Finite Group Databasebeta

← #49170311

Subgroups — #49170311

Order 37504.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627649C211yesmaximal, characteristic, complemented
2C2×#22627644C211yesmaximal, characteristic, complemented
3#34036775C211yesmaximal, characteristic, complemented
4#34036521C211yesmaximal, characteristic, complemented
5#34036385C211yesmaximal, characteristic, complemented
6#34036360C211yesmaximal, characteristic, complemented
7#34036344C211yesmaximal, characteristic, complemented
8#34036331C211yesmaximal, characteristic, complemented
9#34036291C211yesmaximal, characteristic, complemented
10#34036289C211yesmaximal, characteristic, complemented
11#34036279C211yesmaximal, characteristic, complemented
12#34036274C211yesmaximal, characteristic, complemented
13#34036239C211yesmaximal, characteristic, complemented
14#34036201C211yesmaximal, characteristic, complemented
15#547×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18#22627649—24no—
19#22627644—24no—
20C2×#17989792C2211yescharacteristic, complemented
21C2×#17989792C2211yescharacteristic, complemented
22C2×#17989790C2211yescharacteristic, complemented
23C2×#17989789C2211yescharacteristic
24C2×#17989789C2211yescharacteristic
25C2×#17989789C2211yescharacteristic
26C2×#17989789C2211yescharacteristic
27C2×#17989787C2211yescharacteristic
28C2×#17989787C2211yescharacteristic
29C2×#17989786C2211yescharacteristic
30C22×#14556873C2211yescharacteristic, complemented
31#22627612C2211yescharacteristic
32#22627609—21no—
33#22627603C2211yescharacteristic
34#22627598C2211yescharacteristic
35#22627597—21no—
36#22627592—21no—
37C4×D1172C2211yescharacteristic
38C22⋊C4×D293—21no—
39#22627555C2211yescharacteristic, complemented
40C4⋊#14556873C2211yescharacteristic, complemented
41#22627548C2211yescharacteristic, complemented
42#22627548C2211yescharacteristic, complemented
43#22627547C2211yescharacteristic, complemented
44#22627545C2211yescharacteristic, complemented
45#22627544C2211yescharacteristic
46#22627544C2211yescharacteristic
47C22⋊#14556873C2211yescharacteristic
48C22⋊#14556873C2211yescharacteristic, complemented
49#22627535C2211yescharacteristic, complemented
50C2×#41×C293C2211yescharacteristic
51C2×#41×C293C2211yescharacteristic, complemented
52C4⋊Q8×C293C2211yescharacteristic, complemented
53#123×C293—21no—
54#123×C293—21no—
55C22⋊Q8×C293C2211yescharacteristic
56C22⋊Q8×C293C2211yescharacteristic, complemented
57#121×C293—21no—
58C4×D4×C293C2211yescharacteristic
59C4×D4×C293C2211yescharacteristic, complemented
60C2×C4⋊C4×C293—21no—
61C23×D293—21no—
62#17989802—22no—
63#17989802—41no—
64C2×Dic586C2311yescharacteristic
65C2×D1172—22no—
66C2×D1172—21no—
67C2×D1172—41no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C4×D293—21no—
73C2×C4×D293—21no—
74#17989792—22no—
75#17989792—41no—
76#17989790—22no—
77#17989789—22no—
78#17989789—22no—
79#17989789—22no—
80#17989789—41no—
81#17989787—22no—
82#17989787D414yescomplemented
83#17989786—22no—
84C2×#14556873—22no—
85C2×#14556873C2311yescharacteristic
86C2×#14556873C2311yescharacteristic
87C2×#14556873C2311yescharacteristic
88C2×#14556873C2311yescharacteristic
89C4⋊Dic293C2311yescharacteristic
90#17989781—21no—
91C22⋊Dic293—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95#17989779—21no—
96#41×C293—22no—
97#41×C293—22no—
98#41×C293—41no—
99#41×C293—41no—
100C2×Q8×C293C2311yescharacteristic
101C2×Q8×C293C2311yescharacteristic
102C2×D4×C293—21no—
103C2×D4×C293—21no—
104C2×D4×C293D411yescharacteristic
105C2×D4×C293C2311yescharacteristic
106C22×C4×C293—21no—
107C22×C4×C293—21no—
108C22×C4×C293—21no—
109C22×C4×C293C2311yescharacteristic
110C22×C4×C293C2311yescharacteristic
111C4⋊C4×C293—21no—
112C4⋊C4×C293—21no—
113C4⋊C4×C293—21no—
114C4⋊C4×C293—21no—
115C4⋊C4×C293D411yescharacteristic
116C4⋊C4×C293C2311yescharacteristic
117C22⋊C4×C293—21no—
118C22⋊C4×C293—21no—
119C22⋊C4×C293—21no—
120C22⋊C4×C293D411yescharacteristic
121C22⋊C4×C293D411yescharacteristic
122C42×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—42no—
126C22×D293—42no—
127C22×D293—81no—
128Dic586—21no—
129Dic586—21no—
130D1172—21no—
131D1172—21no—
132D1172—21no—
133D1172—42no—
134D1172C2×D412yes—
135C22⋊D293—42no—
136C22⋊D293—81no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C4×D293—42no—
140C4×D293—81no—
141#14556873—21no—
142#14556873—21no—
143#14556873—21no—
144#14556873C2×D412yes—
145C23×C293—21no—
146C23×C293C2×D411yescharacteristic
147C23×C293C2×D411yescharacteristic
148Q8×C293—21no—
149Q8×C293—21no—
150Q8×C293—21no—
151Q8×C293—21no—
152D4×C293—21no—
153D4×C293—41no—
154D4×C293—42no—
155D4×C293—42no—
156D4×C293C2×D412yes—
157C2×C4×C293—21no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—21no—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—42no—
165C2×C4×C293—42no—
166C2×C4×C293—42no—
167C2×C4×C293C2×D411yescharacteristic
168C2×C4×C293C2×D411yescharacteristic
169C2×C4×C293C2×D411yescharacteristic
170C2×C4×C293C2×D411yescharacteristic
171C2×C4×C293C2411yescharacteristic
172C22×C293—21no—
173C22×C293—21no—
174C22×C293—22no—
175C22×C293—21no—
176C22×C293—21no—
177C22×C293—21no—
178C22×C293—21no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293C22×D411yescharacteristic
182C2×D293—22no—
183C2×D293—41no—
184C2×D293—41no—
185C2×D293—41no—
186C2×D293—41no—
187C2×D293—41no—
188C2×D293—84no—
189C4×C293—21no—
190C4×C293—21no—
191C4×C293—21no—
192C4×C293—21no—
193C4×C293—41no—
194C4×C293—41no—
195C4×C293—41no—
196C4×C293C22×D411yescharacteristic
197C4×C293H(5,𝔽2)11yescharacteristic
198Dic293—41no—
199Dic293—41no—
200C2×C293—41no—
201C2×C293—41no—
202C2×C293—42no—
203C2×C293D4211yescharacteristic
204C2×C293#57511yescharacteristic
205C2×C293#57711yescharacteristic
206D293—42no—
207D293—81no—
208C293#326711yescharacteristic, complemented, Sylow
209#3267—2931nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2931no—
211C2×#135—2931no—
212#547D29311yescharacteristic, complemented
213#546—2931no—
214#497—2931no—
215#470—2931no—
216#465—2931no—
217#463—2931no—
218#461—2931no—
219#452—2931no—
220#452—2931no—
221#450—2931no—
222#449—2931no—
223#442—2931no—
224#435—2931no—
225C2×#41—2931no—
226C2×#41C2×D29311yescharacteristic
227C2×#41C2×D29311yescharacteristic, complemented
228C22×D4—2931no—
229AGL(1,ℤ/8)—5864no—
230#135—5864no—
231C2×Q16—2931no—
232C2×Q16—2931no—
233C2×SD16—2931no—
234C2×SD16—2931no—
235C2×SD16—2931no—
236C2×SD16—2931no—
237C2×SD16—2931no—
238C2×D8—2931no—
239C2×D8—2931no—
240C2×M16—2931no—
241C22×C8—2931no—
242C4⋊Q8C2×D29311yescharacteristic, complemented
243#124—2931no—
244#123—21no—
245#123—21no—
246#123—5861no—
247C22⋊Q8—2931no—
248C22⋊Q8C2×D29311yescharacteristic
249C22⋊Q8C2×D29311yescharacteristic, complemented
250#121—21no—
251#121—2931no—
252#121—5861no—
253#120—5861no—
254C4×D4—2931no—
255C4×D4C2×D29311yescharacteristic
256C4×D4C2×D29311yescharacteristic, complemented
257C2×C4⋊C4—21no—
258C2×C22⋊C4—5861no—
259#106—2931no—
260C4⋊C8—2931no—
261#103—2931no—
262#103—2931no—
263#103—2931no—
264#102—2931no—
265#102—2931no—
266#102—2931no—
267C22⋊C8—2931no—
268C22⋊C8—2931no—
269#97—2931no—
270C24—5861no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—5862no—
276#41—11721no—
277C2×Q8—2931no—
278C2×Q8C22×D29311yescharacteristic
279C2×Q8C22×D29311yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2931no—
283C2×D4—2931no—
284C2×D4—5862no—
285C2×D4—5861no—
286C2×D4—5861no—
287C2×D4—5861no—
288C2×D4—11721no—
289C2×D4C22⋊D29311yescharacteristic
290C2×D4C22×D29311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—5861no—
295C22×C4—5861no—
296C22×C4C22×D29311yescharacteristic
297C22×C4C22×D29311yescharacteristic
298Q16—5862no—
299Q16—11721no—
300SD16—5862no—
301SD16—5862no—
302SD16—5862no—
303SD16—5862no—
304SD16—11721no—
305D8—2934no—
306D8—5862no—
307M16—5862no—
308C2×C8—2931no—
309C2×C8—2931no—
310C2×C8—2931no—
311C2×C8—2931no—
312C2×C8—5862no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—2931no—
318C4⋊C4—5861no—
319C4⋊C4C22⋊D29311yescharacteristic
320C4⋊C4C22×D29311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—5861no—
325C22⋊C4—5861no—
326C22⋊C4—5861no—
327C22⋊C4—5861no—
328C22⋊C4—5861no—
329C22⋊C4C22⋊D29311yescharacteristic
330C22⋊C4C22⋊D29311yescharacteristic
331C42C22×D29311yescharacteristic
332C23—21no—
333C23—23441no—
334C23—5861no—
335C23—5861no—
336C23—11722no—
337C23—11722no—
338C23C2×C22⋊D29311yescharacteristic
339C23C2×C22⋊D29311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5861no—
345Q8—5861no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—2932no—
351D4—23441no—
352D4—5861no—
353D4—5861no—
354D4—5861no—
355D4—11722no—
356D4—11722no—
357D4D4×D29312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—23441no—
369C2×C4—5861no—
370C2×C4—5861no—
371C2×C4—11722no—
372C2×C4C2×C22⋊D29311yescharacteristic
373C2×C4C2×C22⋊D29311yescharacteristic
374C2×C4C2×C22⋊D29311yescharacteristic
375C2×C4C2×C22⋊D29311yescharacteristic
376C2×C4C23×D29311yescharacteristic
377C8—2932no—
378C8—5861no—
379C8—5861no—
380C8—5861no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—23444no—
391C22—5862no—
392C22—11721no—
393C22—11721no—
394C22—11721no—
395C22—11721no—
396C22—11721no—
397C22C22×C22⋊D29311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—11721no—
406C4—11721no—
407C4C2×D4×D29311yescharacteristic
408C4#2262766911yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—23441no—
413C2—11722no—
414C2D4×C22⋊D29311yescharacteristic
415C2#3403692711yescharacteristic
416C2#3403693711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#547×C293C2
socleC22×C293C22×D4
hypercenter#547D293
hypocenterC293#3267
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#22627649C2semidirect
2. C2×#22627644C2semidirect
3. #34036775C2semidirect
4. #34036521C2semidirect
5. #34036385C2semidirect
6. #34036360C2semidirect
7. #34036344C2semidirect
8. #34036331C2semidirect
9. #34036291C2semidirect
10. #34036289C2semidirect
11. #34036279C2semidirect
12. #34036274C2semidirect
13. #34036239C2semidirect
14. #34036201C2semidirect
15. #547×C293C2semidirect
20. C2×#17989792C22semidirect
21. C2×#17989792C22semidirect
22. C2×#17989790C22semidirect
30. C22×#14556873C22semidirect
39. #22627555C22semidirect
40. C4⋊#14556873C22semidirect
41. #22627548C22semidirect
42. #22627548C22semidirect
43. #22627547C22semidirect
44. #22627545C22semidirect
48. C22⋊#14556873C22semidirect
49. #22627535C22semidirect
51. C2×#41×C293C22semidirect
52. C4⋊Q8×C293C22semidirect
56. C22⋊Q8×C293C22semidirect
59. C4×D4×C293C22semidirect
82. #17989787D4semidirect
208. C293#3267semidirect
212. #547D293semidirect
227. C2×#41C2×D293semidirect
242. C4⋊Q8C2×D293semidirect
249. C22⋊Q8C2×D293semidirect
256. C4×D4C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276492
C2×#226276442
#340367752
#340365212
#340363852
#340363602
#340363442
#340363312
#340362912
#340362892
#340362792
#340362742
#340362392
#340362012
#547×C2932
#3267293