Finite Group Databasebeta

← C89×#10420 #36388268

Subgroups — C89×#10420 #36388268

Order 22784.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2542C211yesmaximal, 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×C22⋊Q8×C89—21no—
10C2×C22⋊Q8×C89—21no—
11C2×C4×D4×C89C2211yescharacteristic
12C2×#106×C89—21no—
13C2×C4⋊C8×C89C2211yescharacteristic
14C2×C4⋊C8×C89C2211yescharacteristic
15C2×#102×C89—21no—
16C2×C4×C8×C89C2211yescharacteristic
17#399×C89—22no—
18#390×C89—21no—
19#384×C89—21no—
20#384×C89C2211yescharacteristic
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—
37C22⋊Q8×C89—42no—
38C22⋊Q8×C89—42no—
39C22⋊Q8×C89—42no—
40C22⋊Q8×C89—42no—
41C4×D4×C89—21no—
42C4×D4×C89—21no—
43C4×D4×C89—21no—
44C4×D4×C89—21no—
45C2×C4⋊C4×C89—22no—
46C2×C4⋊C4×C89—22no—
47C2×C4⋊C4×C89—21no—
48C2×C4⋊C4×C89—21no—
49C2×C4⋊C4×C89—21no—
50C2×C4⋊C4×C89—21no—
51C2×C4⋊C4×C89D411yescharacteristic
52C2×C4⋊C4×C89D412yes—
53C2×C4⋊C4×C89D411yescharacteristic
54C2×C4⋊C4×C89D411yescharacteristic
55C2×C22⋊C4×C89—22no—
56C2×C22⋊C4×C89—22no—
57C2×C22⋊C4×C89—21no—
58C2×C42×C89—21no—
59C2×C42×C89C2311yescharacteristic
60#106×C89—42no—
61C4⋊C8×C89—21no—
62C4⋊C8×C89—21no—
63C4⋊C8×C89—21no—
64C4⋊C8×C89D412yes—
65#102×C89—42no—
66C4×C8×C89—21no—
67C4×C8×C89—21no—
68#95×C89—21no—
69C24×C89—21no—
70C2×Q8×C89—41no—
71C2×Q8×C89—41no—
72C2×Q8×C89—41no—
73C2×Q8×C89—41no—
74C2×Q8×C89—41no—
75C2×Q8×C89—41no—
76C2×D4×C89—21no—
77C2×D4×C89—21no—
78C2×D4×C89—21no—
79C2×D4×C89—21no—
80C2×D4×C89—21no—
81C2×D4×C89SD1612yes—
82C22×C4×C89—21no—
83C22×C4×C89—21no—
84C22×C4×C89—22no—
85C22×C4×C89—21no—
86C22×C4×C89—41no—
87C22×C4×C89—41no—
88C22×C4×C89—41no—
89C22×C4×C89—41no—
90C22×C4×C89—41no—
91C22×C4×C89—41no—
92C22×C4×C89—41no—
93C22×C4×C89C2×D411yescharacteristic
94C22×C4×C89C2×D411yescharacteristic
95C22×C4×C89C2×D411yescharacteristic
96C2×C8×C89—22no—
97C2×C8×C89—22no—
98C2×C8×C89—41no—
99C2×C8×C89—41no—
100C2×C8×C89—41no—
101C2×C8×C89—41no—
102C2×C8×C89—41no—
103C2×C8×C89—41no—
104C2×C8×C89—41no—
105C4⋊C4×C89—21no—
106C4⋊C4×C89—21no—
107C4⋊C4×C89—21no—
108C4⋊C4×C89—22no—
109C4⋊C4×C89—21no—
110C4⋊C4×C89—22no—
111C4⋊C4×C89—44no—
112C4⋊C4×C89—44no—
113C4⋊C4×C89—42no—
114C4⋊C4×C89—41no—
115C4⋊C4×C89—41no—
116C4⋊C4×C89—41no—
117C4⋊C4×C89—41no—
118C4⋊C4×C89—41no—
119C4⋊C4×C89—41no—
120C4⋊C4×C89D812yes—
121C4⋊C4×C89SD1612yes—
122C22⋊C4×C89—44no—
123C22⋊C4×C89—44no—
124C22⋊C4×C89—41no—
125C22⋊C4×C89—41no—
126C42×C89—41no—
127C42×C89—41no—
128C42×C89C2×D411yescharacteristic
129C42×C89#4111yescharacteristic
130C42×C89#4111yescharacteristic
131C42×C89#4111yescharacteristic
132C23×C89—22no—
133C23×C89—41no—
134C23×C89—41no—
135C23×C89—41no—
136C23×C89—41no—
137C23×C89—41no—
138C23×C89—41no—
139C23×C89#12011yescharacteristic
140Q8×C89—81no—
141Q8×C89—81no—
142Q8×C89—81no—
143Q8×C89—81no—
144D4×C89—22no—
145D4×C89—22no—
146D4×C89—41no—
147D4×C89—41no—
148C2×C4×C89—21no—
149C2×C4×C89—21no—
150C2×C4×C89—22no—
151C2×C4×C89—22no—
152C2×C4×C89—41no—
153C2×C4×C89—41no—
154C2×C4×C89—41no—
155C2×C4×C89—41no—
156C2×C4×C89—41no—
157C2×C4×C89—42no—
158C2×C4×C89—42no—
159C2×C4×C89—41no—
160C2×C4×C89—42no—
161C2×C4×C89—41no—
162C2×C4×C89—81no—
163C2×C4×C89—81no—
164C2×C4×C89—81no—
165C2×C4×C89—81no—
166C2×C4×C89—81no—
167C2×C4×C89—81no—
168C2×C4×C89—81no—
169C2×C4×C89#12111yescharacteristic
170C2×C4×C89#12111yescharacteristic
171C2×C4×C89#12111yescharacteristic
172C2×C4×C89#12311yescharacteristic
173C2×C4×C89#12311yescharacteristic
174C2×C4×C89#12411yescharacteristic
175C2×C4×C89C2×D811yescharacteristic
176C2×C4×C89C2×SD1611yescharacteristic
177C2×C4×C89C2×SD1611yescharacteristic
178C2×C4×C89#13511yescharacteristic
179C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C89#13711yescharacteristic
182C2×C4×C89#13711yescharacteristic
183C8×C89—42no—
184C8×C89—42no—
185C8×C89—81no—
186C22×C89—42no—
187C22×C89—42no—
188C22×C89—42no—
189C22×C89—81no—
190C22×C89—81no—
191C22×C89—81no—
192C22×C89—81no—
193C22×C89#39411yescharacteristic
194C22×C89#44711yescharacteristic
195C22×C89#44811yescharacteristic
196C22×C89#45011yescharacteristic
197C22×C89#45211yescharacteristic
198C22×C89#45411yescharacteristic
199C22×C89#45611yescharacteristic
200C4×C89—21no—
201C4×C89—21no—
202C4×C89—42no—
203C4×C89—42no—
204C4×C89—81no—
205C4×C89—82no—
206C4×C89—161no—
207C4×C89#45911yescharacteristic
208C4×C89#46011yescharacteristic
209C4×C89#46511yescharacteristic
210C4×C89#47011yescharacteristic
211C4×C89#48311yescharacteristic
212C4×C89#48511yescharacteristic
213C4×C89#48611yescharacteristic
214C4×C89#49011yescharacteristic
215#10420C8911yesmaximal, characteristic, direct product factor, Sylow
216C2×C89—82no—
217C2×C89#162711yescharacteristic
218C2×C89#163511yescharacteristic
219C2×C89#164711yescharacteristic
220C2×C89#165111yescharacteristic
221C2×C89#203511yescharacteristic
222C2×C89#203811yescharacteristic
223C2×C89#204111yescharacteristic
224#2542C2×C8911yescharacteristic
225#1990C2×C8911yescharacteristic, complemented
226#1948C2×C8911yescharacteristic, complemented
227#1802C2×C8911yescharacteristic
228C2×#332C2×C8911yescharacteristic, complemented
229C2×D4⋊C8C2×C8911yescharacteristic
230#1279C2×C8911yescharacteristic, complemented
231C89#1042011yescharacteristic, direct product factor, Sylow
232C2×C4⋊Q8C22×C8911yescharacteristic
233C2×C22⋊Q8—21no—
234C2×C22⋊Q8—21no—
235C2×C4×D4C22×C8911yescharacteristic
236C2×#106—21no—
237C2×C4⋊C8C22×C8911yescharacteristic
238C2×C4⋊C8C22×C8911yescharacteristic
239C2×#102—21no—
240C2×C4×C8C22×C8911yescharacteristic
241#399—22no—
242#390—21no—
243#384—21no—
244#384C22×C8911yescharacteristic
245C4×C4⋊C4C22×C8911yescharacteristic
246#340—21no—
247#340—21no—
248#340—22no—
249#332—22no—
250D4⋊C8—22no—
251C22×Q8—21no—
252C22×D4D4×C8911yescharacteristic
253C23×C4—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C22⋊Q8—42no—
262C22⋊Q8—42no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4D4×C8911yescharacteristic
276C2×C4⋊C4D4×C8912yes—
277C2×C4⋊C4D4×C8911yescharacteristic
278C2×C4⋊C4D4×C8911yescharacteristic
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C42—21no—
283C2×C42C23×C8911yescharacteristic
284#106—42no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8D4×C8912yes—
289#102—42no—
290C4×C8—21no—
291C4×C8—21no—
292#95—21no—
293C24—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4SD16×C8912yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4C2×D4×C8911yescharacteristic
318C22×C4C2×D4×C8911yescharacteristic
319C22×C4C2×D4×C8911yescharacteristic
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4D8×C8912yes—
345C4⋊C4SD16×C8912yes—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C42—41no—
351C42—41no—
352C42C2×D4×C8911yescharacteristic
353C42#41×C8911yescharacteristic
354C42#41×C8911yescharacteristic
355C42#41×C8911yescharacteristic
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23#120×C8911yescharacteristic
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368D4—22no—
369D4—22no—
370D4—41no—
371D4—41no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4#121×C8911yescharacteristic
394C2×C4#121×C8911yescharacteristic
395C2×C4#121×C8911yescharacteristic
396C2×C4#123×C8911yescharacteristic
397C2×C4#123×C8911yescharacteristic
398C2×C4#124×C8911yescharacteristic
399C2×C4C2×D8×C8911yescharacteristic
400C2×C4C2×SD16×C8911yescharacteristic
401C2×C4C2×SD16×C8911yescharacteristic
402C2×C4#135×C8911yescharacteristic
403C2×C4AGL(1,ℤ/8)×C8911yescharacteristic
404C2×C4AGL(1,ℤ/8)×C8911yescharacteristic
405C2×C4#137×C8911yescharacteristic
406C2×C4#137×C8911yescharacteristic
407C8—42no—
408C8—42no—
409C8—81no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#394×C8911yescharacteristic
418C22#447×C8911yescharacteristic
419C22#448×C8911yescharacteristic
420C22#450×C8911yescharacteristic
421C22#452×C8911yescharacteristic
422C22#454×C8911yescharacteristic
423C22#456×C8911yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—42no—
428C4—81no—
429C4—82no—
430C4—161no—
431C4#459×C8911yescharacteristic
432C4#460×C8911yescharacteristic
433C4#465×C8911yescharacteristic
434C4#470×C8911yescharacteristic
435C4#483×C8911yescharacteristic
436C4#485×C8911yescharacteristic
437C4#486×C8911yescharacteristic
438C4#490×C8911yescharacteristic
439C2—82no—
440C2C89×#162711yescharacteristic
441C2C89×#163511yescharacteristic
442C2C89×#164711yescharacteristic
443C2C89×#165111yescharacteristic
444C2C89×#203511yescharacteristic
445C2C89×#203811yescharacteristic
446C2C89×#204111yescharacteristic
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
215. #10420C89direct (N × H)
225. #1990C2×C89semidirect
226. #1948C2×C89semidirect
228. C2×#332C2×C89semidirect
230. #1279C2×C89semidirect
231. C89#10420direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#25422
C89×#19902
C89×#19482
C89×#18022
C2×#332×C892
C2×D4⋊C8×C892
C89×#12792
#1042089