Finite Group Databasebeta

← #49170321

Subgroups — #49170321

Order 37504.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627650C211yesmaximal, characteristic, complemented
2C2×#22627644C211yesmaximal, characteristic, complemented
3#34036800C211yesmaximal, characteristic, complemented
4#34036519C211yesmaximal, characteristic, complemented
5#34036377C211yesmaximal, characteristic, complemented
6#34036363C211yesmaximal, characteristic, complemented
7#34036348C211yesmaximal, characteristic, complemented
8#34036324C211yesmaximal, characteristic, complemented
9#34036289C211yesmaximal, characteristic, complemented
10#34036274C211yesmaximal, characteristic, complemented
11#34036272C211yesmaximal, characteristic, complemented
12#34036263C211yesmaximal, characteristic, complemented
13#34036229C211yesmaximal, characteristic, complemented
14#34036201C211yesmaximal, characteristic, complemented
15#546×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C2×#17989802C2211yescharacteristic, complemented
18#22627650—24no—
19#22627644—24no—
20C2×#17989792C2211yescharacteristic, complemented
21C2×#17989790C2211yescharacteristic
22C2×#17989790C2211yescharacteristic
23C2×#17989789C2211yescharacteristic, complemented
24C2×#17989789C2211yescharacteristic, complemented
25C2×#17989787C2211yescharacteristic
26C2×#17989787C2211yescharacteristic
27C2×#17989787C2211yescharacteristic
28C2×#17989787C2211yescharacteristic
29C2×#17989786C2211yescharacteristic
30C22×#14556873C2211yescharacteristic, complemented
31#22627621C2211yescharacteristic, complemented
32#22627620—21no—
33#22627598C2211yescharacteristic
34#22627598C2211yescharacteristic, complemented
35#22627597—21no—
36#22627597—21no—
37C4×Dic586C2211yescharacteristic, complemented
38#22627586—21no—
39#22627557C2211yescharacteristic, complemented
40C4⋊#14556873C2211yescharacteristic, complemented
41#22627548C2211yescharacteristic, complemented
42#22627547C2211yescharacteristic, complemented
43#22627545C2211yescharacteristic
44#22627545C2211yescharacteristic
45#22627544C2211yescharacteristic, complemented
46#22627544C2211yescharacteristic, complemented
47C22⋊#14556873C2211yescharacteristic
48C22⋊#14556873C2211yescharacteristic, complemented
49#22627535C2211yescharacteristic, complemented
50C2×#41×C293C2211yescharacteristic
51C22×D4×C293C2211yescharacteristic
52#124×C293C2211yescharacteristic
53#123×C293—21no—
54C22⋊Q8×C293C2211yescharacteristic
55#121×C293—21no—
56#121×C293C2211yescharacteristic
57#120×C293—21no—
58C4×D4×C293C2211yescharacteristic
59C4×D4×C293C2211yescharacteristic
60C2×C22⋊C4×C293—21no—
61#17989802—22no—
62#17989802—22no—
63#17989802—41no—
64#17989802—41no—
65C2×Dic586C2311yescharacteristic
66C2×D1172—21no—
67C2×D1172—21no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293—21no—
74C2×C4×D293—21no—
75C2×C4×D293—21no—
76#17989792—22no—
77#17989790—22no—
78#17989790D414yescomplemented
79#17989789—22no—
80#17989789—41no—
81#17989787—22no—
82#17989787—22no—
83#17989787—22no—
84#17989787—41no—
85#17989786—22no—
86C2×#14556873—22no—
87C2×#14556873C2311yescharacteristic
88C2×#14556873C2311yescharacteristic
89C2×#14556873C2311yescharacteristic
90C2×#14556873C2311yescharacteristic
91C4⋊Dic293C2311yescharacteristic
92#17989781—21no—
93#17989779—21no—
94#17989779—21no—
95C4×Dic293—21no—
96C24×C293—21no—
97#41×C293—22no—
98#41×C293—41no—
99C2×Q8×C293C2311yescharacteristic
100C2×D4×C293—22no—
101C2×D4×C293—21no—
102C2×D4×C293—21no—
103C2×D4×C293—21no—
104C2×D4×C293—41no—
105C2×D4×C293D411yescharacteristic
106C2×D4×C293C2311yescharacteristic
107C2×D4×C293C2311yescharacteristic
108C22×C4×C293—21no—
109C22×C4×C293—21no—
110C22×C4×C293C2311yescharacteristic
111C22×C4×C293C2311yescharacteristic
112C4⋊C4×C293—21no—
113C4⋊C4×C293D411yescharacteristic
114C4⋊C4×C293C2311yescharacteristic
115C22⋊C4×C293—21no—
116C22⋊C4×C293—21no—
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—
125Dic586—21no—
126Dic586C2×D412yes—
127D1172—21no—
128D1172—21no—
129D1172—21no—
130D1172—21no—
131D1172—42no—
132D1172—81no—
133C22⋊D293—42no—
134C22⋊D293—42no—
135C22⋊D293—81no—
136C22⋊D293—81no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C4×D293—42no—
140C4×D293—42no—
141#14556873—21no—
142#14556873—21no—
143#14556873—21no—
144#14556873C2×D412yes—
145C23×C293—21no—
146C23×C293—21no—
147C23×C293—42no—
148C23×C293—42no—
149C23×C293—81no—
150C23×C293C2×D411yescharacteristic
151C23×C293C2×D411yescharacteristic
152Q8×C293—21no—
153Q8×C293—21no—
154D4×C293—21no—
155D4×C293—21no—
156D4×C293—21no—
157D4×C293—41no—
158D4×C293—42no—
159D4×C293—42no—
160D4×C293—81no—
161D4×C293C2×D412yes—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293—21no—
168C2×C4×C293—42no—
169C2×C4×C293—81no—
170C2×C4×C293C2×D411yescharacteristic
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2×D411yescharacteristic
173C2×C4×C293C2×D411yescharacteristic
174C2×C4×C293C2411yescharacteristic
175C22×C293—21no—
176C22×C293—21no—
177C22×C293—22no—
178C22×C293—21no—
179C22×C293—21no—
180C22×C293—21no—
181C22×C293—21no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293—41no—
185C22×C293—41no—
186C22×C293—41no—
187C22×C293—84no—
188C22×C293C22×D411yescharacteristic
189C2×D293—41no—
190C2×D293—41no—
191C2×D293—41no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—41no—
195C4×C293—21no—
196C4×C293—21no—
197C4×C293—21no—
198C4×C293—21no—
199C4×C293—41no—
200C4×C293—41no—
201C4×C293C22×D411yescharacteristic
202C4×C293H(5,𝔽2)11yescharacteristic
203Dic293—22no—
204Dic293—41no—
205C2×C293—41no—
206C2×C293—41no—
207C2×C293—42no—
208C2×C293—81no—
209C2×C293D4211yescharacteristic
210C2×C293#57511yescharacteristic
211C2×C293#57611yescharacteristic
212D293—81no—
213D293—81no—
214C293#326811yescharacteristic, complemented, Sylow
215#3268—2931nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2931no—
217C2×#135—2931no—
218#550—2931no—
219#546D29311yescharacteristic, complemented
220#496—2931no—
221#468—2931no—
222#466—2931no—
223#464—2931no—
224#459—2931no—
225#452—2931no—
226#449—2931no—
227#449—2931no—
228#447—2931no—
229#440—2931no—
230#435—2931no—
231C2×#41—2931no—
232C2×#41—2931no—
233C2×#41C2×D29311yescharacteristic
234C22×D4C2×D29311yescharacteristic
235AGL(1,ℤ/8)—5864no—
236#135—5864no—
237C2×Q16—2931no—
238C2×SD16—2931no—
239C2×SD16—2931no—
240C2×SD16—2931no—
241C2×SD16—2931no—
242C2×D8—2931no—
243C2×D8—2931no—
244C2×D8—2931no—
245C2×D8—2931no—
246C2×M16—2931no—
247C22×C8—2931no—
248#127—2931no—
249#127—5861no—
250#124C2×D29311yescharacteristic
251#123—21no—
252C22⋊Q8C2×D29311yescharacteristic
253#121—21no—
254#121—2931no—
255#121—2931no—
256#121—5861no—
257#121—5861no—
258#121C2×D29311yescharacteristic
259#120—21no—
260C4×Q8—2931no—
261C4×D4—5861no—
262C4×D4C2×D29311yescharacteristic
263C4×D4C2×D29311yescharacteristic
264C2×C22⋊C4—21no—
265#107—2931no—
266C4⋊C8—2931no—
267#103—2931no—
268#103—2931no—
269#102—2931no—
270#102—2931no—
271#102—2931no—
272#102—2931no—
273C22⋊C8—2931no—
274C22⋊C8—2931no—
275#97—2931no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—5862no—
280#41—5862no—
281#41—11721no—
282#41—11721no—
283C2×Q8—2931no—
284C2×Q8C22×D29311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2931no—
291C2×D4—2931no—
292C2×D4—5861no—
293C2×D4—5861no—
294C2×D4—5861no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—5861no—
298C2×D4C22⋊D29311yescharacteristic
299C2×D4C22×D29311yescharacteristic
300C2×D4C22×D29311yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—5861no—
304C22×C4—5861no—
305C22×C4C22×D29311yescharacteristic
306C22×C4C22×D29311yescharacteristic
307Q16—5862no—
308SD16—2934no—
309SD16—5862no—
310SD16—5862no—
311SD16—11721no—
312D8—5862no—
313D8—5862no—
314D8—5862no—
315D8—11721no—
316M16—5862no—
317C2×C8—2931no—
318C2×C8—2931no—
319C2×C8—2931no—
320C2×C8—2931no—
321C2×C8—5862no—
322C4⋊C4—21no—
323C4⋊C4—2931no—
324C4⋊C4—5861no—
325C4⋊C4C22⋊D29311yescharacteristic
326C4⋊C4C22×D29311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—5861no—
333C22⋊C4—5861no—
334C22⋊C4C22⋊D29311yescharacteristic
335C22⋊C4C22⋊D29311yescharacteristic
336C42—5861no—
337C42C22×D29311yescharacteristic
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—5861no—
344C23—5861no—
345C23C2×C22⋊D29311yescharacteristic
346C23C2×C22⋊D29311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—2932no—
350Q8—5861no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—42no—
356D4—42no—
357D4—81no—
358D4—23441no—
359D4—23441no—
360D4—23441no—
361D4—5861no—
362D4—5861no—
363D4—5861no—
364D4—5861no—
365D4—11722no—
366D4—11722no—
367D4—11722no—
368D4D4×D29312yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—5861no—
378C2×C4—5861no—
379C2×C4—11722no—
380C2×C4—11722no—
381C2×C4C2×C22⋊D29311yescharacteristic
382C2×C4C2×C22⋊D29311yescharacteristic
383C2×C4C2×C22⋊D29311yescharacteristic
384C2×C4C2×C22⋊D29311yescharacteristic
385C2×C4C23×D29311yescharacteristic
386C8—2932no—
387C8—5861no—
388C8—5861no—
389C8—5861no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—11721no—
404C22—11721no—
405C22—11721no—
406C22—11721no—
407C22—11721no—
408C22—11721no—
409C22C22×C22⋊D29311yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—5862no—
417C4—11721no—
418C4C2×D4×D29311yescharacteristic
419C4#2262766911yescharacteristic
420C2—41no—
421C2—41no—
422C2—42no—
423C2—81no—
424C2—23441no—
425C2—23441no—
426C2D4×C22⋊D29311yescharacteristic
427C2#3403692711yescharacteristic
428C2#3403693211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#546×C293C2
socleC22×C293C22×D4
hypercenter#546D293
hypocenterC293#3268
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#22627650C2semidirect
2. C2×#22627644C2semidirect
3. #34036800C2semidirect
4. #34036519C2semidirect
5. #34036377C2semidirect
6. #34036363C2semidirect
7. #34036348C2semidirect
8. #34036324C2semidirect
9. #34036289C2semidirect
10. #34036274C2semidirect
11. #34036272C2semidirect
12. #34036263C2semidirect
13. #34036229C2semidirect
14. #34036201C2semidirect
15. #546×C293C2semidirect
17. C2×#17989802C22semidirect
20. C2×#17989792C22semidirect
23. C2×#17989789C22semidirect
24. C2×#17989789C22semidirect
30. C22×#14556873C22semidirect
31. #22627621C22semidirect
34. #22627598C22semidirect
37. C4×Dic586C22semidirect
39. #22627557C22semidirect
40. C4⋊#14556873C22semidirect
41. #22627548C22semidirect
42. #22627547C22semidirect
45. #22627544C22semidirect
46. #22627544C22semidirect
48. C22⋊#14556873C22semidirect
49. #22627535C22semidirect
78. #17989790D4semidirect
214. C293#3268semidirect
219. #546D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276502
C2×#226276442
#340368002
#340365192
#340363772
#340363632
#340363482
#340363242
#340362892
#340362742
#340362722
#340362632
#340362292
#340362012
#546×C2932
#3268293