Finite Group Databasebeta

← #36365339

Subgroups — #36365339

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#26012308C211yesmaximal, characteristic, complemented
2#26012305C211yesmaximal, characteristic, complemented
3#26012303C211yesmaximal, characteristic, complemented
4#26012298C211yesmaximal, characteristic, complemented
5#26011885C211yesmaximal, characteristic, complemented
6#26011767C211yesmaximal, characteristic, complemented
7#26011766C211yesmaximal, characteristic, complemented
8#26011755C211yesmaximal, characteristic, complemented
9#26011754C211yesmaximal, characteristic, complemented
10#26011675C211yesmaximal, characteristic, complemented
11#26011675C211yesmaximal, characteristic, complemented
12#26011670C211yesmaximal, characteristic, complemented
13#26011665C211yesmaximal, characteristic, complemented
14#26011614C211yesmaximal, characteristic, complemented
15#444×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic
17#19149726C2211yescharacteristic
18#19149726C2211yescharacteristic, complemented
19#19149725C2211yescharacteristic
20#19149712—21no—
21#19149712—21no—
22#19149712—21no—
23#19149712C2211yescharacteristic
24#19149712C2211yescharacteristic
25#19149712C2211yescharacteristic, complemented
26#19149711—21no—
27#19149709C2211yescharacteristic
28#19149709C2211yescharacteristic
29#19149709C2211yescharacteristic, complemented
30#19149708—21no—
31#19149708—21no—
32#19149707—21no—
33#19149707C2211yescharacteristic
34#19149704C2211yescharacteristic
35#19149703—21no—
36#19149701—21no—
37#19149701—21no—
38#19149701C2211yescharacteristic
39#19149700—21no—
40#19149698—21no—
41C2×Dic692—21no—
42C2×Dic692C2211yescharacteristic, complemented
43C2×Dic692C2211yescharacteristic, complemented
44Q16×D173—21no—
45C2×#14583815—21no—
46C2×#14583815C2211yescharacteristic
47C2×#14583815C2211yescharacteristic
48SD16×D173—21no—
49#19149685C2211yescharacteristic
50#19149685C2211yescharacteristic, complemented
51#19149679C2211yescharacteristic, complemented
52#19149667C2211yescharacteristic
53#19149615C2211yescharacteristic, complemented
54#19149607C2211yescharacteristic
55#19149607C2211yescharacteristic, complemented
56#19149597C2211yescharacteristic, complemented
57#19149597C2211yescharacteristic, complemented
58#19149597C2211yescharacteristic, complemented
59#19149595C2211yescharacteristic
60#135×C173C2211yescharacteristic
61#131×C173C2211yescharacteristic
62#131×C173C2211yescharacteristic, complemented
63#108×C173C2211yescharacteristic, complemented
64#104×C173C2211yescharacteristic
65#104×C173C2211yescharacteristic, complemented
66#97×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583826—21no—
69#14583826—21no—
70#14583826—21no—
71#14583826—21no—
72#14583826C2311yescharacteristic
73#14583826C2311yescharacteristic
74#14583825—21no—
75#14583825—21no—
76#14583825—21no—
77#14583825—41no—
78#14583825—41no—
79#14583825—41no—
80C2×Dic346—21no—
81C2×Dic346—21no—
82C2×Dic346—21no—
83C2×Dic346C2311yescharacteristic
84C2×Dic346C2311yescharacteristic
85C2×Dic346C2311yescharacteristic
86Q8×D173—21no—
87Q8×D173—21no—
88Q8×D173—21no—
89C2×D692—21no—
90C2×D692C2311yescharacteristic
91D4×D173—21no—
92D4×D173—41no—
93Dic692—21no—
94Dic692—21no—
95Dic692—21no—
96Dic692—21no—
97Dic692—21no—
98Dic692—21no—
99Dic692—21no—
100Dic692D411yescharacteristic
101#14583816—21no—
102#14583816—21no—
103#14583816—21no—
104#14583815—21no—
105#14583815—21no—
106#14583815—21no—
107#14583815—21no—
108#14583815—21no—
109#14583815—21no—
110#14583815—21no—
111#14583815—21no—
112#14583815D411yescharacteristic
113#14583815D411yescharacteristic
114#14583814—21no—
115#14583814—21no—
116#14583814—21no—
117#14583813—21no—
118D1384—21no—
119D1384D411yescharacteristic
120#14583811—21no—
121#14583810C2311yescharacteristic
122#14583810C2311yescharacteristic
123#14583809—21no—
124#14583809—21no—
125C8×D173—21no—
126C8×D173—21no—
127C4⋊Dic173—41no—
128#14583803—41no—
129#41×C173C2311yescharacteristic
130#41×C173C2311yescharacteristic
131Q16×C173D411yescharacteristic
132SD16×C173D411yescharacteristic
133SD16×C173D411yescharacteristic
134D8×C173D411yescharacteristic
135M16×C173—21no—
136M16×C173—21no—
137M16×C173C2311yescharacteristic
138M16×C173C2311yescharacteristic
139C2×C8×C173—21no—
140C2×C8×C173—21no—
141C2×C8×C173C2311yescharacteristic
142C2×C8×C173C2311yescharacteristic
143C42×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146Dic346—21no—
147Dic346—21no—
148Dic346—21no—
149Dic346—21no—
150Dic346—21no—
151Dic346—21no—
152Dic346—41no—
153Dic346—41no—
154Dic346—41no—
155Dic346C2×D411yescharacteristic
156Dic346C2×D411yescharacteristic
157D692—21no—
158D692—21no—
159D692—41no—
160D692C2×D411yescharacteristic
161D692C2×D411yescharacteristic
162C22⋊D173—21no—
163C22⋊D173—21no—
164C22⋊D173—41no—
165C22⋊D173—41no—
166C22⋊D173—41no—
167C22⋊D173—41no—
168C2×Dic173—21no—
169C2×Dic173—21no—
170C2×Dic173—21no—
171C2×Dic173—41no—
172C2×Dic173—41no—
173C2×Dic173—41no—
174C4×D173—21no—
175C4×D173—21no—
176C4×D173—41no—
177C4×D173—41no—
178C4×D173—41no—
179C4×D173—41no—
180#13063436—21no—
181#13063436—21no—
182Q8×C173C2×D411yescharacteristic
183Q8×C173C2×D411yescharacteristic
184D4×C173—21no—
185D4×C173—21no—
186D4×C173C2×D411yescharacteristic
187D4×C173C2×D411yescharacteristic
188C2×C4×C173—21no—
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173C2411yescharacteristic
192C8×C173—21no—
193C8×C173—21no—
194C8×C173C2×D411yescharacteristic
195C8×C173C2×D411yescharacteristic
196C8×C173C2×D411yescharacteristic
197C8×C173C2×D411yescharacteristic
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×C173C22×D411yescharacteristic
210C4×C173C22×D411yescharacteristic
211Dic173—21no—
212Dic173—21no—
213Dic173—41no—
214Dic173—41no—
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#456—1731no—
236#455—1731no—
237#454—1731no—
238#444—1731no—
239#444D17311yescharacteristic, complemented
2402−1+4—1731no—
2412−1+4—1731no—
2422−1+4—1731no—
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#137—1731no—
255#137—1731no—
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#135C2×D17311yescharacteristic
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#131C2×D17311yescharacteristic
277#131C2×D17311yescharacteristic, complemented
278C4⋊Q8—1731no—
279#124—1731no—
280#108—1731no—
281#108C2×D17311yescharacteristic, complemented
282#104—1731no—
283#104—1731no—
284#104C2×D17311yescharacteristic
285#104C2×D17311yescharacteristic, complemented
286#101—1731no—
287#101—1731no—
288#101—1731no—
289#100—1731no—
290#97C2×D17311yescharacteristic, complemented
291#41—3461no—
292#41—3461no—
293#41—3461no—
294#41—3461no—
295#41—3461no—
296#41—3461no—
297#41—3461no—
298#41—3461no—
299#41—1731no—
300#41—1731no—
301#41—6921no—
302#41—6921no—
303#41—6921no—
304#41C22×D17311yescharacteristic
305#41C22×D17311yescharacteristic
306C2×Q8—3461no—
307C2×Q8—3461no—
308C2×Q8—3461no—
309C2×Q8—3461no—
310C2×Q8—3461no—
311C2×Q8—3461no—
312C2×Q8—1731no—
313C2×Q8—1731no—
314C2×Q8—1731no—
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—1731no—
318C2×D4—6921no—
319Q16—3461no—
320Q16—3461no—
321Q16—3461no—
322Q16—3461no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—3461no—
329Q16—1731no—
330Q16D69211yescharacteristic
331SD16—3461no—
332SD16—3461no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—3461no—
343SD16—1731no—
344SD16—1731no—
345SD16D69211yescharacteristic
346SD16D69211yescharacteristic
347D8—3461no—
348D8—3461no—
349D8—1731no—
350D8D69211yescharacteristic
351M16—21no—
352M16—21no—
353M16—3461no—
354M16—3461no—
355M16—1731no—
356M16—1731no—
357M16C22×D17311yescharacteristic
358M16C22×D17311yescharacteristic
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—3461no—
362C2×C8—3461no—
363C2×C8C22×D17311yescharacteristic
364C2×C8C22×D17311yescharacteristic
365C4⋊C4—6921no—
366C22⋊C4—6921no—
367C42C22×D17311yescharacteristic
368C23—3461no—
369C23—6921no—
370Q8—3461no—
371Q8—3461no—
372Q8—3461no—
373Q8—3461no—
374Q8—3461no—
375Q8—3461no—
376Q8—1731no—
377Q8—1731no—
378Q8—6921no—
379Q8—6921no—
380Q8—6921no—
381Q8C2×D69211yescharacteristic
382Q8C2×D69211yescharacteristic
383D4—21no—
384D4—21no—
385D4—3461no—
386D4—3461no—
387D4—3461no—
388D4—3461no—
389D4—1731no—
390D4—1731no—
391D4—6921no—
392D4—6921no—
393D4—6921no—
394D4—6921no—
395D4—6921no—
396D4C2×D69211yescharacteristic
397D4C2×D69211yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—3461no—
402C2×C4—3461no—
403C2×C4—3461no—
404C2×C4—3461no—
405C2×C4—3461no—
406C2×C4—6921no—
407C2×C4—6921no—
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—
418C8D4×D17311yescharacteristic
419C8D4×D17311yescharacteristic
420C8C2×D69211yescharacteristic
421C8C2×D69211yescharacteristic
422C22—21no—
423C22—21no—
424C22—3461no—
425C22—3461no—
426C22—13841no—
427C22—6921no—
428C22—27681no—
429C22#1914972511yescharacteristic
430C4—21no—
431C4—21no—
432C4—41no—
433C4—3461no—
434C4—3461no—
435C4—6921no—
436C4—6921no—
437C4—6921no—
438C4C2×D4×D17311yescharacteristic
439C4C22×D69211yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—13841no—
444C2—6921no—
445C2—6921no—
446C2D4×D69211yescharacteristic
subgroupG/H
centerC2D4×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#444×C173C2
socleC2×C173D42
hypercenter#444D173
hypocenterC173#3289
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #26012308C2semidirect
2. #26012305C2semidirect
3. #26012303C2semidirect
4. #26012298C2semidirect
5. #26011885C2semidirect
6. #26011767C2semidirect
7. #26011766C2semidirect
8. #26011755C2semidirect
9. #26011754C2semidirect
10. #26011675C2semidirect
11. #26011675C2semidirect
12. #26011670C2semidirect
13. #26011665C2semidirect
14. #26011614C2semidirect
15. #444×C173C2semidirect
18. #19149726C22semidirect
25. #19149712C22semidirect
29. #19149709C22semidirect
42. C2×Dic692C22semidirect
43. C2×Dic692C22semidirect
50. #19149685C22semidirect
51. #19149679C22semidirect
53. #19149615C22semidirect
55. #19149607C22semidirect
56. #19149597C22semidirect
57. #19149597C22semidirect
58. #19149597C22semidirect
62. #131×C173C22semidirect
63. #108×C173C22semidirect
65. #104×C173C22semidirect
66. #97×C173C22semidirect
223. C173#3289semidirect
239. #444D173semidirect
277. #131C2×D173semidirect
281. #108C2×D173semidirect
285. #104C2×D173semidirect
290. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123082
#260123052
#260123032
#260122982
#260118852
#260117672
#260117662
#260117552
#260117542
#260116752
#260116702
#260116652
#260116142
#444×C1732
#3289173