Finite Group Databasebeta

← #36363097

Subgroups — #36363097

Order 22144.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C211yesmaximal, characteristic, complemented
2C2×#19149698C211yesmaximal, characteristic, complemented
3#26012054C211yesmaximal, characteristic, complemented
4#26011970C211yesmaximal, characteristic, complemented
5#26011714C211yesmaximal, characteristic, complemented
6#26011712C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011690C211yesmaximal, characteristic, complemented
10#26011688C211yesmaximal, characteristic, complemented
11#26011684C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#459×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149703—24no—
20#19149698—24no—
21C2×Dic692C2211yescharacteristic, complemented
22C2×#14583815C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24C2×#14583814C2211yescharacteristic, complemented
25C2×#14583814C2211yescharacteristic, complemented
26C2×#14583811C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149675C2211yescharacteristic
30#19149667C2211yescharacteristic
31#19149666C2211yescharacteristic
32#19149651—21no—
33#19149651—21no—
34#19149647—21no—
35#19149647—21no—
36C4×Dic346C2211yescharacteristic, complemented
37#19149644C2211yescharacteristic, complemented
38D4×Dic173C2211yescharacteristic, complemented
39C4×C22⋊D173—21no—
40#19149639—21no—
41#19149636C2211yescharacteristic
42#19149635—21no—
43C2×#14583805—21no—
44C4⋊C4×D173C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149608C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149602C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic
51#19149600C2211yescharacteristic
52#19149600C2211yescharacteristic, complemented
53#19149599C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic, complemented
56#127×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583825—22no—
63#14583825—22no—
64#14583825—22no—
65#14583825D414yescomplemented
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74D4×D173—22no—
75D4×D173—22no—
76C22×Dic173—21no—
77C22×Dic173—21no—
78C22×Dic173—21no—
79C2×C4×D173—21no—
80C2×C4×D173C2311yescharacteristic
81Dic692—22no—
82#14583815—22no—
83#14583814—22no—
84#14583814—22no—
85#14583814—22no—
86#14583811—22no—
87#14583809—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C8×D173—22no—
91C4⋊Dic173C2311yescharacteristic
92#14583805—21no—
93#14583805—21no—
94#14583805—21no—
95#14583805D411yescharacteristic
96#14583805D411yescharacteristic
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99C22⋊Dic173—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803D411yescharacteristic
103#14583803D411yescharacteristic
104C4×Dic173—21no—
105C4×Dic173—21no—
106C4×Dic173C2311yescharacteristic
107C2×D4×C173—21no—
108C2×D4×C173—21no—
109C2×D4×C173C2311yescharacteristic
110C2×D4×C173C2311yescharacteristic
111C2×D4×C173C2311yescharacteristic
112C22×C4×C173—21no—
113D8×C173—22no—
114D8×C173—22no—
115C2×C8×C173C2311yescharacteristic
116C2×C8×C173C2311yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173—21no—
121C22×D173—42no—
122C22×D173—81no—
123C22×D173—81no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346C2×D412yes—
129D692—21no—
130D692—21no—
131C22⋊D173—24no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C22⋊D173—81no—
135C22⋊D173—81no—
136C2×Dic173—21no—
137C2×Dic173—24no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—42no—
141C2×Dic173—81no—
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C4×D173—81no—
146C4×D173C2×D412yes—
147C4×D173#4112yes—
148#13063436—21no—
149#13063436—21no—
150C23×C173—21no—
151C23×C173—21no—
152C23×C173—21no—
153D4×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—81no—
159D4×C173—81no—
160D4×C173C2×D412yes—
161C2×C4×C173—21no—
162C2×C4×C173—81no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2411yescharacteristic
166C8×C173—21no—
167C8×C173—21no—
168C22×C173—22no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173C22×D411yescharacteristic
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C2×D173#12112yes—
185C4×C173—21no—
186C4×C173—21no—
187C4×C173—41no—
188C4×C173C22×D411yescharacteristic
189C4×C173C2×#4111yescharacteristic
190Dic173—22no—
191Dic173—21no—
192Dic173—21no—
193Dic173—41no—
194Dic173—41no—
195Dic173#12112yes—
196C2×C173—42no—
197C2×C173—81no—
198C2×C173—81no—
199C2×C173C2×#12111yescharacteristic
200C2×C173#57611yescharacteristic
201C2×C173#57711yescharacteristic
202D173—22no—
203D173—81no—
204C173#304511yescharacteristic, complemented, Sylow
205#3045—1731nomaximal, Sylow
206C2×AGL(1,ℤ/8)—1731no—
207C2×#135—1731no—
208#535—1731no—
209#520—1731no—
210#464—1731no—
211#464—1731no—
212#463—1731no—
213#463—1731no—
214#460—1731no—
215#460—1731no—
216#459—1731no—
217#459D17311yescharacteristic, complemented
218#424—1731no—
219#417—1731no—
220C2×#102—1731no—
221C2×#41—1731no—
222C2×#41—1731no—
223C22×D4—1731no—
224AGL(1,ℤ/8)—3464no—
225#135—3464no—
226C2×Q16—1731no—
227C2×SD16—1731no—
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16—1731no—
231C2×D8—1731no—
232C2×D8C2×D17311yescharacteristic
233C2×D8C2×D17311yescharacteristic, complemented
234C2×M16—1731no—
235C22×C8—1731no—
236#127—1731no—
237#127C2×D17311yescharacteristic
238#124—1731no—
239#124—1731no—
240#121—3461no—
241#121—3461no—
242#120—3461no—
243#120—3461no—
244C4×Q8—1731no—
245C4×Q8—1731no—
246C4×D4—3461no—
247C4×D4—3461no—
248C4×D4—1731no—
249C4×D4C2×D17311yescharacteristic, complemented
250#117—3461no—
251#117—1731no—
252C2×C4⋊C4—3461no—
253C2×C4⋊C4—1731no—
254C4⋊C8—1731no—
255C4⋊C8—1731no—
256C4⋊C8—1731no—
257C4⋊C8C2×D17311yescharacteristic, complemented
258#103—1731no—
259#103—1731no—
260#102—1731no—
261#102—1731no—
262#102—1731no—
263#102—1731no—
264#102C2×D17311yescharacteristic
265#102C2×D17311yescharacteristic, complemented
266C24—3461no—
267#41—3462no—
268#41—3462no—
269#41—3462no—
270#41—1734no—
271C2×Q8—1731no—
272C2×Q8—1731no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3462no—
278C2×D4—3462no—
279C2×D4—3461no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—1731no—
283C2×D4C22×D17311yescharacteristic
284C2×D4C22×D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—21no—
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292Q16—3462no—
293SD16—3462no—
294SD16—3462no—
295SD16—3462no—
296SD16—3462no—
297D8—22no—
298D8—22no—
299D8—3462no—
300M16—3462no—
301C2×C8—3462no—
302C2×C8—1731no—
303C2×C8—1731no—
304C2×C8C22×D17311yescharacteristic
305C2×C8C22×D17311yescharacteristic
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4—1731no—
312C4⋊C4C22×D17311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—1731no—
320C22⋊C4—1731no—
321C42—3461no—
322C42—3461no—
323C42—1731no—
324C42C22×D17311yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—3461no—
329C23—13841no—
330C23—13841no—
331C23—1731no—
332C23—6922no—
333Q8—3461no—
334Q8—3461no—
335Q8—3461no—
336Q8—1732no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—81no—
343D4—81no—
344D4—3464no—
345D4—3461no—
346D4—3461no—
347D4—13841no—
348D4—13841no—
349D4—6922no—
350D4—6922no—
351D4D4×D17312yes—
352C2×C4—21no—
353C2×C4—81no—
354C2×C4—3461no—
355C2×C4—3464no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—13841no—
359C2×C4—13841no—
360C2×C4—1731no—
361C2×C4—1731no—
362C2×C4—1731no—
363C2×C4—1732no—
364C2×C4—1732no—
365C2×C4—6922no—
366C2×C4D4×D17311yescharacteristic
367C2×C4D4×D17311yescharacteristic
368C2×C4C23×D17311yescharacteristic
369C8—21no—
370C8—21no—
371C8—3461no—
372C8—3461no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—3461no—
383C22—3461no—
384C22—13844no—
385C22—1732no—
386C22—6921no—
387C22—6921no—
388C22—6921no—
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—3462no—
394C4—3461no—
395C4—3461no—
396C4—1732no—
397C4—6921no—
398C4—6921no—
399C4C2×D4×D17311yescharacteristic
400C4#41×D17311yescharacteristic
401C2—42no—
402C2—81no—
403C2—81no—
404C2—3462no—
405C2—13841no—
406C2#121×D17311yescharacteristic
407C2#2601229211yescharacteristic
408C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3045
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×#19149698C2semidirect
3. #26012054C2semidirect
4. #26011970C2semidirect
5. #26011714C2semidirect
6. #26011712C2semidirect
7. #26011706C2semidirect
8. #26011704C2semidirect
9. #26011690C2semidirect
10. #26011688C2semidirect
11. #26011684C2semidirect
12. #26011516C2semidirect
13. #26011489C2semidirect
14. #102×D173C2semidirect
15. #459×C173C2semidirect
21. C2×Dic692C22semidirect
24. C2×#14583814C22semidirect
25. C2×#14583814C22semidirect
36. C4×Dic346C22semidirect
37. #19149644C22semidirect
38. D4×Dic173C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149602C22semidirect
52. #19149600C22semidirect
55. C2×D8×C173C22semidirect
57. C4×D4×C173C22semidirect
58. C4⋊C8×C173C22semidirect
60. #102×C173C22semidirect
65. #14583825D4semidirect
204. C173#3045semidirect
217. #459D173semidirect
233. C2×D8C2×D173semidirect
249. C4×D4C2×D173semidirect
257. C4⋊C8C2×D173semidirect
265. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C2×#191496982
#260120542
#260119702
#260117142
#260117122
#260117062
#260117042
#260116902
#260116882
#260116842
#260115162
#260114892
#102×D1732
#459×C1732
#3045173