Finite Group Databasebeta

← #49130578

Subgroups — #49130578

Order 36224.

712 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#22583657C211yesmaximal, characteristic, complemented
2C2×#22583643C211yesmaximal, characteristic, complemented
3C2×#22583630C211yesmaximal, characteristic, complemented
4#31827046C211yesmaximal, characteristic, complemented
5#31826628C212yesmaximal, complemented
6#31826622C212yesmaximal, complemented
7#31826611C212yesmaximal, complemented
8#31826605C212yesmaximal, complemented
9C2×#22583533C211yesmaximal, characteristic, complemented
10#31826433C211yesmaximal, characteristic, complemented
11#417×C283C211yesmaximal, characteristic, complemented
12#22583657—24no—
13#22583657—24no—
14C2×#17985392—21no—
15C2×#17985392C2211yescharacteristic
16C2×#17985392C2211yescharacteristic
17C2×#17985391—22no—
18C2×#17985391C2212yes—
19C22×Dic566—21no—
20C22×Dic566C2211yescharacteristic
21C2×Q8×D283C2212yes—
22#22583643—24no—
23#22583630—24no—
24C2×Dic1132C2212yes—
25C2×#17985382C2211yescharacteristic
26C2×#17985381C2212yes—
27C2×#17985380C2211yescharacteristic
28C2×#17985379C2211yescharacteristic
29C2×#17985377C2211yescharacteristic
30C22×C283⋊C8C2211yescharacteristic
31#22583590—21no—
32#22583590C2212yes—
33#22583585—21no—
34#22583585C2212yes—
35#22583584—22no—
36#22583533C2212yes—
37#22583533C2212yes—
38(C2×D283)⋊C8C2212yes—
39(C2×D283)⋊C8C2212yes—
40C2×#41×C283C2211yescharacteristic
41C2×M16×C283C2211yescharacteristic
42C2×C4⋊C4×C283C2211yescharacteristic
43#103×C283C2212yes—
44#102×C283C2212yes—
45#17985392—28no—
46#17985392—22no—
47#17985392—22no—
48#17985392—22no—
49#17985392—22no—
50#17985391—216no—
51#17985391—24no—
52#17985391—24no—
53C2×Dic566—28no—
54C2×Dic566—22no—
55C2×Dic566—21no—
56C2×Dic566—22no—
57C2×Dic566D411yescharacteristic
58C2×Dic566D414yescomplemented
59C2×Dic566C2311yescharacteristic
60C2×Dic566C2312yes—
61Q8×D283—24no—
62Q8×D283—24no—
63C2×D1132—22no—
64C2×D1132D411yescharacteristic
65C2×D1132C2311yescharacteristic
66C2×C22⋊D283—22no—
67C2×C22⋊D283—21no—
68C2×C22⋊D283D412yes—
69C22×Dic283—22no—
70C22×Dic283—21no—
71C2×C4×D283—22no—
72C2×C4×D283—21no—
73C2×C4×D283C2312yes—
74Dic1132—24no—
75#17985382—22no—
76#17985381—24no—
77#17985380—22no—
78#17985379—22no—
79#17985377—22no—
80C2×C283⋊C8—22no—
81C2×C283⋊C8C2311yescharacteristic
82C2×C283⋊C8C2311yescharacteristic
83C4⋊Dic283—22no—
84(C2×D283)⋊C4—22no—
85(C2×D283)⋊C4—22no—
86#41×C283—22no—
87#41×C283D414yescomplemented
88C2×Q8×C283C2311yescharacteristic
89C2×D4×C283—21no—
90C2×D4×C283C2311yescharacteristic
91C22×C4×C283—21no—
92C22×C4×C283—21no—
93C22×C4×C283C2311yescharacteristic
94M16×C283—22no—
95C2×C8×C283C2312yes—
96C4⋊C4×C283—21no—
97C4⋊C4×C283C2312yes—
98C22×D283—21no—
99C22×D283C2×D412yes—
100Dic566—28no—
101Dic566—22no—
102Dic566—24no—
103Dic566—22no—
104Dic566—21no—
105Dic566—21no—
106Dic566C2×D414yes—
107D1132—22no—
108D1132—21no—
109D1132—21no—
110D1132—82no—
111C22⋊D283—28no—
112C22⋊D283—24no—
113C22⋊D283—42no—
114C2×Dic283—28no—
115C2×Dic283—24no—
116C2×Dic283—22no—
117C2×Dic283—21no—
118C2×Dic283C2×D412yes—
119C4×D283—28no—
120C4×D283—22no—
121C4×D283—22no—
122C4×D283—42no—
123C283⋊C8—21no—
124C283⋊C8—21no—
125C23×C283—21no—
126C23×C283C2×D411yescharacteristic
127Q8×C283—21no—
128Q8×C283C2×D412yes—
129D4×C283—24no—
130D4×C283—21no—
131D4×C283C2×D412yes—
132C2×C4×C283—21no—
133C2×C4×C283—24no—
134C2×C4×C283—22no—
135C2×C4×C283—81no—
136C2×C4×C283C2×D411yescharacteristic
137C2×C4×C283C2×D412yes—
138C2×C4×C283C2×D412yes—
139C2×C4×C283C2411yescharacteristic
140C8×C283—22no—
141C22×C283—21no—
142C22×C283—21no—
143C22×C283—22no—
144C22×C283—41no—
145C22×C283—41no—
146C22×C283#12012yes—
147C22×C283C22×D411yescharacteristic
148C2×D283—22no—
149C2×D283—22no—
150C2×D283—22no—
151C2×D283—41no—
152C2×D283—41no—
153C2×D283—41no—
154C4×C283—22no—
155C4×C283—42no—
156C4×C283#12012yes—
157C4×C283C22×D411yescharacteristic
158C4×C283C22×D411yescharacteristic
159Dic283—24no—
160Dic283—22no—
161Dic283—41no—
162C2×C283—22no—
163C2×C283—42no—
164C2×C283C2×#12011yescharacteristic
165C2×C283#57711yescharacteristic
166C2×C283#57811yescharacteristic
167D283—42no—
168D283—81no—
169C283#301311yescharacteristic, complemented, Sylow
170#3013—2831nomaximal, Sylow
171C2×2−1+4—2831no—
172C2×#137—2831no—
173C2×#135—2831no—
174#537—2831no—
175#452—2832no—
176#451—2832no—
177#449—2832no—
178#448—2832no—
179#417D28311yescharacteristic, complemented
180C2×#103—2831no—
181#408—2831no—
1822−1+4—5664no—
1832−1+4—5664no—
184C2×#41—2831no—
185C2×#41—2832no—
186C2×#41—2831no—
187C2×#41—5661no—
188C2×#41—5662no—
189C2×#41C2×D28311yescharacteristic
190C22×Q8—2832no—
191C22×Q8—2831no—
192C22×Q8—5661no—
193#137—5664no—
194#135—5664no—
195C2×Q16—2831no—
196C2×Q16—2832no—
197C2×SD16—2831no—
198C2×SD16—2831no—
199C2×SD16—2832no—
200C2×D8—2831no—
201C2×M16C2×D28311yescharacteristic
202C22×C8—2831no—
203C22⋊Q8—2832no—
204C22⋊Q8—5661no—
205#121—2832no—
206#121—5661no—
207#121—5662no—
208C2×C4⋊C4C2×D28311yescharacteristic
209#103—2832no—
210#103—2832no—
211#103C2×D28312yes—
212#102C2×D28312yes—
213C22⋊C8—2832no—
214C22⋊C8—2832no—
215#41—22no—
216#41—5668no—
217#41—5664no—
218#41—56616no—
219#41—5664no—
220#41—5662no—
221#41—5662no—
222#41—5662no—
223#41—5662no—
224#41D113214yescomplemented
225C2×Q8—2831no—
226C2×Q8—2834no—
227C2×Q8—2832no—
228C2×Q8—2831no—
229C2×Q8—5662no—
230C2×Q8—5668no—
231C2×Q8—5664no—
232C2×Q8—5664no—
233C2×Q8—5661no—
234C2×Q8—5662no—
235C2×Q8C22×D28311yescharacteristic
236C2×D4—21no—
237C2×D4—2832no—
238C2×D4—2831no—
239C2×D4—2831no—
240C2×D4—5662no—
241C2×D4—5662no—
242C2×D4—5661no—
243C2×D4C22×D28311yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—2832no—
247C22×C4—5662no—
248C22×C4—5662no—
249C22×C4—5661no—
250C22×C4—5661no—
251C22×C4C22×D28311yescharacteristic
252Q16—5664no—
253Q16—5662no—
254SD16—5664no—
255SD16—5662no—
256SD16—5662no—
257D8—5662no—
258M16—22no—
259C2×C8—2831no—
260C2×C8—2831no—
261C2×C8—5662no—
262C2×C8C22×D28312yes—
263C4⋊C4—21no—
264C4⋊C4—5662no—
265C4⋊C4C22×D28312yes—
266C22⋊C4—5662no—
267C22⋊C4—5662no—
268C23—21no—
269C23—2832no—
270C23—5661no—
271C23D4×D28311yescharacteristic
272Q8—21no—
273Q8—2834no—
274Q8—5668no—
275Q8—5662no—
276Q8—5664no—
277Q8—5662no—
278Q8—5661no—
279Q8—5661no—
280Q8C2×D113212yes—
281D4—24no—
282D4—21no—
283D4—5668no—
284D4—5664no—
285D4—5662no—
286D4—5661no—
287D4—5661no—
288D4—11322no—
289D4—22642no—
290D4C2×D113212yes—
291C2×C4—21no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—81no—
295C2×C4—2832no—
296C2×C4—5668no—
297C2×C4—5668no—
298C2×C4—5664no—
299C2×C4—5662no—
300C2×C4—5662no—
301C2×C4—5662no—
302C2×C4—5661no—
303C2×C4—11322no—
304C2×C4D4×D28311yescharacteristic
305C2×C4D4×D28312yes—
306C2×C4C2×D113212yes—
307C2×C4C23×D28311yescharacteristic
308C8—22no—
309C8—5661no—
310C8—5661no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—41no—
315C22—41no—
316C22—5662no—
317C22—5662no—
318C22—5662no—
319C22—11321no—
320C22—11321no—
321C22—11321no—
322C22#2258358012yes—
323C22C2×D4×D28311yescharacteristic
324C4—22no—
325C4—42no—
326C4—5662no—
327C4—5664no—
328C4—11321no—
329C4#2258358012yes—
330C4C2×D4×D28311yescharacteristic
331C4C22×D113211yescharacteristic
332C2—22no—
333C2—42no—
334C2—11322no—
335C2—22641no—
336C2C2×#2258358011yescharacteristic
337C2#3182727511yescharacteristic
338C2#3182728111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
hypercenter#417D283
hypocenterC283#3013
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22583657C2semidirect
2. C2×#22583643C2semidirect
3. C2×#22583630C2semidirect
4. #31827046C2semidirect
5. #31826628C2semidirect
6. #31826622C2semidirect
7. #31826611C2semidirect
8. #31826605C2semidirect
9. C2×#22583533C2semidirect
10. #31826433C2semidirect
11. #417×C283C2semidirect
58. C2×Dic566D4semidirect
87. #41×C283D4semidirect
169. C283#3013semidirect
179. #417D283semidirect
224. #41D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836572
C2×#225836432
C2×#225836302
#318270462
#318266282
#318266222
#318266112
#318266052
C2×#225835332
#318264332
#417×C2832
#3013283