Finite Group Databasebeta

← C89×#29211 #36407059

Subgroups — C89×#29211 #36407059

Order 22784.

726 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2862C211yesmaximal, characteristic
2C4×C4⋊Q8×C89C211yesmaximal, characteristic
3C89×#2058C212yesmaximal
4C89×#2057C212yesmaximal
5C89×#1993C212yesmaximal
6C89×#1949C214yesmaximal
7C89×#1847C212yesmaximal
8C89×#1767C211yesmaximal, characteristic
9C2×C4⋊Q8×C89C2211yescharacteristic
10C2×C4⋊Q8×C89C2211yescharacteristic
11C2×C4⋊Q8×C89C2211yescharacteristic
12C2×C4×Q8×C89—21no—
13C2×#107×C89C2214yes—
14C2×#106×C89C2214yes—
15C2×C4⋊C8×C89C2212yes—
16C2×#97×C89C2212yes—
17#391×C89—21no—
18#391×C89—21no—
19#391×C89—21no—
20#391×C89—21no—
21#391×C89—21no—
22#389×C89—21no—
23#389×C89C2212yes—
24#384×C89C2211yescharacteristic
25#382×C89C2212yes—
26#382×C89C2212yes—
27C4×C4⋊C4×C89—21no—
28C4×C4⋊C4×C89C2212yes—
29C43×C89C2211yescharacteristic
30#340×C89C2214yes—
31#340×C89C2214yes—
32#336×C89C2212yes—
33C22×Q8×C89—21no—
34C22×Q8×C89—21no—
35C22×Q8×C89—21no—
36C22×C8×C89C2314yes—
37C4⋊Q8×C89—22no—
38C4⋊Q8×C89—42no—
39C4⋊Q8×C89—42no—
40C4⋊Q8×C89Q814yescomplemented
41C4×Q8×C89—44no—
42C2×C4⋊C4×C89—21no—
43C2×C4⋊C4×C89—21no—
44C2×C4⋊C4×C89—21no—
45C2×C4⋊C4×C89—22no—
46C2×C4⋊C4×C89—21no—
47C2×C4⋊C4×C89—21no—
48C2×C4⋊C4×C89C2312yes—
49C2×C4⋊C4×C89C2312yes—
50C2×C4⋊C4×C89C2312yes—
51C2×C4⋊C4×C89C2312yes—
52C2×C42×C89—22no—
53C2×C42×C89D411yescharacteristic
54C2×C42×C89D411yescharacteristic
55C2×C42×C89D411yescharacteristic
56C2×C42×C89D411yescharacteristic
57C2×C42×C89C2311yescharacteristic
58C2×C42×C89C2311yescharacteristic
59C2×C42×C89C2311yescharacteristic
60#107×C89—44no—
61#106×C89—44no—
62C4⋊C8×C89Q818yescomplemented
63#97×C89—24no—
64#95×C89—22no—
65#95×C89—22no—
66#95×C89—22no—
67#95×C89—22no—
68#95×C89—22no—
69#95×C89—22no—
70C2×Q8×C89—24no—
71C2×Q8×C89—42no—
72C2×Q8×C89—42no—
73C2×Q8×C89—42no—
74C2×Q8×C89—42no—
75C2×Q8×C89—42no—
76C2×Q8×C89—42no—
77C2×Q8×C89—42no—
78C2×Q8×C89—42no—
79C22×C4×C89—22no—
80C22×C4×C89—22no—
81C22×C4×C89—22no—
82C22×C4×C89—22no—
83C22×C4×C89C2×D411yescharacteristic
84C22×C4×C89C2×D411yescharacteristic
85C22×C4×C89C2×D411yescharacteristic
86C22×C4×C89C2×D411yescharacteristic
87C22×C4×C89C2×D411yescharacteristic
88C22×C4×C89C2×D411yescharacteristic
89C22×C4×C89C2411yescharacteristic
90C2×C8×C89—24no—
91C2×C8×C89—24no—
92C2×C8×C89C2×Q818yes—
93C4⋊C4×C89—24no—
94C4⋊C4×C89—22no—
95C4⋊C4×C89—24no—
96C4⋊C4×C89—24no—
97C4⋊C4×C89—24no—
98C4⋊C4×C89—42no—
99C4⋊C4×C89—42no—
100C4⋊C4×C89—42no—
101C4⋊C4×C89—42no—
102C4⋊C4×C89—44no—
103C4⋊C4×C89C2×Q814yes—
104C42×C89—22no—
105C42×C89—22no—
106C42×C89—22no—
107C42×C89—41no—
108C42×C89—41no—
109C42×C89—44no—
110C42×C89C2×Q812yes—
111C42×C89C2×Q814yes—
112C42×C89#4112yes—
113C23×C89C22×D411yescharacteristic
114Q8×C89—48no—
115Q8×C89—84no—
116Q8×C89—84no—
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—21no—
124C2×C4×C89—21no—
125C2×C4×C89—21no—
126C2×C4×C89—21no—
127C2×C4×C89—21no—
128C2×C4×C89—21no—
129C2×C4×C89—24no—
130C2×C4×C89—42no—
131C2×C4×C89—42no—
132C2×C4×C89—42no—
133C2×C4×C89—42no—
134C2×C4×C89—42no—
135C2×C4×C89—42no—
136C2×C4×C89—42no—
137C2×C4×C89—42no—
138C2×C4×C89—42no—
139C2×C4×C89—42no—
140C2×C4×C89—42no—
141C2×C4×C89C22⋊Q812yes—
142C2×C4×C89C22⋊Q812yes—
143C2×C4×C89C4⋊Q812yes—
144C2×C4×C89C4⋊Q812yes—
145C2×C4×C89AGL(1,ℤ/8)12yes—
146C2×C4×C89#13712yes—
147C2×C4×C89C22×Q812yes—
148C2×C4×C89C22×Q811yescharacteristic
149C2×C4×C89C2×#4111yescharacteristic
150C2×C4×C89H(5,𝔽2)12yes—
151C8×C89—28no—
152C22×C89C2×C22⋊Q811yescharacteristic
153C22×C89C2×C4⋊Q811yescharacteristic
154C22×C89#53511yescharacteristic
155C22×C89C2×AGL(1,ℤ/8)11yescharacteristic
156C22×C89C2×#13711yescharacteristic
157C22×C89#57511yescharacteristic
158C22×C89#57511yescharacteristic
159C4×C89—22no—
160C4×C89—22no—
161C4×C89—22no—
162C4×C89—22no—
163C4×C89—22no—
164C4×C89—22no—
165C4×C89—44no—
166C4×C89—82no—
167C4×C89—82no—
168C4×C89—82no—
169C4×C89#55412yes—
170C4×C89Q8212yes—
171#29211C8911yesmaximal, characteristic, direct product factor, Sylow
172C2×C89#256711yescharacteristic
173C2×C89#307711yescharacteristic
174C2×C89#307811yescharacteristic
175C2×C89#316211yescharacteristic
176C2×C89#316511yescharacteristic
177C2×C89#321311yescharacteristic
178C2×C89#321511yescharacteristic
179#2862C2×C8911yescharacteristic
180C4×C4⋊Q8C2×C8911yescharacteristic
181#2058C2×C8912yes—
182#2057C2×C8912yes—
183#1993C2×C8912yes—
184#1949C2×C8914yes—
185#1847C2×C8912yes—
186#1767C2×C8911yescharacteristic
187C89#2921111yescharacteristic, direct product factor, Sylow
188C2×C4⋊Q8C22×C8911yescharacteristic
189C2×C4⋊Q8C22×C8911yescharacteristic
190C2×C4⋊Q8C22×C8911yescharacteristic
191C2×C4×Q8—21no—
192C2×#107C22×C8914yes—
193C2×#106C22×C8914yes—
194C2×C4⋊C8C22×C8912yes—
195C2×#97C22×C8912yes—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#389—21no—
202#389C22×C8912yes—
203#384C22×C8911yescharacteristic
204#382C22×C8912yes—
205#382C22×C8912yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C22×C8912yes—
208C43C22×C8911yescharacteristic
209#340C22×C8914yes—
210#340C22×C8914yes—
211#336C22×C8912yes—
212C22×Q8—21no—
213C22×Q8—21no—
214C22×Q8—21no—
215C22×C8C23×C8914yes—
216C4⋊Q8—22no—
217C4⋊Q8—42no—
218C4⋊Q8—42no—
219C4⋊Q8Q8×C8914yescomplemented
220C4×Q8—44no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C23×C8912yes—
228C2×C4⋊C4C23×C8912yes—
229C2×C4⋊C4C23×C8912yes—
230C2×C4⋊C4C23×C8912yes—
231C2×C42—22no—
232C2×C42D4×C8911yescharacteristic
233C2×C42D4×C8911yescharacteristic
234C2×C42D4×C8911yescharacteristic
235C2×C42D4×C8911yescharacteristic
236C2×C42C23×C8911yescharacteristic
237C2×C42C23×C8911yescharacteristic
238C2×C42C23×C8911yescharacteristic
239#107—44no—
240#106—44no—
241C4⋊C8Q8×C8918yescomplemented
242#97—24no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249C2×Q8—24no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4C2×D4×C8911yescharacteristic
263C22×C4C2×D4×C8911yescharacteristic
264C22×C4C2×D4×C8911yescharacteristic
265C22×C4C2×D4×C8911yescharacteristic
266C22×C4C2×D4×C8911yescharacteristic
267C22×C4C2×D4×C8911yescharacteristic
268C22×C4C24×C8911yescharacteristic
269C2×C8—24no—
270C2×C8—24no—
271C2×C8C2×Q8×C8918yes—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4C2×Q8×C8914yes—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—41no—
287C42—41no—
288C42—44no—
289C42C2×Q8×C8912yes—
290C42C2×Q8×C8914yes—
291C42#41×C8912yes—
292C23C22×D4×C8911yescharacteristic
293Q8—48no—
294Q8—84no—
295Q8—84no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4C22⋊Q8×C8912yes—
321C2×C4C22⋊Q8×C8912yes—
322C2×C4C4⋊Q8×C8912yes—
323C2×C4C4⋊Q8×C8912yes—
324C2×C4AGL(1,ℤ/8)×C8912yes—
325C2×C4#137×C8912yes—
326C2×C4C22×Q8×C8912yes—
327C2×C4C22×Q8×C8911yescharacteristic
328C2×C4C2×#41×C8911yescharacteristic
329C2×C4H(5,𝔽2)×C8912yes—
330C8—28no—
331C22C2×C22⋊Q8×C8911yescharacteristic
332C22C2×C4⋊Q8×C8911yescharacteristic
333C22#535×C8911yescharacteristic
334C22C2×AGL(1,ℤ/8)×C8911yescharacteristic
335C22C2×#137×C8911yescharacteristic
336C22#575×C8911yescharacteristic
337C22#575×C8911yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—44no—
345C4—82no—
346C4—82no—
347C4—82no—
348C4#554×C8912yes—
349C4Q82×C8912yes—
350C2C89×#256711yescharacteristic
351C2C89×#307711yescharacteristic
352C2C89×#307811yescharacteristic
353C2C89×#316211yescharacteristic
354C2C89×#316511yescharacteristic
355C2C89×#321311yescharacteristic
356C2C89×#321511yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C89C22×D4
commutator, Frattini, cosocleC22×C4C24×C89
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C89 ◁ C43×C89 ◁ G

Complements

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

NH ≅ G/Ntype
40. C4⋊Q8×C89Q8semidirect
62. C4⋊C8×C89Q8semidirect
171. #29211C89direct (N × H)
187. C89#29211direct (N × H)
219. C4⋊Q8Q8×C89semidirect
241. C4⋊C8Q8×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#28622
C4×C4⋊Q8×C892
C89×#20582
C89×#20572
C89×#19932
C89×#19492
C89×#18472
C89×#17672
#2921189