Finite Group Databasebeta

← #36363099

Subgroups — #36363099

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×#19149706C211yesmaximal, characteristic, complemented
2C2×#19149700C211yesmaximal, characteristic, complemented
3#26012054C211yesmaximal, characteristic, complemented
4#26011973C211yesmaximal, characteristic, complemented
5#26011713C211yesmaximal, characteristic, complemented
6#26011711C211yesmaximal, characteristic, complemented
7#26011707C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011689C211yesmaximal, characteristic, complemented
10#26011685C211yesmaximal, characteristic, complemented
11#26011683C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#460×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149706—24no—
20#19149700—24no—
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583815C2211yescharacteristic, complemented
23C2×#14583813C2211yescharacteristic
24C2×D1384C2211yescharacteristic
25C2×#14583811C2211yescharacteristic
26C2×#14583811C2211yescharacteristic, complemented
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149675C2211yescharacteristic
30#19149667C2211yescharacteristic
31#19149666C2211yescharacteristic
32#19149651—21no—
33#19149651—21no—
34#19149647—21no—
35#19149647—21no—
36Q8×Dic173C2211yescharacteristic, complemented
37C4×D692—21no—
38C4×D692C2211yescharacteristic, complemented
39#19149641—21no—
40#19149641C2211yescharacteristic, complemented
41#19149636—21no—
42#19149636C2211yescharacteristic
43C4⋊C4×D173—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×SD16×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic, complemented
56#127×C173C2211yescharacteristic
57C4×Q8×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827D414yescomplemented
64#14583825—22no—
65#14583825—22no—
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
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—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81C2×C4×D173C2311yescharacteristic
82#14583816—22no—
83#14583815—22no—
84#14583813—22no—
85D1384—22no—
86#14583811—22no—
87#14583811—22no—
88#14583809—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C8×D173—22no—
92C4⋊Dic173—21no—
93C4⋊Dic173C2311yescharacteristic
94#14583805—21no—
95#14583805D411yescharacteristic
96#14583805D411yescharacteristic
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803D411yescharacteristic
104#14583803D411yescharacteristic
105C4×Dic173—21no—
106C4×Dic173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173—21no—
109C2×D4×C173—21no—
110C2×D4×C173C2311yescharacteristic
111C2×D4×C173C2311yescharacteristic
112SD16×C173—22no—
113SD16×C173—22no—
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173—21no—
117C4⋊C4×C173C2311yescharacteristic
118C42×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173—21no—
121C22×D173—21no—
122C22×D173—42no—
123C22×D173—81no—
124C22×D173—81no—
125C22×D173C2×D411yescharacteristic
126Dic346—21no—
127Dic346—21no—
128D692—24no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692C2×D412yes—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C22⋊D173—81no—
136C22⋊D173—81no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C4×D173—24no—
144C4×D173—81no—
145C4×D173—81no—
146C4×D173—81no—
147C4×D173C2×D412yes—
148C4×D173#4112yes—
149#13063436—21no—
150#13063436—21no—
151C23×C173—21no—
152C23×C173—21no—
153Q8×C173—21no—
154Q8×C173C2×D412yes—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—81no—
160D4×C173—81no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2411yescharacteristic
166C8×C173—21no—
167C8×C173—21no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173C22×D411yescharacteristic
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—22no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C2×D173#12112yes—
185C4×C173—22no—
186C4×C173—21no—
187C4×C173—21no—
188C4×C173—41no—
189C4×C173C22×D411yescharacteristic
190C4×C173C2×#4111yescharacteristic
191Dic173—21no—
192Dic173—21no—
193Dic173—41no—
194Dic173—41no—
195Dic173#12112yes—
196C2×C173—81no—
197C2×C173—81no—
198C2×C173C2×#12111yescharacteristic
199C2×C173#57611yescharacteristic
200C2×C173#57711yescharacteristic
201D173—22no—
202D173—42no—
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#460D17311yescharacteristic, complemented
216#459—1731no—
217#459—1731no—
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×SD16C2×D17311yescharacteristic
230C2×SD16C2×D17311yescharacteristic, complemented
231C2×D8—1731no—
232C2×D8—1731no—
233C2×D8—1731no—
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×Q8C2×D17311yescharacteristic, complemented
246C4×D4—3461no—
247C4×D4—3461no—
248C4×D4—1731no—
249C4×D4—1731no—
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×Q8C22×D17311yescharacteristic
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×D4—1731no—
284C2×D4C22×D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292Q16—3462no—
293SD16—22no—
294SD16—22no—
295SD16—3462no—
296SD16—3462no—
297D8—3462no—
298D8—3462no—
299D8—3462no—
300M16—3462no—
301C2×C8—3462no—
302C2×C8—1731no—
303C2×C8—1731no—
304C2×C8C22×D17311yescharacteristic
305C2×C8C22×D17311yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4—1731no—
312C4⋊C4C22×D17311yescharacteristic
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—1731no—
320C22⋊C4—1731no—
321C42—21no—
322C42—3461no—
323C42—1731no—
324C42C22×D17311yescharacteristic
325C23—21no—
326C23—21no—
327C23—3461no—
328C23—3461no—
329C23—13841no—
330C23—13841no—
331C23—1731no—
332C23—6922no—
333Q8—21no—
334Q8—3461no—
335Q8—3461no—
336Q8D4×D17312yes—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—81no—
342D4—81no—
343D4—3464no—
344D4—3461no—
345D4—3461no—
346D4—3461no—
347D4—13841no—
348D4—13841no—
349D4—1732no—
350D4—6922no—
351D4—6922no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—3464no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—13841no—
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—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—3461no—
380C22—3461no—
381C22—3462no—
382C22—13844no—
383C22—1732no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22—6921no—
388C22—6921no—
389C22C2×D4×D17311yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—3461no—
395C4—3461no—
396C4—1732no—
397C4—6921no—
398C4—6921no—
399C4C2×D4×D17311yescharacteristic
400C4#41×D17311yescharacteristic
401C2—81no—
402C2—81no—
403C2—3462no—
404C2—13841no—
405C2—6922no—
406C2#121×D17311yescharacteristic
407C2#2601229511yescharacteristic
408C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#460×C173C2
socleC22×C173C22×D4
hypercenter#460D173
hypocenterC173#3045
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×#19149700C2semidirect
3. #26012054C2semidirect
4. #26011973C2semidirect
5. #26011713C2semidirect
6. #26011711C2semidirect
7. #26011707C2semidirect
8. #26011705C2semidirect
9. #26011689C2semidirect
10. #26011685C2semidirect
11. #26011683C2semidirect
12. #26011516C2semidirect
13. #26011489C2semidirect
14. #102×D173C2semidirect
15. #460×C173C2semidirect
21. C2×#14583816C22semidirect
22. C2×#14583815C22semidirect
26. C2×#14583811C22semidirect
36. Q8×Dic173C22semidirect
38. C4×D692C22semidirect
40. #19149641C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149608C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149602C22semidirect
52. #19149600C22semidirect
55. C2×SD16×C173C22semidirect
57. C4×Q8×C173C22semidirect
58. C4⋊C8×C173C22semidirect
60. #102×C173C22semidirect
63. #14583827D4semidirect
204. C173#3045semidirect
215. #460D173semidirect
230. C2×SD16C2×D173semidirect
245. C4×Q8C2×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×#191497062
C2×#191497002
#260120542
#260119732
#260117132
#260117112
#260117072
#260117052
#260116892
#260116852
#260116832
#260115162
#260114892
#102×D1732
#460×C1732
#3045173