Finite Group Databasebeta

← #36362835

Subgroups — #36362835

Order 22144.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149724C211yesmaximal, characteristic
2C2×#19149708C211yesmaximal, characteristic, complemented
3C2×SD16×D173C211yesmaximal, characteristic, complemented
4#26012064C211yesmaximal, characteristic, complemented
5#26011655C211yesmaximal, characteristic, complemented
6#26011654C211yesmaximal, characteristic, complemented
7#26011651C211yesmaximal, characteristic, complemented
8#26011650C211yesmaximal, characteristic, complemented
9#26011643C211yesmaximal, characteristic, complemented
10#26011642C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011494C211yesmaximal, characteristic, complemented
13#26011484C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16#19149724—24no—
17#19149724—24no—
18C2×#14583826C2211yescharacteristic
19C2×#14583826C2211yescharacteristic
20C2×#14583825—21no—
21C2×#14583825C2211yescharacteristic
22C2×#14583825C2211yescharacteristic
23C2×Q8×D173C2211yescharacteristic, complemented
24C22×C22⋊D173—21no—
25C22×C22⋊D173—21no—
26C2×D4×D173—21no—
27C2×D4×D173C2211yescharacteristic
28C2×D4×D173C2211yescharacteristic
29#19149708—24no—
30C2×Dic692C2211yescharacteristic, complemented
31C2×#14583816C2211yescharacteristic, complemented
32C2×#14583815C2211yescharacteristic, complemented
33C2×#14583814C2211yescharacteristic, complemented
34C2×#14583814C2211yescharacteristic, complemented
35C2×#14583813C2211yescharacteristic, complemented
36SD16×D173—24no—
37C2×#14583809C2211yescharacteristic, complemented
38C2×C8×D173C2211yescharacteristic, complemented
39#19149678—21no—
40#19149665—21no—
41#19149664—21no—
42#19149661—21no—
43#19149659C2211yescharacteristic, complemented
44#19149658C2211yescharacteristic, complemented
45#19149655C2211yescharacteristic, complemented
46#19149637C2211yescharacteristic, complemented
47#19149604C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic, complemented
50#19149600C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic, complemented
52#19149599C2211yescharacteristic, complemented
53C22⋊#13063436C2211yescharacteristic, complemented
54#19149591C2211yescharacteristic, complemented
55#19149591C2211yescharacteristic, complemented
56C22×D4×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic, complemented
58C2×SD16×C173C2211yescharacteristic, complemented
59C22⋊Q8×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61#102×C173C2211yescharacteristic, complemented
62C22⋊C8×C173C2211yescharacteristic, complemented
63C23×D173—21no—
64C23×D173—21no—
65#14583826—22no—
66#14583826—22no—
67#14583826—22no—
68#14583826—22no—
69#14583825—28no—
70#14583825—22no—
71#14583825—22no—
72#14583825—22no—
73#14583825D414yes—
74C2×Dic346—21no—
75C2×Dic346D411yescharacteristic
76C2×Dic346C2311yescharacteristic
77C2×Dic346C2311yescharacteristic
78Q8×D173—22no—
79Q8×D173—22no—
80C2×D692D411yescharacteristic
81C2×D692C2311yescharacteristic
82C2×C22⋊D173—28no—
83C2×C22⋊D173—28no—
84C2×C22⋊D173—21no—
85C2×C22⋊D173—21no—
86C2×C22⋊D173—21no—
87C2×C22⋊D173—21no—
88C2×C22⋊D173—21no—
89C2×C22⋊D173D411yescharacteristic
90C2×C22⋊D173D411yescharacteristic
91D4×D173—28no—
92D4×D173—22no—
93D4×D173—22no—
94D4×D173—22no—
95D4×D173D414yes—
96C22×Dic173—21no—
97C22×Dic173—21no—
98C2×C4×D173—21no—
99C2×C4×D173—21no—
100C2×C4×D173C2311yescharacteristic
101C2×C4×D173C2311yescharacteristic
102Dic692—22no—
103#14583816—22no—
104#14583815—22no—
105#14583814—22no—
106#14583814—22no—
107#14583813—22no—
108#14583809—22no—
109C2×#13063436C2311yescharacteristic
110C2×#13063436C2311yescharacteristic
111C8×D173—22no—
112C4⋊Dic173—21no—
113C4⋊Dic173C2311yescharacteristic
114#14583805—21no—
115#14583805—21no—
116C22⋊Dic173—21no—
117#14583803—21no—
118#14583803—21no—
119#14583803—21no—
120C4×Dic173—21no—
121C24×C173—21no—
122C2×Q8×C173C2311yescharacteristic
123C2×D4×C173—22no—
124C2×D4×C173—22no—
125C2×D4×C173—21no—
126C2×D4×C173C2311yescharacteristic
127C2×D4×C173C2311yescharacteristic
128C22×C4×C173C2311yescharacteristic
129SD16×C173—22no—
130SD16×C173—22no—
131C2×C8×C173C2311yescharacteristic
132C2×C8×C173C2311yescharacteristic
133C4⋊C4×C173—21no—
134C4⋊C4×C173C2311yescharacteristic
135C22⋊C4×C173—21no—
136C22×D173—21no—
137C22×D173—24no—
138C22×D173—24no—
139C22×D173—81no—
140C22×D173—81no—
141C22×D173—81no—
142C22×D173—81no—
143C22×D173C2×D411yescharacteristic
144C22×D173C2×D411yescharacteristic
145Dic346—21no—
146Dic346—22no—
147Dic346—21no—
148Dic346—21no—
149Dic346—42no—
150Dic346C2×D412yes—
151D692—22no—
152D692—21no—
153D692C2×D412yes—
154C22⋊D173—24no—
155C22⋊D173—24no—
156C22⋊D173—24no—
157C22⋊D173—24no—
158C22⋊D173—22no—
159C22⋊D173—24no—
160C22⋊D173—22no—
161C2×Dic173—24no—
162C2×Dic173—21no—
163C2×Dic173—24no—
164C2×Dic173—21no—
165C2×Dic173—21no—
166C2×Dic173C2×D411yescharacteristic
167C2×Dic173C2×D411yescharacteristic
168C4×D173—24no—
169C4×D173—21no—
170C4×D173—21no—
171C4×D173—42no—
172C4×D173C2×D412yes—
173C4×D173C2×D412yes—
174#13063436—21no—
175#13063436—21no—
176C23×C173—24no—
177C23×C173—21no—
178C23×C173—21no—
179C23×C173—81no—
180C23×C173—81no—
181C23×C173C2×D411yescharacteristic
182Q8×C173—21no—
183Q8×C173—21no—
184D4×C173—24no—
185D4×C173—21no—
186D4×C173—21no—
187D4×C173C2×D412yes—
188D4×C173C2×D412yes—
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—22no—
199C22×C173—22no—
200C22×C173—21no—
201C22×C173—21no—
202C22×C173—41no—
203C22×C173—41no—
204C22×C173—41no—
205C22×C173—41no—
206C22×C173—84no—
207C22×C173C22×D411yescharacteristic
208C2×D173—21no—
209C2×D173—21no—
210C2×D173—21no—
211C2×D173—22no—
212C2×D173—21no—
213C2×D173—21no—
214C2×D173—41no—
215C2×D173—41no—
216C2×D173—84no—
217C2×D173—84no—
218C2×D173#12012yes—
219C4×C173—21no—
220C4×C173—41no—
221C4×C173—41no—
222C4×C173C22×D411yescharacteristic
223C4×C173C22×D411yescharacteristic
224Dic173—22no—
225Dic173—21no—
226Dic173—41no—
227Dic173—41no—
228Dic173#12012yes—
229C2×C173—42no—
230C2×C173—42no—
231C2×C173—41no—
232C2×C173C2×#12011yescharacteristic
233C2×C173#57711yescharacteristic
234C2×C173#57711yescharacteristic
235D173—22no—
236D173—41no—
237D173—42no—
238C173#300911yescharacteristic, complemented, Sylow
239#3009—1731nomaximal, Sylow
240C2×H(5,𝔽2)—1731no—
241C2×#137—1731no—
242C22×SD16—1731no—
243#536—1731no—
244#452—1731no—
245#452—1731no—
246#452—1731no—
247#452—1731no—
248#450—1731no—
249#450—1731no—
250#450—1731no—
251#450D17311yescharacteristic, complemented
252#418—1731no—
253#416—1731no—
254#408—1731no—
255H(5,𝔽2)—3464no—
256H(5,𝔽2)—3464no—
257C2×#41—3461no—
258C2×#41—1731no—
259C2×#41—1731no—
260C2×#41—1731no—
261C2×#41—1731no—
262C22×Q8—1731no—
263C22×D4—3461no—
264C22×D4—3461no—
265C22×D4—3461no—
266C22×D4—1731no—
267C22×D4—1731no—
268C22×D4C2×D17311yescharacteristic
269#137—3464no—
270C2×Q16—1731no—
271C2×Q16—1731no—
272C2×SD16—3464no—
273C2×SD16—1731no—
274C2×SD16—1731no—
275C2×SD16—1731no—
276C2×SD16—1731no—
277C2×SD16C2×D17311yescharacteristic, complemented
278C2×SD16C2×D17311yescharacteristic, complemented
279C2×M16—1731no—
280C22×C8—1731no—
281C4⋊Q8—3461no—
282#124—3461no—
283#123—3461no—
284#123—3461no—
285C22⋊Q8—1731no—
286C22⋊Q8—1731no—
287C22⋊Q8—1731no—
288C22⋊Q8C2×D17311yescharacteristic, complemented
289#117—1731no—
290#103—1731no—
291#103—1731no—
292#102—1731no—
293#102—1731no—
294#102—1731no—
295#102—1731no—
296#102C2×D17311yescharacteristic, complemented
297#102C2×D17311yescharacteristic, complemented
298C22⋊C8—1731no—
299C22⋊C8—1731no—
300C22⋊C8—1731no—
301C22⋊C8C2×D17311yescharacteristic, complemented
302C24—21no—
303C24—3461no—
304C24—3461no—
305#41—3462no—
306#41—3462no—
307#41—3462no—
308#41—3462no—
309#41—3468no—
310#41—3462no—
311#41—3462no—
312#41—3462no—
313#41—1734no—
314C2×Q8—3461no—
315C2×Q8—3462no—
316C2×Q8—3462no—
317C2×Q8—1731no—
318C2×Q8—1731no—
319C2×Q8—1731no—
320C2×Q8C22×D17311yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—3468no—
325C2×D4—3462no—
326C2×D4—3468no—
327C2×D4—3468no—
328C2×D4—3461no—
329C2×D4—3461no—
330C2×D4—3461no—
331C2×D4—3462no—
332C2×D4—3462no—
333C2×D4—3461no—
334C2×D4—3461no—
335C2×D4—1731no—
336C2×D4—1731no—
337C2×D4—1734no—
338C2×D4—1731no—
339C2×D4—1731no—
340C2×D4C22×D17311yescharacteristic
341C2×D4C22×D17311yescharacteristic
342C22×C4—3461no—
343C22×C4—3461no—
344C22×C4—3461no—
345C22×C4—3461no—
346C22×C4—1731no—
347C22×C4—1731no—
348C22×C4C22×D17311yescharacteristic
349Q16—3462no—
350Q16—3462no—
351SD16—22no—
352SD16—22no—
353SD16—3462no—
354SD16—3462no—
355SD16—3462no—
356SD16—3462no—
357M16—3462no—
358C2×C8—3462no—
359C2×C8—1731no—
360C2×C8—1731no—
361C2×C8C22×D17311yescharacteristic
362C2×C8C22×D17311yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—3461no—
365C4⋊C4—3461no—
366C4⋊C4—3461no—
367C4⋊C4—1731no—
368C4⋊C4C22×D17311yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—3461no—
371C22⋊C4—3461no—
372C22⋊C4—3461no—
373C22⋊C4—3461no—
374C42—3461no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—3464no—
381C23—3464no—
382C23—3461no—
383C23—13841no—
384C23—13841no—
385C23—13841no—
386C23—13841no—
387C23—1731no—
388C23—1731no—
389C23D4×D17311yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—3462no—
393Q8—3461no—
394Q8—3461no—
395Q8—3461no—
396Q8—1732no—
397Q8—6922no—
398D4—24no—
399D4—21no—
400D4—21no—
401D4—3462no—
402D4—3464no—
403D4—3464no—
404D4—3464no—
405D4—3464no—
406D4—3462no—
407D4—3462no—
408D4—3464no—
409D4—3461no—
410D4—1732no—
411D4D4×D17312yes—
412D4D4×D17312yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—3464no—
418C2×C4—3464no—
419C2×C4—3461no—
420C2×C4—3464no—
421C2×C4—3461no—
422C2×C4—3461no—
423C2×C4—3461no—
424C2×C4—3461no—
425C2×C4—1731no—
426C2×C4—1731no—
427C2×C4—1732no—
428C2×C4—1732no—
429C2×C4—6922no—
430C2×C4D4×D17311yescharacteristic
431C2×C4C23×D17311yescharacteristic
432C8—21no—
433C8—21no—
434C8—3461no—
435C8—3461no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—84no—
446C22—3461no—
447C22—3461no—
448C22—3461no—
449C22—3462no—
450C22—3461no—
451C22—3461no—
452C22—13844no—
453C22—13844no—
454C22—1732no—
455C22—6921no—
456C22—6921no—
457C22C2×D4×D17311yescharacteristic
458C4—21no—
459C4—41no—
460C4—41no—
461C4—3462no—
462C4—3461no—
463C4—1732no—
464C4—6921no—
465C4—6921no—
466C4C2×D4×D17311yescharacteristic
467C4C2×D4×D17311yescharacteristic
468C2—41no—
469C2—42no—
470C2—42no—
471C2—3462no—
472C2—6921no—
473C2—6922no—
474C2#120×D17311yescharacteristic
475C2#2601229711yescharacteristic
476C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149724C2
hypercenter#450D173
hypocenterC173#3009
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#19149708C2semidirect
3. C2×SD16×D173C2semidirect
4. #26012064C2semidirect
5. #26011655C2semidirect
6. #26011654C2semidirect
7. #26011651C2semidirect
8. #26011650C2semidirect
9. #26011643C2semidirect
10. #26011642C2semidirect
11. #26011640C2semidirect
12. #26011494C2semidirect
13. #26011484C2semidirect
14. #26011456C2semidirect
15. #450×C173C2semidirect
23. C2×Q8×D173C22semidirect
30. C2×Dic692C22semidirect
31. C2×#14583816C22semidirect
32. C2×#14583815C22semidirect
33. C2×#14583814C22semidirect
34. C2×#14583814C22semidirect
35. C2×#14583813C22semidirect
37. C2×#14583809C22semidirect
38. C2×C8×D173C22semidirect
43. #19149659C22semidirect
44. #19149658C22semidirect
45. #19149655C22semidirect
46. #19149637C22semidirect
47. #19149604C22semidirect
48. #19149602C22semidirect
49. #19149601C22semidirect
50. #19149600C22semidirect
51. #19149600C22semidirect
52. #19149599C22semidirect
53. C22⋊#13063436C22semidirect
54. #19149591C22semidirect
55. #19149591C22semidirect
57. C2×SD16×C173C22semidirect
58. C2×SD16×C173C22semidirect
59. C22⋊Q8×C173C22semidirect
60. #102×C173C22semidirect
61. #102×C173C22semidirect
62. C22⋊C8×C173C22semidirect
238. C173#3009semidirect
251. #450D173semidirect
277. C2×SD16C2×D173semidirect
278. C2×SD16C2×D173semidirect
288. C22⋊Q8C2×D173semidirect
296. #102C2×D173semidirect
297. #102C2×D173semidirect
301. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497242
C2×#191497082
C2×SD16×D1732
#260120642
#260116552
#260116542
#260116512
#260116502
#260116432
#260116422
#260116402
#260114942
#260114842
#260114562
#450×C1732
#3009173