Finite Group Databasebeta

← #49170740

Subgroups — #49170740

Order 37504.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036775C211yesmaximal, characteristic, complemented
2#34036491C211yesmaximal, characteristic, complemented
3#34036459C211yesmaximal, characteristic, complemented
4#34036447C211yesmaximal, characteristic, complemented
5#34036416C211yesmaximal, characteristic, complemented
6#34036380C211yesmaximal, characteristic, complemented
7#34036356C211yesmaximal, characteristic, complemented
8#34036344C211yesmaximal, characteristic, complemented
9#34036272C211yesmaximal, characteristic, complemented
10#34036262C211yesmaximal, characteristic, complemented
11#34036239C211yesmaximal, characteristic, complemented
12#34036201C211yesmaximal, characteristic, complemented
13#34036128C211yesmaximal, characteristic, complemented
14C2×#22627544C211yesmaximal, characteristic, complemented
15#547×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18C2×#17989790C2211yescharacteristic, complemented
19C2×#17989787C2211yescharacteristic
20C2×#17989787C2211yescharacteristic
21C2×#17989786C2211yescharacteristic, complemented
22C22×#14556873C2211yescharacteristic
23#22627612C2211yescharacteristic
24#22627609—21no—
25#22627603C2211yescharacteristic
26#22627598C2211yescharacteristic
27#22627597—21no—
28#22627592—21no—
29C4×D1172C2211yescharacteristic
30C22⋊C4×D293—21no—
31#22627555C2211yescharacteristic, complemented
32#22627555C2211yescharacteristic, complemented
33#22627555C2211yescharacteristic, complemented
34C4⋊#14556873C2211yescharacteristic, complemented
35#22627547C2211yescharacteristic, complemented
36#22627547C2211yescharacteristic, complemented
37#22627547C2211yescharacteristic, complemented
38#22627545C2211yescharacteristic, complemented
39#22627544C2211yescharacteristic
40#22627544C2211yescharacteristic
41#22627544C2211yescharacteristic
42#22627544C2211yescharacteristic
43#22627544C2211yescharacteristic
44#22627544C2211yescharacteristic
45C22⋊#14556873C2211yescharacteristic
46C22⋊#14556873C2211yescharacteristic, complemented
47#22627535C2211yescharacteristic, complemented
48C2×#41×C293—21no—
49C4⋊Q8×C293C2211yescharacteristic, complemented
50#123×C293—21no—
51#123×C293—21no—
52C22⋊Q8×C293—21no—
53#121×C293C2211yescharacteristic
54#121×C293C2211yescharacteristic, complemented
55C4×D4×C293C2211yescharacteristic
56C4×D4×C293C2211yescharacteristic, complemented
57C2×C4⋊C4×C293C2211yescharacteristic
58C2×C4⋊C4×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60#17989802—22no—
61#17989802—41no—
62C2×Dic586C2311yescharacteristic
63C2×D1172—22no—
64C2×D1172—21no—
65C2×D1172—41no—
66C2×D1172C2311yescharacteristic
67C2×D1172C2311yescharacteristic
68C2×C22⋊D293—21no—
69C2×C22⋊D293—21no—
70C2×C4×D293—21no—
71C2×C4×D293—21no—
72#17989790—41no—
73#17989787—22no—
74#17989787—41no—
75#17989786—41no—
76C2×#14556873—41no—
77C2×#14556873C2311yescharacteristic
78C2×#14556873C2311yescharacteristic
79C2×#14556873C2311yescharacteristic
80C2×#14556873C2311yescharacteristic
81C4⋊Dic293C2311yescharacteristic
82#17989781—21no—
83C22⋊Dic293—21no—
84#17989779—21no—
85#17989779—21no—
86#17989779—21no—
87#17989779—21no—
88#41×C293—44no—
89C2×Q8×C293—21no—
90C2×D4×C293—21no—
91C2×D4×C293—21no—
92C2×D4×C293D411yescharacteristic
93C2×D4×C293C2311yescharacteristic
94C22×C4×C293—21no—
95C22×C4×C293—21no—
96C22×C4×C293—21no—
97C22×C4×C293C2311yescharacteristic
98C22×C4×C293C2311yescharacteristic
99C4⋊C4×C293—21no—
100C4⋊C4×C293—21no—
101C4⋊C4×C293—21no—
102C4⋊C4×C293D411yescharacteristic
103C4⋊C4×C293C2311yescharacteristic
104C4⋊C4×C293C2311yescharacteristic
105C4⋊C4×C293C2311yescharacteristic
106C22⋊C4×C293—21no—
107C22⋊C4×C293—21no—
108C22⋊C4×C293—21no—
109C22⋊C4×C293D411yescharacteristic
110C22⋊C4×C293D411yescharacteristic
111C42×C293C2311yescharacteristic
112C22×D293—21no—
113C22×D293—21no—
114C22×D293—42no—
115C22×D293—42no—
116C22×D293—81no—
117Dic586—21no—
118Dic586—21no—
119D1172—21no—
120D1172—21no—
121D1172—21no—
122D1172—42no—
123D1172#4112yescomplemented
124C22⋊D293—42no—
125C22⋊D293—81no—
126C2×Dic293—21no—
127C2×Dic293—21no—
128C4×D293—42no—
129C4×D293—81no—
130#14556873—21no—
131#14556873—21no—
132#14556873—21no—
133#14556873—21no—
134C23×C293—21no—
135C23×C293C2×D411yescharacteristic
136C23×C293C2×D411yescharacteristic
137Q8×C293—42no—
138D4×C293—22no—
139D4×C293—21no—
140D4×C293—21no—
141D4×C293—42no—
142D4×C293—42no—
143C2×C4×C293—21no—
144C2×C4×C293—21no—
145C2×C4×C293—21no—
146C2×C4×C293—21no—
147C2×C4×C293—21no—
148C2×C4×C293—21no—
149C2×C4×C293—21no—
150C2×C4×C293—42no—
151C2×C4×C293—42no—
152C2×C4×C293—42no—
153C2×C4×C293C2×D411yescharacteristic
154C2×C4×C293C2×D411yescharacteristic
155C2×C4×C293C2×D411yescharacteristic
156C2×C4×C293C2×D411yescharacteristic
157C2×C4×C293C2411yescharacteristic
158C22×C293—21no—
159C22×C293—21no—
160C22×C293—21no—
161C22×C293—21no—
162C22×C293—21no—
163C22×C293—41no—
164C22×C293—41no—
165C22×C293—41no—
166C22×C293AGL(1,ℤ/8)12yes—
167C22×C293C22×D411yescharacteristic
168C2×D293—22no—
169C2×D293—41no—
170C2×D293—41no—
171C2×D293—41no—
172C2×D293—41no—
173C2×D293—41no—
174C2×D293—84no—
175C4×C293—21no—
176C4×C293—21no—
177C4×C293—21no—
178C4×C293—21no—
179C4×C293—41no—
180C4×C293—41no—
181C4×C293—41no—
182C4×C293C2×#4111yescharacteristic
183C4×C293H(5,𝔽2)11yescharacteristic
184Dic293—41no—
185Dic293—41no—
186C2×C293—22no—
187C2×C293—41no—
188C2×C293—81no—
189C2×C293#54611yescharacteristic
190C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C293#57711yescharacteristic
192D293—42no—
193D293—81no—
194C293#330611yescharacteristic, complemented, Sylow
195#3306—2931nomaximal, Sylow
196#547D29311yescharacteristic, complemented
197#546—2931no—
198#489—2931no—
199#483—2931no—
200#481—2931no—
201#476—2931no—
202#469—2931no—
203#465—2931no—
204#463—2931no—
205#449—2931no—
206#447—2931no—
207#442—2931no—
208#435—2931no—
209#417—2931no—
210C2×#102—2931no—
211C2×#41—21no—
212C2×#41—2931no—
213C22×D4—2931no—
214C2×SD16—2931no—
215C2×D8—2931no—
216C2×D8—2931no—
217C2×M16—2931no—
218C22×C8—2931no—
219C4⋊Q8C2×D29311yescharacteristic, complemented
220#124—2931no—
221#123—21no—
222#123—21no—
223#123—5861no—
224C22⋊Q8—21no—
225C22⋊Q8—2931no—
226#121—2931no—
227#121—5861no—
228#121C2×D29311yescharacteristic
229#121C2×D29311yescharacteristic, complemented
230#120—5861no—
231C4×D4—2931no—
232C4×D4C2×D29311yescharacteristic
233C4×D4C2×D29311yescharacteristic, complemented
234C2×C4⋊C4C2×D29311yescharacteristic
235C2×C4⋊C4C2×D29311yescharacteristic, complemented
236C2×C22⋊C4—5861no—
237#106—2931no—
238#106—2931no—
239#106—2931no—
240C4⋊C8—2931no—
241#103—2931no—
242#103—2931no—
243#103—2931no—
244#102—2931no—
245#102—2931no—
246#102—2931no—
247#102—2931no—
248#102—2931no—
249#102—2931no—
250#102—2931no—
251C22⋊C8—2931no—
252C22⋊C8—2931no—
253#97—2931no—
254C24—5861no—
255#41—44no—
256#41—5862no—
257#41—11721no—
258C2×Q8—21no—
259C2×Q8—2931no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—2931no—
263C2×D4—2931no—
264C2×D4—5862no—
265C2×D4—5861no—
266C2×D4—5861no—
267C2×D4—5861no—
268C2×D4—11721no—
269C2×D4C22⋊D29311yescharacteristic
270C2×D4C22×D29311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—5861no—
275C22×C4—5861no—
276C22×C4C22×D29311yescharacteristic
277C22×C4C22×D29311yescharacteristic
278SD16—11721no—
279D8—5862no—
280D8—11721no—
281M16—11721no—
282C2×C8—2931no—
283C2×C8—2931no—
284C2×C8—2931no—
285C2×C8—2931no—
286C2×C8—11721no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—2931no—
291C4⋊C4—5861no—
292C4⋊C4C22⋊D29311yescharacteristic
293C4⋊C4C22×D29311yescharacteristic
294C4⋊C4C22×D29311yescharacteristic
295C4⋊C4C22×D29311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—5861no—
300C22⋊C4—5861no—
301C22⋊C4—5861no—
302C22⋊C4—5861no—
303C22⋊C4—5861no—
304C22⋊C4C22⋊D29311yescharacteristic
305C22⋊C4C22⋊D29311yescharacteristic
306C42C22×D29311yescharacteristic
307C23—21no—
308C23—23441no—
309C23—5861no—
310C23—5861no—
311C23—11722no—
312C23—11722no—
313C23C2×C22⋊D29311yescharacteristic
314C23C2×C22⋊D29311yescharacteristic
315Q8—42no—
316Q8—5861no—
317Q8—5861no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—2932no—
324D4—23441no—
325D4—5861no—
326D4—5861no—
327D4—5861no—
328D4—11722no—
329D4—11722no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—23441no—
341C2×C4—5861no—
342C2×C4—5861no—
343C2×C4—11722no—
344C2×C4C2×C22⋊D29311yescharacteristic
345C2×C4C2×C22⋊D29311yescharacteristic
346C2×C4C2×C22⋊D29311yescharacteristic
347C2×C4C2×C22⋊D29311yescharacteristic
348C2×C4C23×D29311yescharacteristic
349C8—5861no—
350C8—5861no—
351C8—5861no—
352C8—5861no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—23444no—
362C22—5862no—
363C22—11721no—
364C22—11721no—
365C22—11721no—
366C22—11721no—
367C22—11721no—
368C22#2262765012yes—
369C22C22×C22⋊D29311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—11721no—
378C4—11721no—
379C4#41×D29311yescharacteristic
380C4#2262767011yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—23441no—
385C2—11722no—
386C2#3403676711yescharacteristic
387C2C2×#2262765011yescharacteristic
388C2#3403693711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#547×C293C2
socleC22×C293C22×D4
hypercenter#547D293
hypocenterC293#3306
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. #34036775C2semidirect
2. #34036491C2semidirect
3. #34036459C2semidirect
4. #34036447C2semidirect
5. #34036416C2semidirect
6. #34036380C2semidirect
7. #34036356C2semidirect
8. #34036344C2semidirect
9. #34036272C2semidirect
10. #34036262C2semidirect
11. #34036239C2semidirect
12. #34036201C2semidirect
13. #34036128C2semidirect
14. C2×#22627544C2semidirect
15. #547×C293C2semidirect
18. C2×#17989790C22semidirect
21. C2×#17989786C22semidirect
31. #22627555C22semidirect
32. #22627555C22semidirect
33. #22627555C22semidirect
34. C4⋊#14556873C22semidirect
35. #22627547C22semidirect
36. #22627547C22semidirect
37. #22627547C22semidirect
38. #22627545C22semidirect
46. C22⋊#14556873C22semidirect
47. #22627535C22semidirect
49. C4⋊Q8×C293C22semidirect
54. #121×C293C22semidirect
56. C4×D4×C293C22semidirect
58. C2×C4⋊C4×C293C22semidirect
123. D1172#41semidirect
194. C293#3306semidirect
196. #547D293semidirect
219. C4⋊Q8C2×D293semidirect
229. #121C2×D293semidirect
233. C4×D4C2×D293semidirect
235. C2×C4⋊C4C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340367752
#340364912
#340364592
#340364472
#340364162
#340363802
#340363562
#340363442
#340362722
#340362622
#340362392
#340362012
#340361282
C2×#226275442
#547×C2932
#3306293