Finite Group Databasebeta

← #36362844

Subgroups — #36362844

Order 22144.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149727C211yesmaximal, characteristic, complemented
2C2×#19149706C211yesmaximal, characteristic, complemented
3C2×SD16×D173C211yesmaximal, characteristic, complemented
4#26012057C211yesmaximal, characteristic, complemented
5#26011639C211yesmaximal, characteristic, complemented
6#26011638C211yesmaximal, characteristic, complemented
7#26011635C211yesmaximal, characteristic, complemented
8#26011634C211yesmaximal, characteristic, complemented
9#26011631C211yesmaximal, characteristic, complemented
10#26011630C211yesmaximal, characteristic, complemented
11#26011628C211yesmaximal, characteristic, complemented
12#26011498C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#448×C173C211yesmaximal, characteristic, complemented
16#19149727—24no—
17#19149727—24no—
18C2×#14583827—21no—
19C2×#14583827C2211yescharacteristic
20C2×#14583827C2211yescharacteristic, complemented
21C2×#14583826—21no—
22C2×#14583826—21no—
23C2×#14583826C2211yescharacteristic
24C2×#14583826C2211yescharacteristic, complemented
25C2×Q8×D173—21no—
26C2×Q8×D173C2211yescharacteristic
27C2×Q8×D173C2211yescharacteristic, complemented
28C2×D4×D173C2211yescharacteristic
29#19149706—24no—
30C2×#14583815C2211yescharacteristic
31C2×#14583814C2211yescharacteristic
32C2×#14583813C2211yescharacteristic
33C2×#14583813C2211yescharacteristic
34SD16×D173—24no—
35C2×D1384C2211yescharacteristic
36C2×#14583811C2211yescharacteristic
37C2×#14583809C2211yescharacteristic
38C2×C8×D173C2211yescharacteristic
39#19149675—21no—
40#19149665—21no—
41#19149654C2211yescharacteristic
42#19149653C2211yescharacteristic
43#19149650C2211yescharacteristic
44#19149649—21no—
45#19149647—21no—
46#19149637C2211yescharacteristic
47#19149604C2211yescharacteristic, complemented
48#19149603C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52#19149599C2211yescharacteristic, complemented
53C22⋊#13063436C2211yescharacteristic, complemented
54#19149591C2211yescharacteristic, complemented
55#19149591C2211yescharacteristic, complemented
56C22×Q8×C173C2211yescharacteristic, complemented
57C2×SD16×C173C2211yescharacteristic
58C2×SD16×C173C2211yescharacteristic
59#121×C173C2211yescharacteristic
60#103×C173C2211yescharacteristic, complemented
61#103×C173C2211yescharacteristic, complemented
62C22⋊C8×C173C2211yescharacteristic, complemented
63C23×D173—21no—
64#14583827—28no—
65#14583827—22no—
66#14583827—22no—
67#14583827—22no—
68#14583827D414yescomplemented
69#14583826—28no—
70#14583826—28no—
71#14583826—22no—
72#14583826—22no—
73#14583826—22no—
74#14583826—22no—
75C2×Dic346—21no—
76C2×Dic346—21no—
77C2×Dic346D411yescharacteristic
78C2×Dic346C2311yescharacteristic
79Q8×D173—28no—
80Q8×D173—22no—
81Q8×D173—22no—
82Q8×D173—22no—
83Q8×D173D414yescomplemented
84C2×D692—21no—
85C2×D692—21no—
86C2×D692—21no—
87C2×D692D411yescharacteristic
88C2×D692C2311yescharacteristic
89C2×D692C2311yescharacteristic
90C2×C22⋊D173—21no—
91C2×C22⋊D173—21no—
92C2×C22⋊D173—21no—
93C2×C22⋊D173—21no—
94C2×C22⋊D173D411yescharacteristic
95C2×C22⋊D173D411yescharacteristic
96D4×D173—22no—
97D4×D173—22no—
98C2×C4×D173—21no—
99C2×C4×D173—21no—
100C2×C4×D173—21no—
101C2×C4×D173—21no—
102C2×C4×D173—21no—
103C2×C4×D173C2311yescharacteristic
104C2×C4×D173C2311yescharacteristic
105#14583815—22no—
106#14583814—22no—
107#14583813—22no—
108#14583813—22no—
109D1384—22no—
110#14583811—22no—
111#14583809—22no—
112C2×#13063436C2311yescharacteristic
113C2×#13063436C2311yescharacteristic
114C8×D173—22no—
115C4⋊Dic173C2311yescharacteristic
116C22⋊Dic173—21no—
117#14583803—21no—
118#14583803—21no—
119#14583803—21no—
120C4×Dic173—21no—
121C2×Q8×C173—22no—
122C2×Q8×C173—22no—
123C2×Q8×C173—21no—
124C2×Q8×C173C2311yescharacteristic
125C2×Q8×C173C2311yescharacteristic
126C2×D4×C173—21no—
127C2×D4×C173C2311yescharacteristic
128C22×C4×C173—21no—
129C22×C4×C173C2311yescharacteristic
130SD16×C173—22no—
131SD16×C173—22no—
132C2×C8×C173C2311yescharacteristic
133C2×C8×C173C2311yescharacteristic
134C4⋊C4×C173C2311yescharacteristic
135C22⋊C4×C173—21no—
136C22×D173—21no—
137C22×D173—21no—
138C22×D173—42no—
139C22×D173—81no—
140C22×D173—81no—
141C22×D173C2×D411yescharacteristic
142C22×D173C2×D411yescharacteristic
143Dic346—24no—
144Dic346—21no—
145Dic346—22no—
146Dic346—24no—
147Dic346C2×D412yes—
148D692—24no—
149D692—24no—
150D692—22no—
151D692—21no—
152D692—21no—
153D692—21no—
154D692—81no—
155D692C2×D412yes—
156C22⋊D173—24no—
157C22⋊D173—22no—
158C22⋊D173—22no—
159C22⋊D173—42no—
160C22⋊D173—81no—
161C22⋊D173—81no—
162C2×Dic173—21no—
163C2×Dic173—21no—
164C2×Dic173C2×D411yescharacteristic
165C2×Dic173C2×D411yescharacteristic
166C4×D173—24no—
167C4×D173—24no—
168C4×D173—24no—
169C4×D173—24no—
170C4×D173—21no—
171C4×D173—21no—
172C4×D173—24no—
173C4×D173C2×D412yes—
174C4×D173C2×D412yes—
175#13063436—21no—
176#13063436—21no—
177C23×C173—21no—
178C23×C173C2×D411yescharacteristic
179Q8×C173—24no—
180Q8×C173—21no—
181Q8×C173—21no—
182Q8×C173C2×D412yes—
183Q8×C173C2×D412yes—
184D4×C173—21no—
185D4×C173—21no—
186D4×C173—81no—
187C2×C4×C173—24no—
188C2×C4×C173—21no—
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173—21no—
193C2×C4×C173C2×D411yescharacteristic
194C2×C4×C173C2411yescharacteristic
195C8×C173—21no—
196C8×C173—21no—
197C22×C173—21no—
198C22×C173—21no—
199C22×C173—21no—
200C22×C173—41no—
201C22×C173—41no—
202C22×C173—41no—
203C22×C173C22×D411yescharacteristic
204C2×D173—21no—
205C2×D173—21no—
206C2×D173—21no—
207C2×D173—22no—
208C2×D173—21no—
209C2×D173—21no—
210C2×D173—41no—
211C2×D173—41no—
212C2×D173—41no—
213C2×D173—41no—
214C2×D173—41no—
215C2×D173—84no—
216C2×D173#12012yes—
217C4×C173—22no—
218C4×C173—22no—
219C4×C173—21no—
220C4×C173—41no—
221C4×C173C22×D411yescharacteristic
222C4×C173C22×D411yescharacteristic
223Dic173—22no—
224Dic173—21no—
225Dic173—41no—
226Dic173#12012yes—
227C2×C173—41no—
228C2×C173—81no—
229C2×C173C2×#12011yescharacteristic
230C2×C173#57711yescharacteristic
231C2×C173#57711yescharacteristic
232D173—22no—
233D173—41no—
234D173—42no—
235D173—81no—
236C173#301011yescharacteristic, complemented, Sylow
237#3010—1731nomaximal, Sylow
238C2×2−1+4—1731no—
239C2×AGL(1,ℤ/8)—1731no—
240C22×SD16—1731no—
241#535—1731no—
242#449—1731no—
243#449—1731no—
244#449—1731no—
245#449—1731no—
246#448—1731no—
247#448—1731no—
248#448—1731no—
249#448D17311yescharacteristic, complemented
250#419—1731no—
251#416—1731no—
252#408—1731no—
2532−1+4—3464no—
2542−1+4—3464no—
255C2×#41—3461no—
256C2×#41—3461no—
257C2×#41—3461no—
258C2×#41—1731no—
259C2×#41—1731no—
260C2×#41—1731no—
261C2×#41—1731no—
262C22×Q8—3461no—
263C22×Q8—1731no—
264C22×Q8—1731no—
265C22×Q8C2×D17311yescharacteristic, complemented
266C22×D4—1731no—
267AGL(1,ℤ/8)—3464no—
268C2×SD16—3464no—
269C2×SD16—1731no—
270C2×SD16—1731no—
271C2×SD16—1731no—
272C2×SD16—1731no—
273C2×SD16C2×D17311yescharacteristic
274C2×SD16C2×D17311yescharacteristic
275C2×D8—1731no—
276C2×D8—1731no—
277C2×M16—1731no—
278C22×C8—1731no—
279#127—3461no—
280#124—3461no—
281#121—1731no—
282#121—1731no—
283#121—1731no—
284#121C2×D17311yescharacteristic
285#120—3461no—
286#120—3461no—
287#117—1731no—
288#103—1731no—
289#103—1731no—
290#103—1731no—
291#103—1731no—
292#103C2×D17311yescharacteristic, complemented
293#103C2×D17311yescharacteristic, complemented
294#102—1731no—
295#102—1731no—
296C22⋊C8—1731no—
297C22⋊C8—1731no—
298C22⋊C8—1731no—
299C22⋊C8C2×D17311yescharacteristic, complemented
300C24—3461no—
301#41—3468no—
302#41—3468no—
303#41—3462no—
304#41—3462no—
305#41—3462no—
306#41—3462no—
307#41—3468no—
308#41—3462no—
309#41—3462no—
310#41—3462no—
311#41—1734no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—3461no—
316C2×Q8—3461no—
317C2×Q8—3468no—
318C2×Q8—3462no—
319C2×Q8—3462no—
320C2×Q8—3462no—
321C2×Q8—1734no—
322C2×Q8—1731no—
323C2×Q8—1731no—
324C2×Q8C22×D17311yescharacteristic
325C2×Q8C22×D17311yescharacteristic
326C2×D4—21no—
327C2×D4—3461no—
328C2×D4—3461no—
329C2×D4—3461no—
330C2×D4—3461no—
331C2×D4—3461no—
332C2×D4—3461no—
333C2×D4—3462no—
334C2×D4—3462no—
335C2×D4—3461no—
336C2×D4—1731no—
337C2×D4—1731no—
338C2×D4—1731no—
339C2×D4—1731no—
340C2×D4—1731no—
341C2×D4C22×D17311yescharacteristic
342C22×C4—21no—
343C22×C4—3461no—
344C22×C4—3461no—
345C22×C4—3461no—
346C22×C4—3461no—
347C22×C4—3461no—
348C22×C4—1731no—
349C22×C4—1731no—
350C22×C4C22×D17311yescharacteristic
351SD16—22no—
352SD16—22no—
353SD16—3462no—
354SD16—3462no—
355SD16—3462no—
356SD16—3462no—
357D8—3462no—
358D8—3462no—
359M16—3462no—
360C2×C8—3462no—
361C2×C8—1731no—
362C2×C8—1731no—
363C2×C8C22×D17311yescharacteristic
364C2×C8C22×D17311yescharacteristic
365C4⋊C4—1731no—
366C4⋊C4C22×D17311yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—3461no—
369C22⋊C4—3461no—
370C22⋊C4—3461no—
371C22⋊C4—3461no—
372C42—3461no—
373C23—21no—
374C23—3461no—
375C23—3461no—
376C23—13841no—
377C23—13841no—
378C23—1731no—
379C23—1731no—
380C23—6922no—
381C23D4×D17311yescharacteristic
382Q8—24no—
383Q8—21no—
384Q8—21no—
385Q8—3464no—
386Q8—3464no—
387Q8—3462no—
388Q8—3461no—
389Q8—1732no—
390Q8D4×D17312yes—
391Q8D4×D17312yes—
392D4—21no—
393D4—21no—
394D4—81no—
395D4—3464no—
396D4—3464no—
397D4—3462no—
398D4—3464no—
399D4—3462no—
400D4—3462no—
401D4—3461no—
402D4—3461no—
403D4—3461no—
404D4—13841no—
405D4—13841no—
406D4—13841no—
407D4—1732no—
408D4—6922no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—3464no—
416C2×C4—3464no—
417C2×C4—3464no—
418C2×C4—3464no—
419C2×C4—3464no—
420C2×C4—3461no—
421C2×C4—3461no—
422C2×C4—3461no—
423C2×C4—3461no—
424C2×C4—1731no—
425C2×C4—1731no—
426C2×C4—1732no—
427C2×C4—1732no—
428C2×C4D4×D17311yescharacteristic
429C2×C4C23×D17311yescharacteristic
430C8—21no—
431C8—21no—
432C8—3461no—
433C8—3461no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—3461no—
441C22—3461no—
442C22—3461no—
443C22—3462no—
444C22—3461no—
445C22—3461no—
446C22—13844no—
447C22—1732no—
448C22—6921no—
449C22—6921no—
450C22—6921no—
451C22—6921no—
452C22—6921no—
453C22C2×D4×D17311yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—41no—
458C4—3462no—
459C4—3461no—
460C4—1732no—
461C4—6921no—
462C4C2×D4×D17311yescharacteristic
463C4C2×D4×D17311yescharacteristic
464C2—41no—
465C2—81no—
466C2—3462no—
467C2—13841no—
468C2—6921no—
469C2—6922no—
470C2#120×D17311yescharacteristic
471C2#2601229711yescharacteristic
472C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
hypercenter#448D173
hypocenterC173#3010
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#19149727C2semidirect
2. C2×#19149706C2semidirect
3. C2×SD16×D173C2semidirect
4. #26012057C2semidirect
5. #26011639C2semidirect
6. #26011638C2semidirect
7. #26011635C2semidirect
8. #26011634C2semidirect
9. #26011631C2semidirect
10. #26011630C2semidirect
11. #26011628C2semidirect
12. #26011498C2semidirect
13. #26011486C2semidirect
14. #26011456C2semidirect
15. #448×C173C2semidirect
20. C2×#14583827C22semidirect
24. C2×#14583826C22semidirect
27. C2×Q8×D173C22semidirect
47. #19149604C22semidirect
48. #19149603C22semidirect
49. #19149603C22semidirect
50. #19149602C22semidirect
51. #19149601C22semidirect
52. #19149599C22semidirect
53. C22⋊#13063436C22semidirect
54. #19149591C22semidirect
55. #19149591C22semidirect
56. C22×Q8×C173C22semidirect
60. #103×C173C22semidirect
61. #103×C173C22semidirect
62. C22⋊C8×C173C22semidirect
68. #14583827D4semidirect
83. Q8×D173D4semidirect
236. C173#3010semidirect
249. #448D173semidirect
265. C22×Q8C2×D173semidirect
292. #103C2×D173semidirect
293. #103C2×D173semidirect
299. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497272
C2×#191497062
C2×SD16×D1732
#260120572
#260116392
#260116382
#260116352
#260116342
#260116312
#260116302
#260116282
#260114982
#260114862
#260114562
#448×C1732
#3010173