Finite Group Databasebeta

← #36471492

Subgroups — #36471492

Order 22784.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15667955C211yesmaximal, characteristic
2C2×#19151402C211yesmaximal, characteristic, complemented
3C2×#19151398C211yesmaximal, characteristic
4#26021118C211yesmaximal, characteristic
5#26018343C211yesmaximal, characteristic, complemented
6#26018330C211yesmaximal, characteristic, complemented
7C2×C4×Q16×C89C211yesmaximal, characteristic, complemented
8C22×D712C2211yescharacteristic
9C2×#15667955—22no—
10C2×#15667955—22no—
11C2×#15667955C414yescomplemented
12C22×#13083549C2211yescharacteristic
13C22×#13083549C2211yescharacteristic
14C2×#15667946C2211yescharacteristic
15C2×#15667934—21no—
16#19151500—21no—
17#19151402—22no—
18#19151398—22no—
19C22×Q16×C89C2211yescharacteristic
20C2×C4×Q8×C89—21no—
21C4×Q16×C89—22no—
22C4×Q16×C89—22no—
23C2×#107×C89C2211yescharacteristic
24C2×#103×C89—21no—
25C2×C4×C8×C89C2211yescharacteristic
26C22×D356—21no—
27C2×D712—21no—
28C2×D712—21no—
29C2×D712—21no—
30C2×D712—21no—
31C2×D712—21no—
32C2×D712C2×C412yes—
33C2×C4⋊Dic89—21no—
34C2×#13083551—41no—
35C2×#13083551—41no—
36#15667955—28no—
37C2×#13083549—21no—
38C2×#13083549—21no—
39C2×#13083549—21no—
40C2×#13083549—21no—
41C2×#13083549—21no—
42C2×#13083549C2×C412yes—
43#15667946—21no—
44#15667946—21no—
45#15667934—42no—
46C22×Q8×C89—21no—
47C2×Q16×C89—21no—
48C2×Q16×C89—21no—
49C2×Q16×C89—21no—
50C2×Q16×C89C2×C412yes—
51C2×Q16×C89D412yes—
52C2×Q16×C89D412yes—
53C22×C8×C89D411yescharacteristic
54C22×C8×C89C2311yescharacteristic
55C4×Q8×C89—24no—
56C4×Q8×C89—42no—
57C2×C4⋊C4×C89—21no—
58C2×C4⋊C4×C89—41no—
59C2×C42×C89—41no—
60C2×C42×C89D411yescharacteristic
61#107×C89—21no—
62#107×C89D412yes—
63#103×C89—24no—
64C4×C8×C89—21no—
65C4×C8×C89—21no—
66C23×D89—41no—
67C2×D356—22no—
68C2×D356—41no—
69C2×D356—41no—
70C2×D356—41no—
71C2×D356—41no—
72C2×D356—41no—
73C22×Dic89—41no—
74D712—22no—
75D712—22no—
76D712—41no—
77D712—41no—
78C4⋊Dic89—41no—
79C4⋊Dic89—41no—
80#13083551—82no—
81#13083551—82no—
82#13083549—22no—
83#13083549—41no—
84C2×Q8×C89—22no—
85C2×Q8×C89—22no—
86C2×Q8×C89—22no—
87C2×Q8×C89—41no—
88C2×Q8×C89—41no—
89C2×Q8×C89—41no—
90C22×C4×C89—41no—
91C22×C4×C89—41no—
92C22×C4×C89D811yescharacteristic
93C22×C4×C89D811yescharacteristic
94C22×C4×C89C2×D411yescharacteristic
95Q16×C89—22no—
96Q16×C89—22no—
97Q16×C89—22no—
98Q16×C89C22⋊C414yescomplemented
99C2×C8×C89—21no—
100C2×C8×C89—21no—
101C2×C8×C89—21no—
102C2×C8×C89C22×C411yescharacteristic
103C2×C8×C89C2×D411yescharacteristic
104C2×C8×C89C2×D411yescharacteristic
105C2×C8×C89C2×D411yescharacteristic
106C2×C8×C89#4111yescharacteristic
107C2×C8×C89#4111yescharacteristic
108C4⋊C4×C89—22no—
109C4⋊C4×C89—44no—
110C4⋊C4×C89—41no—
111C42×C89—21no—
112C42×C89—21no—
113C42×C89—44no—
114C23×C89C2×D811yescharacteristic
115C22×D89—42no—
116C22×D89—81no—
117C22×D89—81no—
118C22×D89—81no—
119C22×D89—81no—
120C22×D89—81no—
121C22×D89—81no—
122Q8×C89—24no—
123Q8×C89—42no—
124Q8×C89—42no—
125Q8×C89—42no—
126C2×C4×C89—21no—
127C2×C4×C89—21no—
128C2×C4×C89—21no—
129C2×C4×C89—21no—
130C2×C4×C89—21no—
131C2×C4×C89—42no—
132C2×C4×C89—42no—
133C2×C4×C89—42no—
134C2×C4×C89—42no—
135C2×C4×C89—81no—
136C2×C4×C89—81no—
137C2×C4×C89SD3212yescomplemented
138C2×C4×C89C4×D411yescharacteristic
139C2×C4×C89#12011yescharacteristic
140C2×C4×C89#12111yescharacteristic
141C2×C4×C89#13511yescharacteristic
142C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
144C2×Dic89—81no—
145C2×Dic89—81no—
146C2×Dic89—81no—
147D356—42no—
148D356—42no—
149D356—81no—
150D356—81no—
151C8×C89—41no—
152C8×C89C2×C22⋊C411yescharacteristic
153C8×C89C4×D411yescharacteristic
154C8×C89#12111yescharacteristic
155C8×C89#12311yescharacteristic
156C22×C89C4×D811yescharacteristic
157C22×C89#44711yescharacteristic
158C22×C89#45911yescharacteristic
159C22×C89C2×SD3211yescharacteristic
160C22×C89#50811yescharacteristic
161C22×C89#50911yescharacteristic
162C22×C89#51011yescharacteristic
163C2×D89—82no—
164C2×D89—82no—
165C2×D89—82no—
166C2×D89—161no—
167C2×D89—161no—
168C2×D89—161no—
169C2×D89—161no—
170C4×C89—22no—
171C4×C89—44no—
172C4×C89—41no—
173C4×C89—82no—
174C4×C89#38611yescharacteristic
175C4×C89#44211yescharacteristic
176C4×C89#44911yescharacteristic
177C4×C89#46311yescharacteristic
178Dic89—161no—
179#13013—891nomaximal, Sylow
180C2×C89#187911yescharacteristic
181C2×C89C4×SD3211yescharacteristic
182C2×C89#217511yescharacteristic
183C2×C89#218511yescharacteristic
184C2×C89#218611yescharacteristic
185C2×C89#220711yescharacteristic
186C2×C89#221111yescharacteristic
187D89—162no—
188C22×SD32—891no—
189C2×C4×Q16D8911yescharacteristic, complemented
190C2×#358—891no—
191C2×#357—891no—
192#1970—891no—
193#1380—891no—
194#1374—891no—
195C89#1301311yescharacteristic, complemented, Sylow
196C22×Q16C2×D8911yescharacteristic
197C22×D8—891no—
198C2×C4×Q8—21no—
199C2×SD32—894no—
200C2×SD32—1782no—
201C2×SD32—1782no—
202C22×C16—891no—
203C22×C16—891no—
204C4×Q16—22no—
205C4×Q16—22no—
206C2×#107—891no—
207C2×#107C2×D8911yescharacteristic
208C2×#103—21no—
209C2×#102—1781no—
210C2×C4×C8C2×D8911yescharacteristic
211#390—1781no—
212#358—1782no—
213#357—1782no—
214C22×Q8—21no—
215C22×D4—1781no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D8912yes—
220C2×Q16D35612yes—
221C2×Q16C22⋊D8912yes—
222C2×D8—892no—
223C2×D8—1781no—
224C2×D8—1781no—
225C2×D8—1781no—
226C2×D8—1781no—
227C2×D8—1781no—
228C22×C8C22⋊D8911yescharacteristic
229C22×C8C22×D8911yescharacteristic
230C4×Q8—24no—
231C4×Q8—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—1781no—
235C2×C22⋊C4—3561no—
236C2×C22⋊C4—3561no—
237C2×C42—41no—
238C2×C42C22⋊D8911yescharacteristic
239SD32—1788no—
240C2×C16—892no—
241C2×C16—1781no—
242C2×C16—1781no—
243C2×C16—1781no—
244C2×C16—1781no—
245C2×C16—1781no—
246#107—21no—
247#107—1781no—
248#107—1781no—
249#107C22⋊D8912yes—
250#103—24no—
251#102—3562no—
252C4×C8—21no—
253C4×C8—21no—
254C24—3561no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—3561no—
262C2×D4—3561no—
263C2×D4—3561no—
264C2×D4—3561no—
265C2×D4—3561no—
266C2×D4—1782no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—3561no—
270C22×C4#1308355911yescharacteristic
271C22×C4#1308355911yescharacteristic
272C22×C4C2×C22⋊D8911yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16#1308355114yescomplemented
277D8—3561no—
278D8—3561no—
279D8—1782no—
280D8—1782no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C2×C4×D8911yescharacteristic
285C2×C8C2×C22⋊D8911yescharacteristic
286C2×C8C2×C22⋊D8911yescharacteristic
287C2×C8C2×D35611yescharacteristic
288C2×C8#1308357411yescharacteristic
289C2×C8#1308357411yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—41no—
293C4⋊C4—3561no—
294C4⋊C4—3561no—
295C22⋊C4—7122no—
296C22⋊C4—7122no—
297C42—21no—
298C42—21no—
299C42—44no—
300C16—3561no—
301C16—1782no—
302C23—3562no—
303C23—7121no—
304C23—7121no—
305C23—7121no—
306C23—7121no—
307C23—7121no—
308C23—7121no—
309C23C2×#1308355911yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—3562no—
315D4—3562no—
316D4—7121no—
317D4—7121no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—7121no—
330C2×C4—7121no—
331C2×C4—7121no—
332C2×C4#1566795512yescomplemented
333C2×C4C4×C22⋊D8911yescharacteristic
334C2×C4#1566798111yescharacteristic
335C2×C4#1566798611yescharacteristic
336C2×C4#1566803211yescharacteristic
337C2×C4#1566803711yescharacteristic
338C2×C4#1566803811yescharacteristic
339C8—41no—
340C8C2×#1308355111yescharacteristic
341C8C4×C22⋊D8911yescharacteristic
342C8#1566798611yescharacteristic
343C8#1566799511yescharacteristic
344C22—14241no—
345C22—14241no—
346C22—14241no—
347C22—14241no—
348C22—7122no—
349C22—7122no—
350C22—7122no—
351C22C4×#1308355911yescharacteristic
352C22#1915173211yescharacteristic
353C22#1915179311yescharacteristic
354C22C2×#1566795511yescharacteristic
355C22#1915202811yescharacteristic
356C22#1915203311yescharacteristic
357C22#1915203911yescharacteristic
358C4—22no—
359C4—41no—
360C4—44no—
361C4—82no—
362C4—14241no—
363C4#1915147711yescharacteristic
364C4#1915170811yescharacteristic
365C4#1915174311yescharacteristic
366C4#1915181311yescharacteristic
367C2—14242no—
368C2#2602060011yescharacteristic
369C2C4×#1566795511yescharacteristic
370C2#2602211411yescharacteristic
371C2#2602216911yescharacteristic
372C2#2602217511yescharacteristic
373C2#2602229211yescharacteristic
374C2#2602231411yescharacteristic
subgroupG/H
centerC23C2×#13083559
commutatorC2×C8×C89C22×C4
FrattiniC22×C8C22×D89
FittingC2×C4×Q16×C89C2
socleC23×C89C2×D8
prime radicalC22×D712C22
cosocleC22×C8×C89C23
hypercenterC2×C4×Q16D89
hypocenterC89#13013
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#19151402C2semidirect
5. #26018343C2semidirect
6. #26018330C2semidirect
7. C2×C4×Q16×C89C2semidirect
11. C2×#15667955C4semidirect
98. Q16×C89C22⋊C4semidirect
137. C2×C4×C89SD32semidirect
189. C2×C4×Q16D89semidirect
195. C89#13013semidirect
276. Q16#13083551semidirect
332. C2×C4#15667955semidirect

Maximal subgroups

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

subgroupindex
C22×#156679552
C2×#191514022
C2×#191513982
#260211182
#260183432
#260183302
C2×C4×Q16×C892
#1301389