Finite Group Databasebeta

← #49170390

Subgroups — #49170390

Order 37504.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627648C211yesmaximal, characteristic, complemented
2#34036773C211yesmaximal, characteristic, complemented
3#34036767C211yesmaximal, characteristic, complemented
4#34036510C211yesmaximal, characteristic, complemented
5#34036375C211yesmaximal, characteristic, complemented
6#34036372C211yesmaximal, characteristic, complemented
7#34036346C211yesmaximal, characteristic, complemented
8#34036342C211yesmaximal, characteristic, complemented
9#34036291C211yesmaximal, characteristic, complemented
10#34036288C211yesmaximal, characteristic, complemented
11#34036263C211yesmaximal, characteristic, complemented
12#34036261C211yesmaximal, characteristic, complemented
13D8×Dic293C211yesmaximal, characteristic, complemented
14#34036205C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C2×D4×D293C2211yescharacteristic, complemented
18#22627648—24no—
19C2×#17989792C2211yescharacteristic, complemented
20C2×#17989791C2211yescharacteristic, complemented
21C2×#17989790C2211yescharacteristic, complemented
22C2×#17989787C2211yescharacteristic
23C2×#17989787C2211yescharacteristic
24C2×#17989785C2211yescharacteristic, complemented
25#22627613C2211yescharacteristic, complemented
26#22627611C2211yescharacteristic, complemented
27#22627607—21no—
28#22627605—21no—
29#22627603C2211yescharacteristic, complemented
30#22627600C2211yescharacteristic, complemented
31#22627598C2211yescharacteristic
32#22627596—21no—
33#22627596—21no—
34#22627595C2211yescharacteristic
35#22627593—21no—
36#22627592—21no—
37D4×Dic293C2211yescharacteristic, complemented
38D4×Dic293C2211yescharacteristic, complemented
39C4×C22⋊D293C2211yescharacteristic
40C2×C22⋊Dic293—21no—
41C2×#17989779—21no—
42#22627558C2211yescharacteristic, complemented
43#22627553C2211yescharacteristic, complemented
44#22627549C2211yescharacteristic, complemented
45#22627548C2211yescharacteristic, complemented
46#22627546C2211yescharacteristic, complemented
47#22627545C2211yescharacteristic, complemented
48#22627545C2211yescharacteristic, complemented
49#22627545C2211yescharacteristic, complemented
50C22⋊#14556873C2211yescharacteristic, complemented
51#22627536C2211yescharacteristic, complemented
52C8×Dic293C2211yescharacteristic, complemented
53C2×#41×C293C2211yescharacteristic
54C22×D4×C293C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C293—24no—
56C2×SD16×C293C2211yescharacteristic, complemented
57C2×SD16×C293C2211yescharacteristic, complemented
58C2×D8×C293C2211yescharacteristic
59C2×D8×C293C2211yescharacteristic
60C2×M16×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989801—22no—
63#17989801—41no—
64C2×Dic586C2311yescharacteristic
65C2×D1172C2311yescharacteristic
66C2×C22⋊D293—21no—
67C2×C22⋊D293—21no—
68C2×C22⋊D293—21no—
69C2×C22⋊D293—21no—
70C2×C22⋊D293D411yescharacteristic
71D4×D293—22no—
72D4×D293—41no—
73C22×Dic293—21no—
74C22×Dic293—21no—
75C22×Dic293—21no—
76C2×C4×D293C2311yescharacteristic
77#17989792—41no—
78#17989791—22no—
79#17989790—22no—
80#17989787—22no—
81#17989787—41no—
82#17989785—22no—
83C2×#14556873C2311yescharacteristic
84C2×#14556873C2311yescharacteristic
85C4⋊Dic293C2311yescharacteristic
86C4⋊Dic293C2311yescharacteristic
87#17989781—21no—
88#17989781—21no—
89#17989781D411yescharacteristic
90C22⋊Dic293—21no—
91C22⋊Dic293—21no—
92C22⋊Dic293—21no—
93C22⋊Dic293—21no—
94C22⋊Dic293—21no—
95C22⋊Dic293—21no—
96C22⋊Dic293D411yescharacteristic
97#17989779—21no—
98#17989779—21no—
99#17989779—21no—
100#17989779—21no—
101#17989779D411yescharacteristic
102C4×Dic293C2311yescharacteristic
103C24×C293—21no—
104#41×C293—22no—
105#41×C293—41no—
106C2×Q8×C293C2311yescharacteristic
107C2×D4×C293—22no—
108C2×D4×C293—21no—
109C2×D4×C293—21no—
110C2×D4×C293—41no—
111C2×D4×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C2×D4×C293C2311yescharacteristic
114C22×C4×C293—21no—
115C22×C4×C293C2311yescharacteristic
116SD16×C293—22no—
117SD16×C293—22no—
118D8×C293—22no—
119D8×C293D414yescomplemented
120M16×C293—22no—
121C2×C8×C293C2311yescharacteristic
122C2×C8×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—42no—
125C22×D293—42no—
126C22×D293—81no—
127C22×D293C2×D411yescharacteristic
128Dic586—21no—
129Dic586—21no—
130D1172—21no—
131D1172—21no—
132C22⋊D293—41no—
133C22⋊D293—42no—
134C22⋊D293—42no—
135C22⋊D293—81no—
136C22⋊D293—81no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C2×Dic293—21no—
140C2×Dic293—42no—
141C2×Dic293—81no—
142C2×Dic293—81no—
143C2×Dic293C2×D411yescharacteristic
144C2×Dic293C2×D411yescharacteristic
145C2×Dic293C2×D411yescharacteristic
146C4×D293—21no—
147C4×D293—21no—
148#14556873—21no—
149#14556873—21no—
150C23×C293—21no—
151C23×C293—21no—
152C23×C293—21no—
153C23×C293—42no—
154C23×C293—42no—
155C23×C293—81no—
156C23×C293C2×D411yescharacteristic
157Q8×C293—21no—
158Q8×C293—21no—
159D4×C293—21no—
160D4×C293—21no—
161D4×C293—21no—
162D4×C293—21no—
163D4×C293—42no—
164D4×C293—42no—
165D4×C293C2×D412yes—
166D4×C293C2×D412yes—
167C2×C4×C293—21no—
168C2×C4×C293—21no—
169C2×C4×C293—21no—
170C2×C4×C293—42no—
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2411yescharacteristic
173C8×C293—21no—
174C8×C293C2×D412yes—
175C22×C293—21no—
176C22×C293—22no—
177C22×C293—22no—
178C22×C293—21no—
179C22×C293—21no—
180C22×C293—41no—
181C22×C293—41no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293—41no—
185C22×C293—41no—
186C22×C293—41no—
187C22×C293—84no—
188C22×C293C22×D411yescharacteristic
189C2×D293—21no—
190C2×D293—21no—
191C2×D293—21no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—41no—
195C2×D293—84no—
196C4×C293—21no—
197C4×C293—41no—
198C4×C293C22×D411yescharacteristic
199C4×C293H(5,𝔽2)11yescharacteristic
200Dic293—21no—
201Dic293—21no—
202Dic293—21no—
203Dic293—41no—
204Dic293—41no—
205Dic293—41no—
206C2×C293—42no—
207C2×C293—41no—
208C2×C293—42no—
209C2×C293—81no—
210C2×C293D4211yescharacteristic
211C2×C293#57511yescharacteristic
212C2×C293#57611yescharacteristic
213D293—41no—
214D293—81no—
215C293#327311yescharacteristic, complemented, Sylow
216#3273—2931nomaximal, Sylow
217C2×AGL(1,ℤ/8)—2931no—
218C2×AGL(1,ℤ/8)D29311yescharacteristic, complemented
219#546—2931no—
220#546—2931no—
221#495—2931no—
222#468—2931no—
223#468—2931no—
224#463—2931no—
225#463—2931no—
226#452—2931no—
227#452—2931no—
228#447—2931no—
229#447—2931no—
230C4×D8—2931no—
231#436—2931no—
232C2×#41—2931no—
233C2×#41C2×D29311yescharacteristic
234C22×D4—2931no—
235C22×D4C2×D29311yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—5864no—
238C2×Q16—2931no—
239C2×SD16—2931no—
240C2×SD16—2931no—
241C2×SD16C2×D29311yescharacteristic, complemented
242C2×SD16C2×D29311yescharacteristic, complemented
243C2×D8—2931no—
244C2×D8—2931no—
245C2×D8C2×D29311yescharacteristic
246C2×D8C2×D29311yescharacteristic
247C2×M16—2931no—
248C2×M16C2×D29311yescharacteristic, complemented
249#124—2931no—
250#124—2931no—
251#123—5861no—
252#123—5861no—
253C22⋊Q8—2931no—
254C22⋊Q8—2931no—
255#121—2931no—
256#121—2931no—
257#121—5861no—
258#121—5861no—
259#120—5861no—
260#120—5861no—
261C4×D4—2931no—
262C4×D4—2931no—
263C4×D4—2931no—
264C2×C22⋊C4—5861no—
265C2×C22⋊C4—5861no—
266#107—2931no—
267C4⋊C8—2931no—
268#103—2931no—
269#103—2931no—
270#102—2931no—
271#102—2931no—
272#102—2931no—
273#102—2931no—
274C22⋊C8—2931no—
275C22⋊C8—2931no—
276C4×C8—2931no—
277C24—21no—
278C24—5861no—
279#41—22no—
280#41—41no—
281#41—5862no—
282#41—11721no—
283C2×Q8—2931no—
284C2×Q8C22×D29311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—2931no—
290C2×D4—2931no—
291C2×D4—5861no—
292C2×D4—5861no—
293C2×D4—5861no—
294C2×D4—5862no—
295C2×D4—5861no—
296C2×D4—11721no—
297C2×D4C22×D29311yescharacteristic
298C2×D4C22×D29311yescharacteristic
299C2×D4C22×D29311yescharacteristic
300C22×C4—21no—
301C22×C4—2931no—
302C22×C4—5861no—
303C22×C4—5861no—
304C22×C4—5861no—
305C22×C4C22×D29311yescharacteristic
306Q16—11721no—
307SD16—22no—
308SD16—22no—
309SD16—5862no—
310SD16—5862no—
311D8—22no—
312D8—5862no—
313D8—11721no—
314D8C22⋊D29314yescomplemented
315M16—22no—
316M16—5862no—
317C2×C8—2931no—
318C2×C8—2931no—
319C2×C8C22×D29311yescharacteristic
320C2×C8C22×D29311yescharacteristic
321C4⋊C4—2931no—
322C4⋊C4—2931no—
323C4⋊C4—2931no—
324C4⋊C4—5861no—
325C4⋊C4—5861no—
326C22⋊C4—2931no—
327C22⋊C4—2931no—
328C22⋊C4—5861no—
329C22⋊C4—5861no—
330C22⋊C4—5861no—
331C22⋊C4—5861no—
332C22⋊C4—5861no—
333C22⋊C4—5861no—
334C22⋊C4—5861no—
335C22⋊C4—5861no—
336C22⋊C4—5861no—
337C22⋊C4—5861no—
338C42—2931no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—2931no—
346C23—23441no—
347C23—5861no—
348C23—11722no—
349C23—11722no—
350C23D4×D29311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—5861no—
354Q8—5861no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—23441no—
362D4—23441no—
363D4—5861no—
364D4—5861no—
365D4—11721no—
366D4—11722no—
367D4—11722no—
368D4C2×C22⋊D29312yes—
369D4C2×C22⋊D29312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—2931no—
375C2×C4—2931no—
376C2×C4—2931no—
377C2×C4—23441no—
378C2×C4—23441no—
379C2×C4—5861no—
380C2×C4—5861no—
381C2×C4—5861no—
382C2×C4—5861no—
383C2×C4—5861no—
384C2×C4—11722no—
385C2×C4D4×D29311yescharacteristic
386C2×C4C23×D29311yescharacteristic
387C8—21no—
388C8—5861no—
389C8—5861no—
390C8C2×C22⋊D29312yes—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—23444no—
405C22—5861no—
406C22—5861no—
407C22—5861no—
408C22—11721no—
409C22—11721no—
410C22—11721no—
411C22C2×D4×D29311yescharacteristic
412C4—21no—
413C4—41no—
414C4—5861no—
415C4—5861no—
416C4—5861no—
417C4—11721no—
418C4—11721no—
419C4—11721no—
420C4C22×C22⋊D29311yescharacteristic
421C4#2262766911yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—23441no—
427C2—11721no—
428C2D4×C22⋊D29311yescharacteristic
429C2#3403692511yescharacteristic
430C2#3403693311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×AGL(1,ℤ/8)×C293C2
socleC22×C293C22×D4
hypercenterC2×AGL(1,ℤ/8)D293
hypocenterC293#3273
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22627648C2semidirect
2. #34036773C2semidirect
3. #34036767C2semidirect
4. #34036510C2semidirect
5. #34036375C2semidirect
6. #34036372C2semidirect
7. #34036346C2semidirect
8. #34036342C2semidirect
9. #34036291C2semidirect
10. #34036288C2semidirect
11. #34036263C2semidirect
12. #34036261C2semidirect
13. D8×Dic293C2semidirect
14. #34036205C2semidirect
15. C2×AGL(1,ℤ/8)×C293C2semidirect
17. C2×D4×D293C22semidirect
19. C2×#17989792C22semidirect
20. C2×#17989791C22semidirect
21. C2×#17989790C22semidirect
24. C2×#17989785C22semidirect
25. #22627613C22semidirect
26. #22627611C22semidirect
29. #22627603C22semidirect
30. #22627600C22semidirect
37. D4×Dic293C22semidirect
38. D4×Dic293C22semidirect
42. #22627558C22semidirect
43. #22627553C22semidirect
44. #22627549C22semidirect
45. #22627548C22semidirect
46. #22627546C22semidirect
47. #22627545C22semidirect
48. #22627545C22semidirect
49. #22627545C22semidirect
50. C22⋊#14556873C22semidirect
51. #22627536C22semidirect
52. C8×Dic293C22semidirect
54. C22×D4×C293C22semidirect
56. C2×SD16×C293C22semidirect
57. C2×SD16×C293C22semidirect
60. C2×M16×C293C22semidirect
119. D8×C293D4semidirect
215. C293#3273semidirect
218. C2×AGL(1,ℤ/8)D293semidirect
235. C22×D4C2×D293semidirect
241. C2×SD16C2×D293semidirect
242. C2×SD16C2×D293semidirect
248. C2×M16C2×D293semidirect
314. D8C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276482
#340367732
#340367672
#340365102
#340363752
#340363722
#340363462
#340363422
#340362912
#340362882
#340362632
#340362612
D8×Dic2932
#340362052
C2×AGL(1,ℤ/8)×C2932
#3273293