Finite Group Databasebeta

← C89×#10417 #36388265

Subgroups — C89×#10417 #36388265

Order 22784.

520 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2547C211yesmaximal, characteristic
2C89×#1990C211yesmaximal, characteristic, complemented
3C89×#1948C211yesmaximal, characteristic, complemented
4C89×#1802C211yesmaximal, characteristic
5C2×#332×C89C211yesmaximal, characteristic, complemented
6C2×D4⋊C8×C89C211yesmaximal, characteristic
7C89×#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C89C2211yescharacteristic
9C2×#124×C89—21no—
10C2×C4×D4×C89C2211yescharacteristic
11C2×#106×C89—21no—
12C2×C4⋊C8×C89C2211yescharacteristic
13C2×C4⋊C8×C89C2211yescharacteristic
14C2×#102×C89—21no—
15C2×C4×C8×C89C2211yescharacteristic
16#397×C89—22no—
17#391×C89—21no—
18#384×C89C2211yescharacteristic
19#380×C89—21no—
20#380×C89—21no—
21C4×C4⋊C4×C89C2211yescharacteristic
22#340×C89—21no—
23#340×C89—21no—
24#340×C89—22no—
25#332×C89—22no—
26D4⋊C8×C89—22no—
27C22×Q8×C89—21no—
28C22×D4×C89D411yescharacteristic
29C23×C4×C89—21no—
30C22×C8×C89—21no—
31C22×C8×C89—21no—
32C22×C8×C89—21no—
33C4⋊Q8×C89—21no—
34C4⋊Q8×C89—21no—
35C4⋊Q8×C89—21no—
36C4⋊Q8×C89—21no—
37#124×C89—42no—
38#124×C89—42no—
39C4×D4×C89—21no—
40C4×D4×C89—21no—
41C4×D4×C89—21no—
42C4×D4×C89—21no—
43C2×C4⋊C4×C89—21no—
44C2×C4⋊C4×C89—21no—
45C2×C4⋊C4×C89—21no—
46C2×C4⋊C4×C89—21no—
47C2×C4⋊C4×C89D411yescharacteristic
48C2×C4⋊C4×C89D411yescharacteristic
49C2×C4⋊C4×C89D411yescharacteristic
50C2×C4⋊C4×C89D412yes—
51C2×C22⋊C4×C89—22no—
52C2×C22⋊C4×C89—22no—
53C2×C22⋊C4×C89—21no—
54C2×C42×C89—21no—
55C2×C42×C89C2311yescharacteristic
56#106×C89—42no—
57C4⋊C8×C89—21no—
58C4⋊C8×C89—21no—
59C4⋊C8×C89—21no—
60C4⋊C8×C89D412yes—
61#102×C89—42no—
62C4×C8×C89—21no—
63C4×C8×C89—21no—
64#95×C89—22no—
65#95×C89—22no—
66#95×C89—21no—
67C24×C89—21no—
68C2×Q8×C89—41no—
69C2×Q8×C89—41no—
70C2×Q8×C89—41no—
71C2×Q8×C89—41no—
72C2×Q8×C89—41no—
73C2×Q8×C89—41no—
74C2×D4×C89—21no—
75C2×D4×C89—21no—
76C2×D4×C89—21no—
77C2×D4×C89—21no—
78C2×D4×C89—21no—
79C2×D4×C89SD1612yes—
80C22×C4×C89—21no—
81C22×C4×C89—21no—
82C22×C4×C89—21no—
83C22×C4×C89—22no—
84C22×C4×C89—41no—
85C22×C4×C89—41no—
86C22×C4×C89—41no—
87C22×C4×C89—41no—
88C22×C4×C89—41no—
89C22×C4×C89—41no—
90C22×C4×C89—41no—
91C22×C4×C89C2×D411yescharacteristic
92C22×C4×C89C2×D411yescharacteristic
93C22×C4×C89C2×D411yescharacteristic
94C2×C8×C89—22no—
95C2×C8×C89—22no—
96C2×C8×C89—41no—
97C2×C8×C89—41no—
98C2×C8×C89—41no—
99C2×C8×C89—41no—
100C2×C8×C89—41no—
101C2×C8×C89—41no—
102C2×C8×C89—41no—
103C4⋊C4×C89—21no—
104C4⋊C4×C89—22no—
105C4⋊C4×C89—21no—
106C4⋊C4×C89—22no—
107C4⋊C4×C89—21no—
108C4⋊C4×C89—21no—
109C4⋊C4×C89—21no—
110C4⋊C4×C89—21no—
111C4⋊C4×C89—42no—
112C4⋊C4×C89—41no—
113C4⋊C4×C89—41no—
114C4⋊C4×C89—41no—
115C4⋊C4×C89—41no—
116C4⋊C4×C89—41no—
117C4⋊C4×C89—41no—
118C22⋊C4×C89—44no—
119C22⋊C4×C89—44no—
120C22⋊C4×C89—41no—
121C22⋊C4×C89—41no—
122C42×C89—41no—
123C42×C89—41no—
124C42×C89C2×D411yescharacteristic
125C42×C89#4111yescharacteristic
126C42×C89#4111yescharacteristic
127C42×C89#4111yescharacteristic
128C23×C89—22no—
129C23×C89—41no—
130C23×C89—41no—
131C23×C89—41no—
132C23×C89—41no—
133C23×C89—41no—
134C23×C89—41no—
135C23×C89#12011yescharacteristic
136Q8×C89—81no—
137Q8×C89—81no—
138Q8×C89—81no—
139Q8×C89—81no—
140D4×C89—22no—
141D4×C89—22no—
142D4×C89—41no—
143D4×C89—41no—
144C2×C4×C89—21no—
145C2×C4×C89—21no—
146C2×C4×C89—41no—
147C2×C4×C89—42no—
148C2×C4×C89—41no—
149C2×C4×C89—41no—
150C2×C4×C89—42no—
151C2×C4×C89—41no—
152C2×C4×C89—41no—
153C2×C4×C89—42no—
154C2×C4×C89—41no—
155C2×C4×C89—41no—
156C2×C4×C89—41no—
157C2×C4×C89—41no—
158C2×C4×C89—81no—
159C2×C4×C89—81no—
160C2×C4×C89—81no—
161C2×C4×C89—81no—
162C2×C4×C89—81no—
163C2×C4×C89—81no—
164C2×C4×C89—81no—
165C2×C4×C89#12111yescharacteristic
166C2×C4×C89#12111yescharacteristic
167C2×C4×C89#12111yescharacteristic
168C2×C4×C89#12311yescharacteristic
169C2×C4×C89#12311yescharacteristic
170C2×C4×C89#12411yescharacteristic
171C2×C4×C89C2×SD1611yescharacteristic
172C2×C4×C89#13511yescharacteristic
173C2×C4×C89#13511yescharacteristic
174C2×C4×C89#13511yescharacteristic
175C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C89#13711yescharacteristic
179C8×C89—42no—
180C8×C89—42no—
181C8×C89—81no—
182C22×C89—42no—
183C22×C89—42no—
184C22×C89—42no—
185C22×C89—81no—
186C22×C89—81no—
187C22×C89—81no—
188C22×C89—81no—
189C22×C89#39411yescharacteristic
190C22×C89#44911yescharacteristic
191C22×C89#44911yescharacteristic
192C22×C89#45011yescharacteristic
193C22×C89#45211yescharacteristic
194C22×C89#45411yescharacteristic
195C22×C89#45611yescharacteristic
196C4×C89—21no—
197C4×C89—21no—
198C4×C89—81no—
199C4×C89—81no—
200C4×C89—82no—
201C4×C89—81no—
202C4×C89—161no—
203C4×C89#46311yescharacteristic
204C4×C89#46411yescharacteristic
205C4×C89#46511yescharacteristic
206C4×C89#46911yescharacteristic
207C4×C89#48111yescharacteristic
208C4×C89#48211yescharacteristic
209C4×C89#48811yescharacteristic
210C4×C89#48911yescharacteristic
211#10417C8911yesmaximal, characteristic, direct product factor, Sylow
212C2×C89—82no—
213C2×C89#162811yescharacteristic
214C2×C89#163511yescharacteristic
215C2×C89#165011yescharacteristic
216C2×C89#165411yescharacteristic
217C2×C89#203411yescharacteristic
218C2×C89#203811yescharacteristic
219C2×C89#204211yescharacteristic
220#2547C2×C8911yescharacteristic
221#1990C2×C8911yescharacteristic, complemented
222#1948C2×C8911yescharacteristic, complemented
223#1802C2×C8911yescharacteristic
224C2×#332C2×C8911yescharacteristic, complemented
225C2×D4⋊C8C2×C8911yescharacteristic
226#1279C2×C8911yescharacteristic, complemented
227C89#1041711yescharacteristic, direct product factor, Sylow
228C2×C4⋊Q8C22×C8911yescharacteristic
229C2×#124—21no—
230C2×C4×D4C22×C8911yescharacteristic
231C2×#106—21no—
232C2×C4⋊C8C22×C8911yescharacteristic
233C2×C4⋊C8C22×C8911yescharacteristic
234C2×#102—21no—
235C2×C4×C8C22×C8911yescharacteristic
236#397—22no—
237#391—21no—
238#384C22×C8911yescharacteristic
239#380—21no—
240#380—21no—
241C4×C4⋊C4C22×C8911yescharacteristic
242#340—21no—
243#340—21no—
244#340—22no—
245#332—22no—
246D4⋊C8—22no—
247C22×Q8—21no—
248C22×D4D4×C8911yescharacteristic
249C23×C4—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257#124—42no—
258#124—42no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4D4×C8911yescharacteristic
268C2×C4⋊C4D4×C8912yes—
269C2×C4⋊C4D4×C8911yescharacteristic
270C2×C4⋊C4D4×C8911yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42C23×C8911yescharacteristic
276#106—42no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8D4×C8912yes—
281#102—42no—
282C4×C8—21no—
283C4×C8—21no—
284#95—22no—
285#95—22no—
286#95—21no—
287C24—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4SD16×C8912yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4C2×D4×C8911yescharacteristic
312C22×C4C2×D4×C8911yescharacteristic
313C22×C4C2×D4×C8911yescharacteristic
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C42—41no—
343C42—41no—
344C42C2×D4×C8911yescharacteristic
345C42#41×C8911yescharacteristic
346C42#41×C8911yescharacteristic
347C42#41×C8911yescharacteristic
348C23—22no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23#120×C8911yescharacteristic
356Q8—81no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360D4—22no—
361D4—22no—
362D4—41no—
363D4—41no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4#121×C8911yescharacteristic
386C2×C4#121×C8911yescharacteristic
387C2×C4#121×C8911yescharacteristic
388C2×C4#123×C8911yescharacteristic
389C2×C4#123×C8911yescharacteristic
390C2×C4#124×C8911yescharacteristic
391C2×C4C2×SD16×C8911yescharacteristic
392C2×C4#135×C8911yescharacteristic
393C2×C4#135×C8911yescharacteristic
394C2×C4#135×C8911yescharacteristic
395C2×C4AGL(1,ℤ/8)×C8911yescharacteristic
396C2×C4AGL(1,ℤ/8)×C8911yescharacteristic
397C2×C4AGL(1,ℤ/8)×C8911yescharacteristic
398C2×C4#137×C8911yescharacteristic
399C8—42no—
400C8—42no—
401C8—81no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22#394×C8911yescharacteristic
410C22#449×C8911yescharacteristic
411C22#449×C8911yescharacteristic
412C22#450×C8911yescharacteristic
413C22#452×C8911yescharacteristic
414C22#454×C8911yescharacteristic
415C22#456×C8911yescharacteristic
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—82no—
422C4—161no—
423C4#463×C8911yescharacteristic
424C4#464×C8911yescharacteristic
425C4#465×C8911yescharacteristic
426C4#469×C8911yescharacteristic
427C4#481×C8911yescharacteristic
428C4#482×C8911yescharacteristic
429C4#488×C8911yescharacteristic
430C4#489×C8911yescharacteristic
431C2—82no—
432C2C89×#162811yescharacteristic
433C2C89×#163511yescharacteristic
434C2C89×#165011yescharacteristic
435C2C89×#165411yescharacteristic
436C2C89×#203411yescharacteristic
437C2C89×#203811yescharacteristic
438C2C89×#204211yescharacteristic
subgroupG/H
center, socleC23×C89#120
commutator, Frattini, cosocleC2×C42C23×C89
prime radicalC22×D4×C89D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C89 ◁ C2×C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#1990C2semidirect
3. C89×#1948C2semidirect
5. C2×#332×C89C2semidirect
7. C89×#1279C2semidirect
211. #10417C89direct (N × H)
221. #1990C2×C89semidirect
222. #1948C2×C89semidirect
224. C2×#332C2×C89semidirect
226. #1279C2×C89semidirect
227. C89#10417direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#25472
C89×#19902
C89×#19482
C89×#18022
C2×#332×C892
C2×D4⋊C8×C892
C89×#12792
#1041789