Finite Group Databasebeta

← #36365302

Subgroups — #36365302

Order 22144.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149699C211yesmaximal, characteristic, complemented
2C2×Q16×D173C211yesmaximal, characteristic, complemented
3#26012151C211yesmaximal, characteristic, complemented
4#26012148C211yesmaximal, characteristic, complemented
5#26011872C211yesmaximal, characteristic, complemented
6#26011730C211yesmaximal, characteristic, complemented
7#26011726C211yesmaximal, characteristic, complemented
8#26011715C211yesmaximal, characteristic, complemented
9#26011712C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011631C211yesmaximal, characteristic, complemented
12#26011628C211yesmaximal, characteristic, complemented
13#26011583C211yesmaximal, characteristic, complemented
14#26011554C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19#19149699—24no—
20C2×Dic692C2211yescharacteristic
21C2×#14583816C2211yescharacteristic
22C2×#14583816C2211yescharacteristic
23Q16×D173—24no—
24C2×#14583815C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583813C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C22×#13063436C2211yescharacteristic
29C2×C8×D173C2211yescharacteristic
30#19149668C2211yescharacteristic
31#19149666—21no—
32#19149666C2211yescharacteristic
33#19149665—21no—
34#19149658—21no—
35#19149658C2211yescharacteristic
36#19149657—21no—
37#19149654C2211yescharacteristic
38#19149652—21no—
39#19149650—21no—
40#19149650C2211yescharacteristic
41Q8×Dic173C2211yescharacteristic
42#19149644C2211yescharacteristic
43#19149641—21no—
44C4×C22⋊D173—21no—
45C4×C22⋊D173C2211yescharacteristic
46#19149612C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149603C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149601C2211yescharacteristic
52#19149591C2211yescharacteristic
53C8×Dic173C2211yescharacteristic
54C2×#41×C173C2211yescharacteristic
55C2×Q16×C173C2211yescharacteristic
56C2×SD16×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic
60C22⋊C8×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—41no—
63C2×Dic346—21no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67Q8×D173—22no—
68Q8×D173—22no—
69Q8×D173—41no—
70Q8×D173—41no—
71C2×D692—21no—
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173D411yescharacteristic
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173C2311yescharacteristic
80Dic692—22no—
81#14583816—22no—
82#14583816D414yescomplemented
83#14583815—41no—
84#14583814—22no—
85#14583813—22no—
86#14583811—22no—
87C2×#13063436—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C8×D173—22no—
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93#14583805—21no—
94#14583805—21no—
95#14583805D411yescharacteristic
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99C22⋊Dic173—21no—
100C22⋊Dic173D411yescharacteristic
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803D411yescharacteristic
105C4×Dic173—21no—
106C4×Dic173—21no—
107C4×Dic173C2311yescharacteristic
108#41×C173—22no—
109#41×C173—41no—
110C2×Q8×C173C2311yescharacteristic
111C2×Q8×C173C2311yescharacteristic
112C2×D4×C173—21no—
113C2×D4×C173C2311yescharacteristic
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116Q16×C173—22no—
117SD16×C173—41no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173—21no—
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—42no—
129Dic346—42no—
130Dic346C2×D412yes—
131D692—21no—
132D692—21no—
133D692—42no—
134C22⋊D173—22no—
135C22⋊D173—42no—
136C22⋊D173—42no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C4×D173—21no—
144C4×D173—21no—
145C4×D173—42no—
146C4×D173—42no—
147C4×D173—42no—
148#13063436—21no—
149#13063436C2×D412yes—
150C23×C173—21no—
151C23×C173C2×D411yescharacteristic
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173—21no—
155Q8×C173C2×D412yes—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—42no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—42no—
165C2×C4×C173C2×D411yescharacteristic
166C2×C4×C173C2411yescharacteristic
167C8×C173—21no—
168C8×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173C22×D411yescharacteristic
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173#13512yescomplemented
182C4×C173—21no—
183C4×C173—22no—
184C4×C173—41no—
185C4×C173—41no—
186C4×C173C22×D411yescharacteristic
187C4×C173H(5,𝔽2)11yescharacteristic
188Dic173—21no—
189Dic173—22no—
190Dic173—21no—
191Dic173—21no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—81no—
196C2×C173D4211yescharacteristic
197C2×C173C2×#13511yescharacteristic
198C2×C173#57811yescharacteristic
199D173—22no—
200D173—81no—
201C173#328511yescharacteristic, complemented, Sylow
202#3285—1731nomaximal, Sylow
203C2×#135—1731no—
204C22×Q16—1731no—
205#548—1731no—
206#548—1731no—
207#495—1731no—
208#467—1731no—
209#466—1731no—
210#464—1731no—
211#464—1731no—
212#452—1731no—
213#452D17311yescharacteristic, complemented
214#448—1731no—
215#448—1731no—
216C4×Q16—1731no—
217C8×D4—1731no—
218C2×#41—1731no—
219C2×#41C2×D17311yescharacteristic
220C22×Q8—1731no—
221C22×Q8—1731no—
222#135—3464no—
223C2×Q16—3464no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×Q16—1731no—
227C2×Q16C2×D17311yescharacteristic
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16—1731no—
231C2×SD16C2×D17311yescharacteristic
232C2×D8—1731no—
233C22×C8—1731no—
234C22×C8—1731no—
235#124—3461no—
236#124—3461no—
237#124—1731no—
238#124—1731no—
239C22⋊Q8—3461no—
240C22⋊Q8—3461no—
241C22⋊Q8—1731no—
242C22⋊Q8C2×D17311yescharacteristic
243#121—3461no—
244#121—3461no—
245#121—1731no—
246#121—1731no—
247C4×Q8—1731no—
248C4×Q8—1731no—
249C4×D4—3461no—
250C4×D4—3461no—
251C4×D4—1731no—
252#107—1731no—
253C4⋊C8—1731no—
254#103—1731no—
255#103—1731no—
256#103—1731no—
257#103C2×D17311yescharacteristic
258#102—1731no—
259#102C2×D17311yescharacteristic
260C22⋊C8—1731no—
261C22⋊C8C2×D17311yescharacteristic
262C4×C8—1731no—
263#41—22no—
264#41—41no—
265#41—3462no—
266#41—6921no—
267C2×Q8—3461no—
268C2×Q8—3461no—
269C2×Q8—3462no—
270C2×Q8—3462no—
271C2×Q8—1731no—
272C2×Q8—1731no—
273C2×Q8—6921no—
274C2×Q8—6921no—
275C2×Q8C22×D17311yescharacteristic
276C2×Q8C22×D17311yescharacteristic
277C2×D4—21no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—3461no—
281C2×D4—1731no—
282C2×D4—1731no—
283C2×D4C22×D17311yescharacteristic
284C22×C4—21no—
285C22×C4—3461no—
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—1731no—
289C22×C4C22×D17311yescharacteristic
290Q16—22no—
291Q16—3462no—
292Q16—3462no—
293Q16—1734no—
294SD16—41no—
295SD16—3462no—
296SD16—3462no—
297SD16—6921no—
298D8—3462no—
299C2×C8—3462no—
300C2×C8—3462no—
301C2×C8—1731no—
302C2×C8—1731no—
303C2×C8C22×D17311yescharacteristic
304C2×C8C22×D17311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4C22×D17311yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—1731no—
321C22⋊C4—1731no—
322C42—3461no—
323C42—3461no—
324C42—1731no—
325C23—21no—
326C23—3461no—
327C23—1731no—
328C23D4×D17311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—3461no—
333Q8—3461no—
334Q8—3461no—
335Q8—1732no—
336Q8—6922no—
337Q8—6922no—
338Q8D4×D17312yes—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—3462no—
343D4—3461no—
344D4—3461no—
345D4—6922no—
346D4—6922no—
347D4—6922no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—1731no—
360C2×C4—1731no—
361C2×C4—1731no—
362C2×C4—6922no—
363C2×C4—6922no—
364C2×C4—6922no—
365C2×C4D4×D17311yescharacteristic
366C2×C4C23×D17311yescharacteristic
367C8—21no—
368C8—21no—
369C8—3461no—
370C8—1732no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3461no—
378C22—3461no—
379C22—1732no—
380C22—6921no—
381C22—6921no—
382C22—6921no—
383C22C2×D4×D17311yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—3461no—
389C4—3461no—
390C4—3461no—
391C4—3462no—
392C4—6921no—
393C4—6921no—
394C4C2×D4×D17311yescharacteristic
395C4#1914972411yescharacteristic
396C2—41no—
397C2—81no—
398C2—3462no—
399C2—13841no—
400C2#2601212311yescharacteristic
401C2#135×D17311yescharacteristic
402C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
hypercenter#452D173
hypocenterC173#3285
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#19149699C2semidirect
2. C2×Q16×D173C2semidirect
3. #26012151C2semidirect
4. #26012148C2semidirect
5. #26011872C2semidirect
6. #26011730C2semidirect
7. #26011726C2semidirect
8. #26011715C2semidirect
9. #26011712C2semidirect
10. #26011655C2semidirect
11. #26011631C2semidirect
12. #26011628C2semidirect
13. #26011583C2semidirect
14. #26011554C2semidirect
15. #452×C173C2semidirect
82. #14583816D4semidirect
181. C2×D173#135semidirect
201. C173#3285semidirect
213. #452D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496992
C2×Q16×D1732
#260121512
#260121482
#260118722
#260117302
#260117262
#260117152
#260117122
#260116552
#260116312
#260116282
#260115832
#260115542
#452×C1732
#3285173