Finite Group Databasebeta

← #36448775

Subgroups — #36448775

Order 22784.

376 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#13083558C211yesmaximal, characteristic
2#26021025C211yesmaximal, characteristic
3#26019007C211yesmaximal, characteristic
4C2×Q8⋊#10623494C211yesmaximal, characteristic
5#26018148C211yesmaximal, characteristic
6#26018088C211yesmaximal, characteristic
7C89×#1487C211yesmaximal, characteristic
8#19151729—22no—
9#19151729—21no—
10Q8×#10623494—21no—
11C2×#15667947—21no—
12#19151622—21no—
13#19151622C2211yescharacteristic
14C2×C4⋊#10623494C2211yescharacteristic
15C4×#13083558—21no—
16C4×#13083558C2211yescharacteristic
17C2×C4×#10623494C2211yescharacteristic
18#19151362—21no—
19#19151348—22no—
20Q8⋊#10623494C411yescharacteristic
21Q8⋊#10623494C411yescharacteristic
22Q8⋊#10623494C411yescharacteristic
23Q8⋊#10623494C411yescharacteristic
24C2×C4×Q8×C89C2211yescharacteristic
25#431×C89C2211yescharacteristic
26#423×C89C2211yescharacteristic
27Q8⋊C8×C89—21no—
28Q8⋊C8×C89—21no—
29C2×#13083558—21no—
30C2×#13083558D411yescharacteristic
31C2×#13083558D411yescharacteristic
32C22×#10623494—21no—
33C22×#10623494—21no—
34#15667947—42no—
35C4⋊#10623494—22no—
36C4⋊#10623494—22no—
37C4⋊#10623494—21no—
38C4⋊#10623494—21no—
39C4⋊#10623494C2×C411yescharacteristic
40C4⋊#10623494C2×C411yescharacteristic
41C4⋊#10623494D412yes—
42#15667923—22no—
43#15667923—21no—
44C4×#10623494—22no—
45C4×#10623494—21no—
46C4×#10623494—21no—
47C4×#10623494C2×C411yescharacteristic
48C4×#10623494C2×C411yescharacteristic
49C22×Q8×C89Q811yescharacteristic
50C2×M16×C89—21no—
51C4×Q8×C89—21no—
52C4×Q8×C89—21no—
53C4×Q8×C89C2×C411yescharacteristic
54C4×Q8×C89C2×C411yescharacteristic
55C4×Q8×C89D411yescharacteristic
56C4×Q8×C89D411yescharacteristic
57C2×C4⋊C4×C89—21no—
58C2×C4⋊C4×C89Q811yescharacteristic
59C2×C42×C89—21no—
60C2×C42×C89C2311yescharacteristic
61C4⋊C8×C89—21no—
62C4⋊C8×C89—21no—
63C4⋊C8×C89—21no—
64C22⋊C8×C89—21no—
65C4×C8×C89—21no—
66#13083558—21no—
67#13083558—21no—
68#13083558—21no—
69#13083558—41no—
70#13083558—41no—
71#13083558SD1612yes—
72C2×#10623494—21no—
73C2×#10623494—21no—
74C2×#10623494—21no—
75C2×#10623494—21no—
76C2×#10623494—21no—
77C2×#10623494—22no—
78C2×#10623494—21no—
79C2×#10623494—42no—
80C2×#10623494—41no—
81C2×#10623494—41no—
82C2×Q8×C89—21no—
83C2×Q8×C89—22no—
84C2×Q8×C89—41no—
85C2×Q8×C89C4⋊C411yescharacteristic
86C2×Q8×C89C4⋊C411yescharacteristic
87C22×C4×C89—21no—
88C22×C4×C89—21no—
89C22×C4×C89C2×D411yescharacteristic
90C22×C4×C89C2×Q811yescharacteristic
91C22×C4×C89#4111yescharacteristic
92M16×C89—41no—
93M16×C89—41no—
94C2×C8×C89—41no—
95C2×C8×C89—41no—
96C4⋊C4×C89—21no—
97C4⋊C4×C89—21no—
98C4⋊C4×C89—21no—
99C4⋊C4×C89—21no—
100C4⋊C4×C89—21no—
101C4⋊C4×C89C4⋊C411yescharacteristic
102C4⋊C4×C89C4⋊C411yescharacteristic
103C42×C89—21no—
104C42×C89—21no—
105C42×C89—21no—
106C42×C89—21no—
107C42×C89C22×C411yescharacteristic
108C42×C89C2×D411yescharacteristic
109C42×C89C2×D411yescharacteristic
110C42×C89#4111yescharacteristic
111C23×C89C22⋊Q811yescharacteristic
112Q8×C89—21no—
113Q8×C89—21no—
114Q8×C89—21no—
115Q8×C89—42no—
116Q8×C89#10812yescomplemented
117C2×C4×C89—21no—
118C2×C4×C89—21no—
119C2×C4×C89—21no—
120C2×C4×C89—21no—
121C2×C4×C89—21no—
122C2×C4×C89—21no—
123C2×C4×C89—41no—
124C2×C4×C89—42no—
125C2×C4×C89—81no—
126C2×C4×C89C2×C4⋊C411yescharacteristic
127C2×C4×C89C4×D411yescharacteristic
128C2×C4×C89C4×D411yescharacteristic
129C2×C4×C89#12011yescharacteristic
130C2×C4×C89C22⋊Q811yescharacteristic
131C2×C4×C89#12311yescharacteristic
132C2×C4×C89C2×SD1611yescharacteristic
133C2×C4×C89#13511yescharacteristic
134C2×C4×C89#13711yescharacteristic
135C2×C4×C89#13711yescharacteristic
136C8×C89—81no—
137C8×C89—81no—
138#10623494—22no—
139#10623494—22no—
140#10623494—41no—
141#10623494—41no—
142#10623494—41no—
143#10623494—41no—
144#10623494—42no—
145C22×C89—21no—
146C22×C89—21no—
147C22×C89#38511yescharacteristic
148C22×C89#47511yescharacteristic
149C22×C89#47911yescharacteristic
150C4×C89—21no—
151C4×C89—22no—
152C4×C89—21no—
153C4×C89—21no—
154C4×C89—41no—
155C4×C89—41no—
156C4×C89—41no—
157C4×C89C2×#10811yescharacteristic
158C4×C89#43011yescharacteristic
159C4×C89C4×SD1611yescharacteristic
160C4×C89#44111yescharacteristic
161C4×C89#44811yescharacteristic
162C4×C89#45211yescharacteristic
163#8909—891nomaximal, Sylow
164C2×C89—21no—
165C2×C89—21no—
166C2×C89#185511yescharacteristic
167C2×C89#186111yescharacteristic
168C2×C89#186511yescharacteristic
169Q8×M16—891no—
170#1950—891no—
171#1566—891no—
172#1487D8911yescharacteristic
173C2×Q8⋊C8—891no—
174#1301—891no—
175#1278—891no—
176C89#890911yescharacteristic, complemented, Sylow
177C2×C4×Q8C2×D8911yescharacteristic
178#446—1782no—
179#446—1781no—
180C8×Q8—1781no—
181#431C2×D8911yescharacteristic
182C2×#108—1781no—
183#423—891no—
184#423—1781no—
185#423C2×D8911yescharacteristic
186C2×C4⋊C8—891no—
187C4×M16—891no—
188C4×M16—1781no—
189C2×C4×C8—891no—
190#341—1781no—
191#334—1782no—
192Q8⋊C8—21no—
193Q8⋊C8—21no—
194Q8⋊C8—891no—
195Q8⋊C8—891no—
196Q8⋊C8—891no—
197Q8⋊C8—891no—
198C22×Q8Dic17811yescharacteristic
199C2×M16—21no—
200C2×M16—891no—
201C2×M16—891no—
202C2×M16—1781no—
203C22×C8—1781no—
204C22×C8—1781no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8C4×D8911yescharacteristic
208C4×Q8C4×D8911yescharacteristic
209C4×Q8C22⋊D8911yescharacteristic
210C4×Q8C22⋊D8911yescharacteristic
211C2×C4⋊C4—21no—
212C2×C4⋊C4Dic17811yescharacteristic
213C2×C42—21no—
214C2×C42C22×D8911yescharacteristic
215#108—3562no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—891no—
220C4⋊C8—891no—
221C4⋊C8—892no—
222C4⋊C8—1782no—
223C4⋊C8—1782no—
224C4⋊C8—1781no—
225C4⋊C8—1781no—
226C22⋊C8—21no—
227#97—1782no—
228#97—1781no—
229C4×C8—21no—
230C4×C8—891no—
231C4×C8—891no—
232C4×C8—1782no—
233C4×C8—1781no—
234C4×C8—1781no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8#1308355311yescharacteristic
239C2×Q8#1308355311yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4D4×D8911yescharacteristic
243C22×C4C2×Dic17811yescharacteristic
244C22×C4#1308357311yescharacteristic
245M16—41no—
246M16—41no—
247M16—892no—
248M16—3561no—
249M16—3561no—
250M16—1781no—
251M16—1781no—
252M16—1781no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—3561no—
256C2×C8—3561no—
257C2×C8—3562no—
258C2×C8—1781no—
259C2×C8—1781no—
260C2×C8—1781no—
261C2×C8—1782no—
262C2×C8—1781no—
263C2×C8—1781no—
264C2×C8—1781no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4#1308355311yescharacteristic
271C4⋊C4#1308355311yescharacteristic
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42C2×C4×D8911yescharacteristic
277C42D4×D8911yescharacteristic
278C42C2×C22⋊D8911yescharacteristic
279C42#1308357311yescharacteristic
280C23#1566798911yescharacteristic
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—42no—
285Q8#1566794712yescomplemented
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4C2×#1308355311yescharacteristic
296C2×C4#1566797211yescharacteristic
297C2×C4#1566797211yescharacteristic
298C2×C4#1566798011yescharacteristic
299C2×C4#1566798911yescharacteristic
300C2×C4#1566799311yescharacteristic
301C2×C4SD16×D8911yescharacteristic
302C2×C4#1566803511yescharacteristic
303C2×C4#1566804111yescharacteristic
304C2×C4#1566804411yescharacteristic
305C8—81no—
306C8—81no—
307C8—3561no—
308C8—3561no—
309C8—3561no—
310C8—3562no—
311C8—3561no—
312C8—1782no—
313C8—1782no—
314C22—21no—
315C22—21no—
316C22#1915147011yescharacteristic
317C22#1915188111yescharacteristic
318C22#1915190711yescharacteristic
319C4—21no—
320C4—22no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4C2×#1566794711yescharacteristic
327C4#1915164611yescharacteristic
328C4#1915168311yescharacteristic
329C4#1915170211yescharacteristic
330C4#1915173511yescharacteristic
331C4#1915175911yescharacteristic
332C2—21no—
333C2—21no—
334C2#2602045611yescharacteristic
335C2#2602049011yescharacteristic
336C2#2602051811yescharacteristic
subgroupG/H
centerC2×C4#15667980
commutatorC42×C89C22×C4
FrattiniC2×C42C22×D89
FittingC89×#1487C2
socleC22×C89#385
prime radicalC23×C89C22⋊Q8
cosocleC2×C42×C89C23
hypercenter#1487D89
hypocenterC89#8909
LCSG ▷ C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
116. Q8×C89#108semidirect
176. C89#8909semidirect
285. Q8#15667947semidirect

Maximal subgroups

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

subgroupindex
Q8×#130835582
#260210252
#260190072
C2×Q8⋊#106234942
#260181482
#260180882
C89×#14872
#890989