Finite Group Databasebeta

← #36363231

Subgroups — #36363231

Order 22144.

484 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149700C211yesmaximal, characteristic, complemented
2C2×#19149687C211yesmaximal, characteristic, complemented
3#26012071C211yesmaximal, characteristic, complemented
4#26012070C211yesmaximal, characteristic, complemented
5#26011738C211yesmaximal, characteristic, complemented
6#26011737C211yesmaximal, characteristic, complemented
7#26011734C211yesmaximal, characteristic, complemented
8#26011731C211yesmaximal, characteristic, complemented
9#26011730C211yesmaximal, characteristic, complemented
10#26011726C211yesmaximal, characteristic, complemented
11#26011725C211yesmaximal, characteristic, complemented
12#107×D173C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14#26011489C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583825C2211yescharacteristic
19#19149700—24no—
20C2×#14583816C2211yescharacteristic
21C2×#14583815C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23#19149687—24no—
24C2×#14583809C2211yescharacteristic
25C2×#14583809C2211yescharacteristic
26C22×#13063436C2211yescharacteristic
27C2×C8×D173C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149659C2211yescharacteristic
30#19149658C2211yescharacteristic
31#19149657—21no—
32#19149656—21no—
33#19149655C2211yescharacteristic
34#19149654—21no—
35#19149654C2211yescharacteristic
36#19149653C2211yescharacteristic
37#19149652—21no—
38#19149652—21no—
39#19149652—21no—
40#19149651—21no—
41#19149650—21no—
42#19149650C2211yescharacteristic
43C4⋊C4×D173C2211yescharacteristic
44C4⋊C4×D173C2211yescharacteristic
45#19149613C2211yescharacteristic
46#19149612C2211yescharacteristic
47#19149612C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149603C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149601C2211yescharacteristic
52#19149600C2211yescharacteristic
53#19149599C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C2×M16×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#107×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—22no—
63#14583826—22no—
64#14583826—22no—
65#14583825—22no—
66#14583825—22no—
67C2×Dic346D411yescharacteristic
68C2×Dic346C2311yescharacteristic
69C2×D692—21no—
70C2×D692—21no—
71C2×D692D411yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173D411yescharacteristic
77C2×C22⋊D173D411yescharacteristic
78C22×Dic173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173C2311yescharacteristic
83C2×C4×D173C2311yescharacteristic
84#14583816—22no—
85#14583815—22no—
86#14583811—22no—
87#14583809—22no—
88#14583809—22no—
89C2×#13063436—22no—
90C2×#13063436C2311yescharacteristic
91C2×#13063436C2311yescharacteristic
92C8×D173—22no—
93C8×D173D414yescomplemented
94C4⋊Dic173—21no—
95C4⋊Dic173C2311yescharacteristic
96C4⋊Dic173C2311yescharacteristic
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173—21no—
109C2×D4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111SD16×C173—22no—
112M16×C173—22no—
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C4⋊C4×C173—21no—
116C4⋊C4×C173C2311yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22×D173—21no—
121C22×D173C2×D411yescharacteristic
122C22×D173C2×D411yescharacteristic
123Dic346—22no—
124Dic346—21no—
125Dic346—21no—
126D692—22no—
127D692—21no—
128D692—21no—
129D692—42no—
130D692—81no—
131C22⋊D173—22no—
132C22⋊D173—22no—
133C22⋊D173—42no—
134C22⋊D173—81no—
135C22⋊D173—81no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C4×D173—21no—
143C4×D173—21no—
144C4×D173—42no—
145C4×D173—81no—
146C4×D173—81no—
147C4×D173C2×D412yes—
148C4×D173#4112yes—
149#13063436—21no—
150#13063436C2×D412yes—
151C23×C173—21no—
152C23×C173C2×D411yescharacteristic
153Q8×C173—21no—
154Q8×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—81no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2411yescharacteristic
165C8×C173—21no—
166C8×C173C2×D412yes—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173C22×D411yescharacteristic
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173#12112yes—
183C4×C173—21no—
184C4×C173—41no—
185C4×C173—41no—
186C4×C173—41no—
187C4×C173C22×D411yescharacteristic
188C4×C173C2×#4111yescharacteristic
189Dic173—21no—
190Dic173—41no—
191Dic173—41no—
192Dic173—41no—
193Dic173#12112yes—
194C2×C173—41no—
195C2×C173—81no—
196C2×C173C2×#12111yescharacteristic
197C2×C173#57611yescharacteristic
198C2×C173#57811yescharacteristic
199D173—22no—
200D173—41no—
201D173—81no—
202C173#306111yescharacteristic, complemented, Sylow
203#3061—1731nomaximal, Sylow
204C2×#135—1731no—
205C2×#131—1731no—
206#537—1731no—
207#537—1731no—
208#468—1731no—
209#468—1731no—
210#468—1731no—
211#468D17311yescharacteristic, complemented
212#467—1731no—
213#467—1731no—
214#466—1731no—
215#466—1731no—
216C2×#107—1731no—
217#417—1731no—
218#417—1731no—
219C2×#41—1731no—
220C2×#41—1731no—
221C2×#41—1731no—
222#135—3464no—
223C2×Q16—1731no—
224C2×SD16—1731no—
225C2×SD16C2×D17311yescharacteristic
226C2×D8—1731no—
227#131—3464no—
228C2×M16—1731no—
229C2×M16—1731no—
230C2×M16C2×D17311yescharacteristic
231C22×C8—1731no—
232C22×C8—1731no—
233C22×C8—1731no—
234C22⋊Q8—3461no—
235C22⋊Q8—3461no—
236C22⋊Q8—1731no—
237C22⋊Q8—1731no—
238C22⋊Q8—1731no—
239C22⋊Q8C2×D17311yescharacteristic
240#121—3461no—
241#121—3461no—
242#121—3461no—
243#121—3461no—
244#121—3461no—
245#121—3461no—
246#121—1731no—
247#121—1731no—
248#121—1731no—
249#121C2×D17311yescharacteristic
250C2×C4⋊C4—1731no—
251C2×C4⋊C4—1731no—
252#107—1731no—
253#107—1731no—
254#107—1731no—
255#107C2×D17311yescharacteristic
256#103—1731no—
257#103—1731no—
258#103—1731no—
259#103C2×D17311yescharacteristic
260#102—1731no—
261#102—1731no—
262#102—1731no—
263#102C2×D17311yescharacteristic
264#41—3462no—
265#41—3462no—
266#41—3462no—
267#41—3462no—
268#41—3462no—
269#41—3462no—
270C2×Q8—1731no—
271C2×Q8—1731no—
272C2×Q8C22×D17311yescharacteristic
273C2×D4—21no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—1731no—
280C2×D4—1731no—
281C2×D4—1731no—
282C2×D4—1731no—
283C2×D4C22×D17311yescharacteristic
284C22×C4—3461no—
285C22×C4—3461no—
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—1731no—
289C22×C4—1731no—
290C22×C4C22×D17311yescharacteristic
291Q16—3462no—
292SD16—22no—
293SD16—3462no—
294D8—3462no—
295M16—22no—
296M16—3462no—
297M16—3462no—
298C2×C8—3462no—
299C2×C8—3462no—
300C2×C8—1731no—
301C2×C8—1731no—
302C2×C8—1734no—
303C2×C8C22×D17311yescharacteristic
304C2×C8C22×D17311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—3461no—
310C4⋊C4—3461no—
311C4⋊C4—1731no—
312C4⋊C4—1731no—
313C4⋊C4C22×D17311yescharacteristic
314C4⋊C4C22×D17311yescharacteristic
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C23—21no—
324C23—3461no—
325C23—1731no—
326C23—1731no—
327C23D4×D17311yescharacteristic
328Q8—21no—
329Q8—21no—
330Q8—3462no—
331Q8—3461no—
332Q8—3461no—
333D4—21no—
334D4—21no—
335D4—81no—
336D4—3462no—
337D4—3462no—
338D4—3462no—
339D4—3461no—
340D4—3461no—
341D4—13841no—
342D4—13841no—
343D4—13841no—
344D4—6922no—
345D4—6922no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—13841no—
357C2×C4—13841no—
358C2×C4—1731no—
359C2×C4—1731no—
360C2×C4—1732no—
361C2×C4—1732no—
362C2×C4—6922no—
363C2×C4—6922no—
364C2×C4D4×D17311yescharacteristic
365C2×C4C23×D17311yescharacteristic
366C8—21no—
367C8—3461no—
368C8—1732no—
369C8D4×D17312yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—3461no—
377C22—3461no—
378C22—3461no—
379C22—3461no—
380C22—3461no—
381C22—1732no—
382C22—6921no—
383C22—6921no—
384C22—6921no—
385C22C2×D4×D17311yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—3461no—
391C4—1732no—
392C4—6921no—
393C4—6921no—
394C4—6921no—
395C4C2×D4×D17311yescharacteristic
396C4#41×D17311yescharacteristic
397C2—41no—
398C2—81no—
399C2—3462no—
400C2—13841no—
401C2—6921no—
402C2#121×D17311yescharacteristic
403C2#2601229511yescharacteristic
404C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3061
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19149700C2semidirect
2. C2×#19149687C2semidirect
3. #26012071C2semidirect
4. #26012070C2semidirect
5. #26011738C2semidirect
6. #26011737C2semidirect
7. #26011734C2semidirect
8. #26011731C2semidirect
9. #26011730C2semidirect
10. #26011726C2semidirect
11. #26011725C2semidirect
12. #107×D173C2semidirect
13. #26011491C2semidirect
14. #26011489C2semidirect
15. #468×C173C2semidirect
93. C8×D173D4semidirect
202. C173#3061semidirect
211. #468D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497002
C2×#191496872
#260120712
#260120702
#260117382
#260117372
#260117342
#260117312
#260117302
#260117262
#260117252
#107×D1732
#260114912
#260114892
#468×C1732
#3061173