristic
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#14583825—22no—
64#14583825—22no—
65C2×Dic346—21no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68Q8×D173—22no—
69Q8×D173D414yescomplemented
70C2×D692—21no—
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×D173D411yescharacteristic
80C2×C4×D173D411yescharacteristic
81C2×C4×D173C2311yescharacteristic
82Dic692—22no—
83#14583816—22no—
84#14583816—22no—
85#14583815—22no—
86#14583814—22no—
87#14583813—22no—
88#14583809—22no—
89#14583809—22no—
90C2×#13063436C2311yescharacteristic
91C2×#13063436C2311yescharacteristic
92C4⋊Dic173—21no—
93C4⋊Dic173C2311yescharacteristic
94#14583805—21no—
95#14583805—21no—
96#14583805—21no—
97#14583805—21no—
98#14583805—21no—
99#14583803—21no—
100#14583803—21no—
101C4×Dic173—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×D173C2×D411yescharacteristic
122Dic346—21no—
123Dic346—24no—
124Dic346—21no—
125Dic346—21no—
126Dic346C2×D412yes—
127D692—21no—
128D692—21no—
129D692—42no—
130C22⋊D173—42no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C2×Dic173—21no—
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×D173C2×D412yes—
146C4×D173#4112yes—
147#13063436—21no—
148#13063436—21no—
149C23×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173C2×D412yes—
154D4×C173—21no—
155D4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2411yescharacteristic
162C8×C173—21no—
163C8×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173#12112yes—
174C4×C173—22no—
175C4×C173—21no—
176C4×C173—21no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173C22×D411yescharacteristic
180C4×C173C2×#4111yescharacteristic
181Dic173—22no—
182Dic173—21no—
183Dic173—41no—
184Dic173—41no—
185Dic173#12112yes—
186Dic173#13712yescomplemented
187C2×C173—81no—
188C2×C173C2×#12111yescharacteristic
189C2×C173C2×#13711yescharacteristic
190C2×C173#57511yescharacteristic
191D173—22no—
192D173—81no—
193C173#303711yescharacteristic, complemented, Sylow
194#3037—1731nomaximal, Sylow
195C2×#137—1731no—
196C2×#137—1731no—
197#532—1731no—
198C2×C4×Q8—1731no—
199#464—1731no—
200#464—1731no—
201#464—1731no—
202#464D17311yescharacteristic, complemented
203#462—1731no—
204#462—1731no—
205#460—1731no—
206#460—1731no—
207#423—1731no—
208#417—1731no—
209#417—1731no—
210C2×#41—1731no—
211C2×#41—1731no—
212C22×Q8—1731no—
213#137—3464no—
214#137—3464no—
215C2×Q16—1731no—
216C2×Q16—1731no—
217C2×Q16—1731no—
218C2×Q16C2×D17311yescharacteristic
219C2×SD16—1731no—
220C2×SD16—1731no—
221C2×SD16—1731no—
222C2×SD16C2×D17311yescharacteristic
223C2×M16—1731no—
224C2×M16—1731no—
225C4⋊Q8—1731no—
226#127—1731no—
227#124—1731no—
228#124C2×D17311yescharacteristic
229#121—3461no—
230#121—3461no—
231C4×Q8—3461no—
232C4×Q8—3461no—
233C4×Q8—1731no—
234C4×Q8—1731no—
235C4×Q8—1731no—
236C4×Q8C2×D17311yescharacteristic
237C4×D4—3461no—
238C4×D4—3461no—
239C2×C4⋊C4—3461no—
240C2×C4⋊C4—1731no—
241C2×C42—3461no—
242C2×C42—1731no—
243C4⋊C8—1731no—
244C4⋊C8—1731no—
245C4⋊C8—1731no—
246C4⋊C8C2×D17311yescharacteristic
247#103—1731no—
248#103—1731no—
249#103—1731no—
250#103C2×D17311yescharacteristic
251#102—1731no—
252#102—1731no—
253#102—1731no—
254#102C2×D17311yescharacteristic
255#41—3462no—
256#41—3462no—
257#41—3462no—
258#41—3462no—
259C2×Q8—3461no—
260C2×Q8—3462no—
261C2×Q8—1734no—
262C2×Q8—1731no—
263C2×Q8—1731no—
264C2×Q8C22×D17311yescharacteristic
265C2×Q8C22×D17311yescharacteristic
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—1731no—
271C2×D4C22×D17311yescharacteristic
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4—3461no—
276C22×C4—1731no—
277C22×C4—1731no—
278C22×C4—1731no—
279Q16—22no—
280Q16—3462no—
281Q16—3462no—
282Q16—3462no—
283SD16—22no—
284SD16—3462no—
285SD16—3462no—
286SD16—3462no—
287M16—3462no—
288M16—3462no—
289C2×C8—1731no—
290C2×C8—1731no—
291C2×C8C22×D17311yescharacteristic
292C2×C8C22×D17311yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—3461no—
297C4⋊C4—3461no—
298C4⋊C4—3461no—
299C4⋊C4—3461no—
300C4⋊C4—1731no—
301C4⋊C4C22×D17311yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C42—21no—
307C42—3461no—
308C42—3461no—
309C42—3461no—
310C42—1731no—
311C42—1731no—
312C42—1731no—
313C42C22×D17311yescharacteristic
314C23—21no—
315C23—3461no—
316C23—1731no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—3464no—
321Q8—3461no—
322Q8—3461no—
323Q8—3461no—
324Q8—1732no—
325Q8D4×D17312yes—
326D4—21no—
327D4—21no—
328D4—3461no—
329D4—3461no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—3464no—
338C2×C4—3462no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—1731no—
343C2×C4—1731no—
344C2×C4—1731no—
345C2×C4—1732no—
346C2×C4—1732no—
347C2×C4—6921no—
348C2×C4—6922no—
349C2×C4—6922no—
350C2×C4—6922no—
351C2×C4D4×D17311yescharacteristic
352C2×C4D4×D17311yescharacteristic
353C2×C4C23×D17311yescharacteristic
354C8—21no—
355C8—21no—
356C8—3461no—
357C8—3461no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3461no—
362C22—3461no—
363C22—1732no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22C2×D4×D17311yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—3462no—
374C4—3461no—
375C4—1732no—
376C4—1732no—
377C4—6921no—
378C4—6921no—
379C4C2×D4×D17311yescharacteristic
380C4#41×D17311yescharacteristic
381C2—81no—
382C2—3462no—
383C2—13841no—
384C2#121×D17311yescharacteristic
385C2#137×D17311yescharacteristic
386C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
prime radical#26012040C2
hypercenter#464D173
hypocenterC173#3037
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×#19149711C2semidirect
2. C2×#19149708C2semidirect
4. C4×Q8×D173C2semidirect
5. #26011714C2semidirect
6. #26011713C2semidirect
7. #26011710C2semidirect
8. #26011700C2semidirect
9. #26011699C2semidirect
10. #26011690C2semidirect
11. #26011689C2semidirect
12. #26011513C2semidirect
13. #26011491C2semidirect
14. #26011489C2semidirect
15. #464×C173C2semidirect
69. Q8×D173D4semidirect
186. Dic173#137semidirect
193. C173#3037semidirect
202. #464D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×#191497082
#260120402
C4×Q8×D1732
#260117142
#260117132
#260117102
#260117002
#260116992
#260116902
#260116892
#260115132
#260114912
#260114892
#464×C1732
#3037173