Finite Group Databasebeta

← #36448906

Subgroups — #36448906

Order 22784.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26028996C211yesmaximal, characteristic
2#26021028C211yesmaximal, characteristic, complemented
3#26019027C211yesmaximal, characteristic, complemented
4#26018703C211yesmaximal, characteristic
5#26018088C211yesmaximal, characteristic, complemented
6#26018086C211yesmaximal, characteristic, complemented
7C89×#1540C211yesmaximal, characteristic
8C2×#15668019—21no—
9#19151676—21no—
10#19151670—22no—
11D4×#10623494—21no—
12C2×#15667948—21no—
13#19151622C2211yescharacteristic
14C2×C4⋊#10623494C2211yescharacteristic
15#19151565—21no—
16C4×#13083558C2211yescharacteristic
17C2×C4×#10623494C2211yescharacteristic
18#19151363—21no—
19#19151351—21no—
20#19151351—21no—
21D4⋊#10623494—22no—
22#532×C89C2211yescharacteristic
23#431×C89C2211yescharacteristic
24C2×C4⋊C8×C89C2211yescharacteristic
25#332×C89C411yescharacteristic
26#332×C89C411yescharacteristic
27#331×C89C411yescharacteristic
28#331×C89C411yescharacteristic
29#15668019—42no—
30#15668019—42no—
31C2×#13083558—22no—
32C2×#13083558—21no—
33C2×#13083558D411yescharacteristic
34C2×#13083558D411yescharacteristic
35C22×#10623494—22no—
36C22×#10623494—21no—
37C22×#10623494—21no—
38#15667948—42no—
39C4⋊#10623494—21no—
40C4⋊#10623494—21no—
41C4⋊#10623494—21no—
42C4⋊#10623494D411yescharacteristic
43C4⋊#10623494D411yescharacteristic
44C22⋊#10623494—22no—
45C22⋊#10623494—22no—
46#15667923—21no—
47C4×#10623494—21no—
48C4×#10623494—21no—
49C4×#10623494—21no—
50C2×#41×C89—21no—
51C22×C8×C89—21no—
52C4⋊Q8×C89C2×C411yescharacteristic
53#127×C89C2×C411yescharacteristic
54#124×C89—21no—
55#121×C89—21no—
56#121×C89Q812yes—
57C4×D4×C89—21no—
58C4×D4×C89D412yes—
59C2×C42×C89C2311yescharacteristic
60C4⋊C8×C89—21no—
61C4⋊C8×C89C2×C411yescharacteristic
62C4⋊C8×C89C2×C411yescharacteristic
63C4⋊C8×C89C2×C411yescharacteristic
64C4⋊C8×C89C2×C411yescharacteristic
65C22⋊C8×C89—21no—
66C4×C8×C89—21no—
67#13083558—21no—
68#13083558—21no—
69#13083558—21no—
70#13083558—44no—
71#13083558—41no—
72#13083558—41no—
73#13083558D812yes—
74C2×#10623494—21no—
75C2×#10623494—22no—
76C2×#10623494—21no—
77C2×#10623494—21no—
78C2×#10623494—21no—
79C2×#10623494—44no—
80C2×#10623494—41no—
81C2×#10623494—41no—
82C2×#10623494—41no—
83C2×#10623494—41no—
84C2×#10623494—41no—
85#41×C89—41no—
86#41×C89—41no—
87#41×C89—41no—
88#41×C89—41no—
89C2×Q8×C89—21no—
90C2×D4×C89—22no—
91C2×D4×C89—21no—
92C2×D4×C89—21no—
93C2×D4×C89C4⋊C412yes—
94C22×C4×C89—22no—
95C22×C4×C89C2×D411yescharacteristic
96C22×C4×C89C2×Q811yescharacteristic
97C22×C4×C89#4111yescharacteristic
98C2×C8×C89—21no—
99C2×C8×C89—21no—
100C2×C8×C89—21no—
101C2×C8×C89—21no—
102C2×C8×C89—41no—
103C2×C8×C89—41no—
104C4⋊C4×C89—21no—
105C4⋊C4×C89C4⋊C412yes—
106C22⋊C4×C89—22no—
107C22⋊C4×C89—22no—
108C42×C89C22×C411yescharacteristic
109C42×C89C2×D411yescharacteristic
110C42×C89C2×D411yescharacteristic
111C42×C89#4111yescharacteristic
112C23×C89—22no—
113C23×C89C22⋊Q811yescharacteristic
114Q8×C89—41no—
115Q8×C89—41no—
116D4×C89—42no—
117D4×C89—42no—
118D4×C89—42no—
119D4×C89—41no—
120D4×C89—41no—
121D4×C89—41no—
122D4×C89—41no—
123C2×C4×C89—21no—
124C2×C4×C89—21no—
125C2×C4×C89—22no—
126C2×C4×C89—41no—
127C2×C4×C89—42no—
128C2×C4×C89—42no—
129C2×C4×C89C2×C4⋊C411yescharacteristic
130C2×C4×C89C4×D411yescharacteristic
131C2×C4×C89C4×D411yescharacteristic
132C2×C4×C89#12011yescharacteristic
133C2×C4×C89C22⋊Q811yescharacteristic
134C2×C4×C89#12311yescharacteristic
135C2×C4×C89C2×D811yescharacteristic
136C2×C4×C89#13511yescharacteristic
137C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C89#13711yescharacteristic
139C8×C89—41no—
140C8×C89—41no—
141C8×C89—41no—
142C8×C89—41no—
143#10623494—22no—
144#10623494—42no—
145#10623494—41no—
146#10623494—41no—
147#10623494—41no—
148#10623494—81no—
149C22×C89—21no—
150C22×C89—21no—
151C22×C89—42no—
152C22×C89—42no—
153C22×C89—42no—
154C22×C89#38511yescharacteristic
155C22×C89#47411yescharacteristic
156C22×C89#47811yescharacteristic
157C4×C89—21no—
158C4×C89—21no—
159C4×C89—21no—
160C4×C89—42no—
161C4×C89C4×D811yescharacteristic
162C4×C89#44011yescharacteristic
163C4×C89#44311yescharacteristic
164C4×C89#44411yescharacteristic
165C4×C89#44711yescharacteristic
166C4×C89#45211yescharacteristic
167#8930—891nomaximal, Sylow
168C2×C89—21no—
169C2×C89—21no—
170C2×C89—82no—
171C2×C89#185811yescharacteristic
172C2×C89#186211yescharacteristic
173C2×C89#186611yescharacteristic
174#2935—891no—
175#1950—891no—
176#1571—891no—
177#1540D8911yescharacteristic
178#1492—891no—
179#1278—891no—
180#1277—891no—
181C89#893011yescharacteristic, complemented, Sylow
182C2×#131—1781no—
183#532C2×D8911yescharacteristic
184#436—1781no—
185#435—1782no—
186C8×D4—1781no—
187#431C2×D8911yescharacteristic
188C2×#108—1781no—
189#423—891no—
190C2×C4⋊C8—891no—
191C2×C4⋊C8C2×D8911yescharacteristic
192#408—1781no—
193C4×M16—891no—
194C2×C4×C8—891no—
195#341—1781no—
196#335—1781no—
197#335—1781no—
198#332Dic8911yescharacteristic
199#332Dic8911yescharacteristic
200#331Dic8911yescharacteristic
201#331Dic8911yescharacteristic
202D4⋊C8—1782no—
203C2×#41—21no—
204#131—3562no—
205#131—3562no—
206C2×M16—891no—
207C2×M16—891no—
208C2×M16—1781no—
209C2×M16—1782no—
210C22×C8—21no—
211C22×C8—1782no—
212C22×C8—1781no—
213C22×C8—1781no—
214C4⋊Q8C2×Dic8911yescharacteristic
215#127C2×Dic8911yescharacteristic
216#124—21no—
217#121—21no—
218#121Dic17812yes—
219C4×D4—21no—
220C4×D4D35612yes—
221C2×C42C22×D8911yescharacteristic
222#108—3562no—
223C4⋊C8—21no—
224C4⋊C8—891no—
225C4⋊C8—891no—
226C4⋊C8—1781no—
227C4⋊C8—1781no—
228C4⋊C8—1781no—
229C4⋊C8C2×Dic8911yescharacteristic
230C4⋊C8C2×Dic8911yescharacteristic
231C4⋊C8C2×Dic8911yescharacteristic
232C4⋊C8C2×Dic8911yescharacteristic
233C22⋊C8—21no—
234C22⋊C8—1782no—
235C22⋊C8—1782no—
236#97—1781no—
237C4×C8—21no—
238C4×C8—1781no—
239C4×C8—1781no—
240C4×C8—1781no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4C4⋊Dic8912yes—
250C22×C4—22no—
251C22×C4D4×D8911yescharacteristic
252C22×C4C2×Dic17811yescharacteristic
253C22×C4#1308357311yescharacteristic
254M16—892no—
255M16—3564no—
256M16—3561no—
257M16—3561no—
258M16—1781no—
259M16—1781no—
260M16—1781no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—3561no—
268C2×C8—3561no—
269C2×C8—3561no—
270C2×C8—3561no—
271C2×C8—3564no—
272C2×C8—3561no—
273C2×C8—1782no—
274C2×C8—1781no—
275C2×C8—1781no—
276C2×C8—1781no—
277C2×C8—1781no—
278C4⋊C4—21no—
279C4⋊C4C4⋊Dic8912yes—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C42C22×Dic8911yescharacteristic
283C42D4×D8911yescharacteristic
284C42C2×D35611yescharacteristic
285C42#1308357311yescharacteristic
286C23—22no—
287C23#1566798911yescharacteristic
288Q8—41no—
289Q8—41no—
290D4—42no—
291D4—42no—
292D4—42no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4C2×C4⋊Dic8911yescharacteristic
304C2×C4D4×Dic8911yescharacteristic
305C2×C4D4×Dic8911yescharacteristic
306C2×C4#1566798211yescharacteristic
307C2×C4#1566798911yescharacteristic
308C2×C4#1566799611yescharacteristic
309C2×C4D8×D8911yescharacteristic
310C2×C4#1566803111yescharacteristic
311C2×C4#1566803911yescharacteristic
312C2×C4#1566804111yescharacteristic
313C8—41no—
314C8—41no—
315C8—41no—
316C8—41no—
317C8—3562no—
318C8—3561no—
319C8—3561no—
320C8—3561no—
321C8—1782no—
322C8—7121no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22#1915147311yescharacteristic
329C22#1915187611yescharacteristic
330C22#1915189611yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—42no—
335C4D8×Dic8911yescharacteristic
336C4#1915169911yescharacteristic
337C4#1915171511yescharacteristic
338C4#1915172011yescharacteristic
339C4#1915173311yescharacteristic
340C4#1915176111yescharacteristic
341C2—21no—
342C2—21no—
343C2—82no—
344C2#2602047411yescharacteristic
345C2#2602050111yescharacteristic
346C2#2602052911yescharacteristic
subgroupG/H
centerC2×C4#15667982
commutatorC42×C89C22×C4
FrattiniC2×C42C22×D89
FittingC89×#1540C2
socleC22×C89#385
prime radical#532×C89C22
cosocleC2×C42×C89C23
hypercenter#1540D89
hypocenterC89#8930
LCSG ▷ C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
2. #26021028C2semidirect
3. #26019027C2semidirect
5. #26018088C2semidirect
6. #26018086C2semidirect
181. C89#8930semidirect

Maximal subgroups

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

subgroupindex
#260289962
#260210282
#260190272
#260187032
#260180882
#260180862
C89×#15402
#893089