Finite Group Databasebeta

← #49133063

Subgroups — #49133063

Order 36224.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827280C212yesmaximal, complemented
2#31827279C211yesmaximal, characteristic, complemented
3#31826843C211yesmaximal, characteristic, complemented
4#31826740C211yesmaximal, characteristic, complemented
5#31826738C212yesmaximal, complemented
6#31826737C211yesmaximal, characteristic, complemented
7#31826648C212yesmaximal, complemented
8#31826646C212yesmaximal, complemented
9#31826582C211yesmaximal, characteristic, complemented
10#31826580C211yesmaximal, characteristic, complemented
11#578×C283C211yesmaximal, characteristic, complemented
12#22583657C2212yes—
13#22583641—22no—
14#22583641C2212yes—
15#22583640—22no—
16#22583640C2212yes—
17#22583640C2212yes—
18#22583639—22no—
19#22583630—22no—
20#22583630C2211yescharacteristic
21#22583629—21no—
22C2×Dic1132C2211yescharacteristic
23C2×#17985382—22no—
24C2×#17985382C2212yes—
25Q16×D283—21no—
26#22583617C2212yes—
27#22583616C2211yescharacteristic
28#22583608C2212yes—
29#22583546C2211yescharacteristic
30#22583545C2211yescharacteristic
31#22583537C2211yescharacteristic
32#22583537C2212yes—
33#22583536C2211yescharacteristic
34#22583529C2212yes—
35#22583528C2212yes—
36C8×Dic283C2211yescharacteristic
372−1+4×C283C2212yes—
38#137×C283—22no—
39#137×C283C2212yes—
40#135×C283—21no—
41#135×C283C2211yescharacteristic
42C2×Q16×C283—21no—
43C2×Q16×C283C2211yescharacteristic
44#131×C283C2211yescharacteristic
45#17985392—22no—
46#17985392C2311yescharacteristic
47#17985391—22no—
48#17985391—42no—
49C2×Dic566—22no—
50C2×Dic566C2312yes—
51Q8×D283—22no—
52Dic1132—21no—
53Dic1132—21no—
54#17985382—22no—
55#17985382—22no—
56#17985382—22no—
57#17985382—22no—
58#17985382—22no—
59#17985382D411yescharacteristic
60#17985380—22no—
61#17985380—22no—
62#17985380D411yescharacteristic
63#17985379—22no—
64#17985379D411yescharacteristic
65#17985377D411yescharacteristic
66#17985376—22no—
67#17985376C2311yescharacteristic
68#17985376C2312yes—
69#17985375—21no—
70C2×C283⋊C8—22no—
71C2×C283⋊C8C2311yescharacteristic
72C8×D283—21no—
73Dic283⋊C4—42no—
74C4×Dic283C2311yescharacteristic
75#41×C283—22no—
76#41×C283—22no—
77#41×C283—42no—
78#41×C283C2312yes—
79#41×C283C2311yescharacteristic
80C2×Q8×C283—22no—
81C2×Q8×C283—22no—
82C2×Q8×C283C2312yes—
83Q16×C283—22no—
84Q16×C283—21no—
85Q16×C283—21no—
86Q16×C283D411yescharacteristic
87SD16×C283—22no—
88SD16×C283D412yes—
89D8×C283D411yescharacteristic
90M16×C283—21no—
91M16×C283C2311yescharacteristic
92C2×C8×C283—21no—
93C2×C8×C283C2311yescharacteristic
94Dic566—22no—
95Dic566—22no—
96Dic566—42no—
97Dic566C2×D411yescharacteristic
98D1132C2×D411yescharacteristic
99C22⋊D283—21no—
100C22⋊D283—42no—
101C2×Dic283—21no—
102C2×Dic283—22no—
103C2×Dic283—42no—
104C4×D283—21no—
105C4×D283—42no—
106C283⋊C8—21no—
107C283⋊C8—22no—
108C283⋊C8C2×D411yescharacteristic
109C283⋊C8C2×D411yescharacteristic
110Q8×C283—22no—
111Q8×C283—22no—
112Q8×C283—42no—
113Q8×C283C2×D411yescharacteristic
114Q8×C283C2×D412yes—
115D4×C283—21no—
116D4×C283—22no—
117D4×C283—42no—
118D4×C283C2×D411yescharacteristic
119D4×C283C2×D412yes—
120C2×C4×C283—21no—
121C2×C4×C283—22no—
122C2×C4×C283—22no—
123C2×C4×C283—42no—
124C2×C4×C283—42no—
125C2×C4×C283C2411yescharacteristic
126C8×C283—21no—
127C8×C283C2×D411yescharacteristic
128C8×C283C2×D411yescharacteristic
129C22×C283—21no—
130C22×C283—22no—
131C22×C283H(5,𝔽2)11yescharacteristic
132C2×D283—21no—
133C4×C283—21no—
134C4×C283—22no—
135C4×C283—42no—
136C4×C283C22×D411yescharacteristic
137C4×C283C22×D411yescharacteristic
138Dic283—22no—
139Dic283—21no—
140Dic283—42no—
141C2×C283—21no—
142C2×C283—41no—
143C2×C283—42no—
144C2×C283D4211yescharacteristic
145D283—41no—
146C283#329311yescharacteristic, complemented, Sylow
147#3293—2831nomaximal, Sylow
148#578—2832no—
149#578—2831no—
150#578D28311yescharacteristic, complemented
151#494—2831no—
152#473—2832no—
153#473—2831no—
154#473—2831no—
155#456—2832no—
156#456—2832no—
157#443—2831no—
158#443—2831no—
1592−1+4—2832no—
1602−1+4C2×D28312yes—
161#137—22no—
162#137—2832no—
163#137—2832no—
164#137—2832no—
165#137—5662no—
166#137—5662no—
167#137—5662no—
168#137C2×D28312yes—
169#135—21no—
170#135—2831no—
171#135—5662no—
172#135—5661no—
173#135C2×D28311yescharacteristic
174C2×Q16—21no—
175C2×Q16—2832no—
176C2×Q16—2831no—
177C2×Q16—5662no—
178C2×Q16—5661no—
179C2×Q16C2×D28311yescharacteristic
180#131—2832no—
181#131—2831no—
182#131C2×D28311yescharacteristic
183C4⋊Q8—2832no—
184#108—2831no—
185#108—2831no—
186#104—2831no—
187#104—2832no—
188#104—2831no—
189#101—2832no—
190#101—2832no—
191C4×C8—2831no—
192#41—22no—
193#41—22no—
194#41—42no—
195#41—2831no—
196#41—5662no—
197#41—5662no—
198#41—11322no—
199#41C22×D28311yescharacteristic
200#41C22×D28312yes—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—2832no—
204C2×Q8—5662no—
205C2×Q8—5662no—
206C2×Q8C22×D28312yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—2831no—
211Q16—5662no—
212Q16—5662no—
213Q16—5662no—
214Q16—5662no—
215Q16—5662no—
216Q16—5661no—
217Q16—5661no—
218Q16C22⋊D28311yescharacteristic
219SD16—22no—
220SD16—2831no—
221SD16—2831no—
222SD16—5662no—
223SD16—5662no—
224SD16—5662no—
225SD16C22⋊D28312yes—
226D8—2831no—
227D8C22⋊D28311yescharacteristic
228M16—21no—
229M16—2832no—
230M16—2831no—
231M16—5662no—
232M16—5661no—
233M16C22×D28311yescharacteristic
234C2×C8—21no—
235C2×C8—2831no—
236C2×C8—5662no—
237C2×C8—5661no—
238C2×C8C22×D28311yescharacteristic
239C4⋊C4—11322no—
240C42—2831no—
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—2831no—
245Q8—5662no—
246Q8—5662no—
247Q8—11322no—
248Q8D4×D28311yescharacteristic
249Q8C2×C22⋊D28312yes—
250D4—21no—
251D4—22no—
252D4—42no—
253D4—2831no—
254D4—5661no—
255D4—11322no—
256D4D4×D28311yescharacteristic
257D4C2×C22⋊D28312yes—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—5661no—
264C2×C4—5661no—
265C2×C4—5662no—
266C2×C4—11322no—
267C2×C4—11322no—
268C2×C4C23×D28311yescharacteristic
269C8—21no—
270C8—2831no—
271C8—2831no—
272C8—5662no—
273C8—5661no—
274C8C2×C22⋊D28311yescharacteristic
275C8C2×C22⋊D28311yescharacteristic
276C22—21no—
277C22—22no—
278C22—5661no—
279C22#2258365511yescharacteristic
280C4—21no—
281C4—22no—
282C4—42no—
283C4—5662no—
284C4—5661no—
285C4—11322no—
286C4C2×D4×D28311yescharacteristic
287C4C22×C22⋊D28311yescharacteristic
288C2—21no—
289C2—41no—
290C2—42no—
291C2—11321no—
292C2D4×C22⋊D28311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#578×C283C2
socleC2×C283D42
hypercenter#578D283
hypocenterC283#3293
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #31827280C2semidirect
2. #31827279C2semidirect
3. #31826843C2semidirect
4. #31826740C2semidirect
5. #31826738C2semidirect
6. #31826737C2semidirect
7. #31826648C2semidirect
8. #31826646C2semidirect
9. #31826582C2semidirect
10. #31826580C2semidirect
11. #578×C283C2semidirect
146. C283#3293semidirect
150. #578D283semidirect

Maximal subgroups

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

subgroupindex
#318272802
#318272792
#318268432
#318267402
#318267382
#318267372
#318266482
#318266462
#318265822
#318265802
#578×C2832
#3293283