Finite Group Databasebeta

← #36365333

Subgroups — #36365333

Order 22144.

448 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
1#26012307C211yesmaximal, characteristic, complemented
2#26012306C211yesmaximal, characteristic, complemented
3#26012300C211yesmaximal, characteristic, complemented
4#26012299C211yesmaximal, characteristic, complemented
5#26011883C211yesmaximal, characteristic, complemented
6#26011765C211yesmaximal, characteristic, complemented
7#26011764C211yesmaximal, characteristic, complemented
8#26011757C211yesmaximal, characteristic, complemented
9#26011756C211yesmaximal, characteristic, complemented
10#26011673C211yesmaximal, characteristic, complemented
11#26011670C211yesmaximal, characteristic, complemented
12#26011665C211yesmaximal, characteristic, complemented
13#26011610C211yesmaximal, characteristic, complemented
14#26011610C211yesmaximal, characteristic, complemented
15#456×C173C211yesmaximal, characteristic, complemented
16#19149727C2211yescharacteristic
17#19149726C2211yescharacteristic, complemented
18#19149725C2211yescharacteristic
19#19149712C2211yescharacteristic
20#19149711—21no—
21#19149711—21no—
22#19149710—21no—
23#19149710—21no—
24#19149710C2211yescharacteristic
25#19149710C2211yescharacteristic, complemented
26#19149709—21no—
27#19149709C2211yescharacteristic
28#19149708—21no—
29#19149707C2211yescharacteristic
30#19149706—21no—
31#19149704—21no—
32#19149704C2211yescharacteristic
33#19149702—21no—
34#19149700—21no—
35#19149699—21no—
36#19149699—21no—
37#19149699C2211yescharacteristic
38#19149699C2211yescharacteristic
39C2×#14583816—21no—
40C2×#14583816C2211yescharacteristic, complemented
41C2×#14583816C2211yescharacteristic, complemented
42Q16×D173—21no—
43C2×#14583813—21no—
44C2×#14583813C2211yescharacteristic
45C2×#14583813C2211yescharacteristic
46SD16×D173—21no—
47#19149687C2211yescharacteristic
48#19149687C2211yescharacteristic, complemented
49#19149686C2211yescharacteristic
50#19149686C2211yescharacteristic, complemented
51#19149667C2211yescharacteristic
52#19149614C2211yescharacteristic, complemented
53#19149614C2211yescharacteristic, complemented
54#19149607C2211yescharacteristic
55#19149607C2211yescharacteristic, complemented
56#19149598C2211yescharacteristic, complemented
57#19149597C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic
59#19149590C2211yescharacteristic, complemented
602−1+4×C173C2211yescharacteristic
61#137×C173C2211yescharacteristic
62#137×C173C2211yescharacteristic, complemented
63C4⋊Q8×C173C2211yescharacteristic, complemented
64#104×C173C2211yescharacteristic
65#104×C173C2211yescharacteristic, complemented
66#101×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583827—21no—
69#14583827—21no—
70#14583826—21no—
71#14583826—21no—
72#14583826—41no—
73#14583826C2311yescharacteristic
74#14583826C2311yescharacteristic
75#14583825—21no—
76#14583825—41no—
77C2×Dic346—21no—
78C2×Dic346C2311yescharacteristic
79Q8×D173—21no—
80Q8×D173—21no—
81Q8×D173—21no—
82C2×D692—21no—
83C2×D692C2311yescharacteristic
84D4×D173—21no—
85D4×D173—41no—
86Dic692—21no—
87#14583816—21no—
88#14583816—21no—
89#14583816—21no—
90#14583816—21no—
91#14583816—21no—
92#14583816—21no—
93#14583816—21no—
94#14583816D411yescharacteristic
95#14583816D411yescharacteristic
96#14583815—21no—
97#14583815—21no—
98#14583814—21no—
99#14583814D411yescharacteristic
100#14583814D411yescharacteristic
101#14583813—21no—
102#14583813—21no—
103#14583813—21no—
104#14583813—21no—
105#14583813—21no—
106#14583813—21no—
107#14583813—21no—
108#14583813D411yescharacteristic
109#14583813D411yescharacteristic
110D1384—21no—
111#14583811—21no—
112#14583811D411yescharacteristic
113#14583811D411yescharacteristic
114#14583810—21no—
115#14583810—21no—
116#14583810C2311yescharacteristic
117#14583810C2311yescharacteristic
118#14583809—21no—
119#14583809—21no—
120C2×#13063436—21no—
121C2×#13063436—21no—
122C2×#13063436C2311yescharacteristic
123C2×#13063436C2311yescharacteristic
124C8×D173—21no—
125C8×D173—21no—
126#14583803—41no—
127#41×C173—21no—
128#41×C173—21no—
129#41×C173—41no—
130#41×C173C2311yescharacteristic
131#41×C173C2311yescharacteristic
132C2×Q8×C173—21no—
133C2×Q8×C173—21no—
134C2×Q8×C173C2311yescharacteristic
135C2×Q8×C173C2311yescharacteristic
136Q16×C173—21no—
137Q16×C173—21no—
138SD16×C173—21no—
139SD16×C173—21no—
140M16×C173C2311yescharacteristic
141M16×C173C2311yescharacteristic
142C4⋊C4×C173—41no—
143C42×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146Dic346—21no—
147Dic346—21no—
148Dic346—41no—
149Dic346—41no—
150Dic346C2×D411yescharacteristic
151Dic346C2×D411yescharacteristic
152D692—21no—
153D692—21no—
154D692—41no—
155D692—41no—
156D692C2×D411yescharacteristic
157D692C2×D411yescharacteristic
158C22⋊D173—21no—
159C22⋊D173—21no—
160C22⋊D173—41no—
161C22⋊D173—41no—
162C2×Dic173—21no—
163C2×Dic173—41no—
164C4×D173—21no—
165C4×D173—21no—
166C4×D173—41no—
167C4×D173—41no—
168C4×D173—41no—
169C4×D173—41no—
170#13063436—21no—
171#13063436—21no—
172#13063436C2×D411yescharacteristic
173#13063436C2×D411yescharacteristic
174#13063436C2×D411yescharacteristic
175#13063436C2×D411yescharacteristic
176Q8×C173—21no—
177Q8×C173—21no—
178Q8×C173—21no—
179Q8×C173—21no—
180Q8×C173—41no—
181Q8×C173C2×D411yescharacteristic
182Q8×C173C2×D411yescharacteristic
183D4×C173—21no—
184D4×C173—21no—
185D4×C173—41no—
186D4×C173C2×D411yescharacteristic
187D4×C173C2×D411yescharacteristic
188C2×C4×C173—21no—
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173—21no—
193C2×C4×C173—41no—
194C2×C4×C173—41no—
195C2×C4×C173C2411yescharacteristic
196C8×C173—21no—
197C8×C173—21no—
198C22×C173—21no—
199C22×C173—21no—
200C22×C173H(5,𝔽2)11yescharacteristic
201C2×D173—21no—
202C2×D173—21no—
203C2×D173—41no—
204C2×D173—81no—
205C2×D173—161no—
206C4×C173—21no—
207C4×C173—21no—
208C4×C173—41no—
209C4×C173—41no—
210C4×C173—41no—
211C4×C173C22×D411yescharacteristic
212C4×C173C22×D411yescharacteristic
213Dic173—21no—
214Dic173—21no—
215Dic173—41no—
216C2×C173—21no—
217C2×C173—41no—
218C2×C173—41no—
219C2×C173D4211yescharacteristic
220D173—41no—
221D173—41no—
222D173—81no—
223C173#328911yescharacteristic, complemented, Sylow
224#3289—1731nomaximal, Sylow
225#578—1731no—
226#578—1731no—
227#577—1731no—
228#577—1731no—
229#497—1731no—
230#473—1731no—
231#473—1731no—
232#471—1731no—
233#471—1731no—
234#456—1731no—
235#456D17311yescharacteristic, complemented
236#455—1731no—
237#454—1731no—
238#444—1731no—
239#444—1731no—
2402−1+4—1731no—
2412−1+4—1731no—
2422−1+4C2×D17311yescharacteristic
243H(5,𝔽2)—1731no—
244#137—3461no—
245#137—3461no—
246#137—3461no—
247#137—3461no—
248#137—3461no—
249#137—3461no—
250#137—1731no—
251#137—1731no—
252#137—1731no—
253#137—1731no—
254#137C2×D17311yescharacteristic
255#137C2×D17311yescharacteristic, complemented
256AGL(1,ℤ/8)—3461no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—1731no—
259AGL(1,ℤ/8)—1731no—
260#135—3461no—
261#135—3461no—
262#135—3461no—
263#135—3461no—
264#135—1731no—
265#135—1731no—
266C2×Q16—3461no—
267C2×Q16—3461no—
268C2×Q16—1731no—
269C2×Q16—1731no—
270C2×SD16—3461no—
271C2×SD16—3461no—
272C2×SD16—1731no—
273C2×SD16—1731no—
274#131—1731no—
275#131—1731no—
276#131—1731no—
277#131—1731no—
278C4⋊Q8C2×D17311yescharacteristic, complemented
279#124—1731no—
280#108—1731no—
281#108—1731no—
282#104—1731no—
283#104—1731no—
284#104C2×D17311yescharacteristic
285#104C2×D17311yescharacteristic, complemented
286#101—1731no—
287#101—1731no—
288#101C2×D17311yescharacteristic, complemented
289#100—1731no—
290#97—1731no—
291#41—21no—
292#41—21no—
293#41—41no—
294#41—3461no—
295#41—3461no—
296#41—3461no—
297#41—3461no—
298#41—3461no—
299#41—3461no—
300#41—1731no—
301#41—1731no—
302#41—6921no—
303#41—6921no—
304#41C22×D17311yescharacteristic
305#41C22×D17311yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—3461no—
309C2×Q8—3461no—
310C2×Q8—3461no—
311C2×Q8—3461no—
312C2×Q8—1731no—
313C2×Q8C22×D17311yescharacteristic
314C2×Q8C22×D17311yescharacteristic
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—1731no—
318C2×D4—6921no—
319Q16—21no—
320Q16—21no—
321Q16—3461no—
322Q16—3461no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—3461no—
329Q16—1731no—
330Q16—1731no—
331SD16—21no—
332SD16—21no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—3461no—
343SD16—1731no—
344SD16—1731no—
345SD16—1731no—
346SD16—1731no—
347D8—3461no—
348D8—3461no—
349D8—1731no—
350D8—1731no—
351M16—3461no—
352M16—3461no—
353M16—3461no—
354M16—3461no—
355M16—1731no—
356M16—1731no—
357M16C22×D17311yescharacteristic
358M16C22×D17311yescharacteristic
359C2×C8—3461no—
360C2×C8—3461no—
361C2×C8—3461no—
362C2×C8—3461no—
363C2×C8—1731no—
364C2×C8—1731no—
365C4⋊C4—41no—
366C22⋊C4—6921no—
367C42C22×D17311yescharacteristic
368C23—3461no—
369C23—6921no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—3461no—
376Q8—3461no—
377Q8—1731no—
378Q8—1731no—
379Q8—6921no—
380Q8—6921no—
381Q8D4×D17311yescharacteristic
382Q8D4×D17311yescharacteristic
383D4—21no—
384D4—21no—
385D4—41no—
386D4—3461no—
387D4—3461no—
388D4—3461no—
389D4—3461no—
390D4—1731no—
391D4—1731no—
392D4—6921no—
393D4—6921no—
394D4—6921no—
395D4—6921no—
396D4D4×D17311yescharacteristic
397D4D4×D17311yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—3461no—
406C2×C4—3461no—
407C2×C4—3461no—
408C2×C4—6921no—
409C2×C4—6921no—
410C2×C4—6921no—
411C2×C4—6921no—
412C2×C4—6921no—
413C2×C4C23×D17311yescharacteristic
414C8—21no—
415C8—21no—
416C8—3461no—
417C8—3461no—
418C8—1731no—
419C8—1731no—
420C8—1731no—
421C8—1731no—
422C22—21no—
423C22—21no—
424C22—3461no—
425C22—3461no—
426C22—13841no—
427C22—6921no—
428C22—27681no—
429C22#1914972411yescharacteristic
430C4—21no—
431C4—21no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—3461no—
436C4—3461no—
437C4—6921no—
438C4C2×D4×D17311yescharacteristic
439C4C2×D4×D17311yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—13841no—
444C2—6921no—
445C2—6921no—
446C2#2601212311yescharacteristic
subgroupG/H
centerC2#26012123
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#456×C173C2
socleC2×C173D42
hypercenter#456D173
hypocenterC173#3289
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26012307C2semidirect
2. #26012306C2semidirect
3. #26012300C2semidirect
4. #26012299C2semidirect
5. #26011883C2semidirect
6. #26011765C2semidirect
7. #26011764C2semidirect
8. #26011757C2semidirect
9. #26011756C2semidirect
10. #26011673C2semidirect
11. #26011670C2semidirect
12. #26011665C2semidirect
13. #26011610C2semidirect
14. #26011610C2semidirect
15. #456×C173C2semidirect
17. #19149726C22semidirect
25. #19149710C22semidirect
40. C2×#14583816C22semidirect
41. C2×#14583816C22semidirect
48. #19149687C22semidirect
50. #19149686C22semidirect
52. #19149614C22semidirect
53. #19149614C22semidirect
55. #19149607C22semidirect
56. #19149598C22semidirect
57. #19149597C22semidirect
59. #19149590C22semidirect
62. #137×C173C22semidirect
63. C4⋊Q8×C173C22semidirect
65. #104×C173C22semidirect
66. #101×C173C22semidirect
223. C173#3289semidirect
235. #456D173semidirect
255. #137C2×D173semidirect
278. C4⋊Q8C2×D173semidirect
285. #104C2×D173semidirect
288. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123072
#260123062
#260123002
#260122992
#260118832
#260117652
#260117642
#260117572
#260117562
#260116732
#260116702
#260116652
#260116102
#456×C1732
#3289173