Finite Group Databasebeta

← #49161516

Subgroups — #49161516

Order 37504.

528 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627601C211yesmaximal, characteristic
2#34036034C211yesmaximal, characteristic, complemented
3#34036033C211yesmaximal, characteristic, complemented
4#34036025C211yesmaximal, characteristic, complemented
5#34036024C211yesmaximal, characteristic, complemented
6#34036014C211yesmaximal, characteristic, complemented
7#34036011C211yesmaximal, characteristic, complemented
8#34036011C211yesmaximal, characteristic, complemented
9#34036010C211yesmaximal, characteristic, complemented
10#34036002C211yesmaximal, characteristic
11#34035987C211yesmaximal, characteristic
12C22⋊C4×Dic293C211yesmaximal, characteristic
13C22⋊C4×Dic293C211yesmaximal, characteristic
14C2×#22627531C211yesmaximal, characteristic
15#385×C293C211yesmaximal, characteristic
16C22×Dic586C2211yescharacteristic
17C23×Dic293C2211yescharacteristic
18C23×Dic293C2211yescharacteristic
19#22627601C418yescomplemented
20C2×C4⋊Dic293C2211yescharacteristic
21C2×#17989781C2211yescharacteristic
22C2×#17989781C2211yescharacteristic
23C2×#17989781C2211yescharacteristic
24C2×#17989781C2211yescharacteristic
25C2×#17989781C2211yescharacteristic
26C2×#17989781C2211yescharacteristic
27C2×C22⋊Dic293C2211yescharacteristic
28C2×C22⋊Dic293C2211yescharacteristic
29C2×C4×Dic293C2211yescharacteristic
30C2×C4×Dic293C2211yescharacteristic
31C2×C4×Dic293C2211yescharacteristic
32C2×C4×Dic293C2211yescharacteristic
33#22627531C2211yescharacteristic
34#22627531C2211yescharacteristic
35#22627531C2211yescharacteristic
36#22627531C2211yescharacteristic
37#22627531C2211yescharacteristic
38#22627531C2211yescharacteristic
39#22627531C2211yescharacteristic
40#22627531C2211yescharacteristic
41#22627531C2211yescharacteristic
42#22627531C2211yescharacteristic
43#22627531C2211yescharacteristic
44#22627531C2211yescharacteristic
45C23×C4×C293C2211yescharacteristic
46C2×C4⋊C4×C293C2211yescharacteristic
47C2×C4⋊C4×C293C2211yescharacteristic
48C2×C22⋊C4×C293C2211yescharacteristic
49C2×C22⋊C4×C293C2211yescharacteristic
50#95×C293C2211yescharacteristic
51#95×C293C2211yescharacteristic
52C2×Dic586—41no—
53C2×Dic586—41no—
54C2×Dic586C2×C414yescomplemented
55C22×Dic293—41no—
56C22×Dic293—41no—
57C22×Dic293—41no—
58C22×Dic293—41no—
59C22×Dic293C2×C414yes—
60C22×Dic293Q814yescomplemented
61C22×Dic293C2311yescharacteristic
62C22×Dic293C2311yescharacteristic
63C22×Dic293C2311yescharacteristic
64C22×Dic293C2311yescharacteristic
65C22×Dic293C2311yescharacteristic
66C22×Dic293C2311yescharacteristic
67C22×Dic293C2311yescharacteristic
68C22×Dic293C2311yescharacteristic
69C4⋊Dic293C2×C414yescomplemented
70#17989781—41no—
71#17989781—41no—
72#17989781—41no—
73#17989781—41no—
74#17989781C2×C414yescomplemented
75#17989781C2×C414yescomplemented
76C22⋊Dic293—22no—
77C22⋊Dic293C2×C414yes—
78C4×Dic293—41no—
79C4×Dic293—41no—
80C4×Dic293—41no—
81C4×Dic293—41no—
82C24×C293C2311yescharacteristic
83C22×C4×C293—22no—
84C22×C4×C293—41no—
85C22×C4×C293—41no—
86C22×C4×C293C2311yescharacteristic
87C22×C4×C293C2311yescharacteristic
88C22×C4×C293C2311yescharacteristic
89C22×C4×C293C2311yescharacteristic
90C22×C4×C293C2311yescharacteristic
91C22×C4×C293C2311yescharacteristic
92C4⋊C4×C293—41no—
93C4⋊C4×C293—41no—
94C22⋊C4×C293—22no—
95C22⋊C4×C293C2×C414yes—
96Dic586—44no—
97C2×Dic293—21no—
98C2×Dic293—21no—
99C2×Dic293—21no—
100C2×Dic293—21no—
101C2×Dic293—21no—
102C2×Dic293—21no—
103C2×Dic293—21no—
104C2×Dic293—21no—
105C2×Dic293—21no—
106C2×Dic293—21no—
107C2×Dic293—21no—
108C2×Dic293—21no—
109C2×Dic293—21no—
110C2×Dic293—21no—
111C2×Dic293—21no—
112C2×Dic293—21no—
113C2×Dic293—21no—
114C2×Dic293—21no—
115C2×Dic293—44no—
116C2×Dic293—44no—
117C2×Dic293C22×C412yes—
118C2×Dic293C22×C412yes—
119C2×Dic293C22×C412yes—
120C2×Dic293C22×C412yes—
121C2×Dic293C2×Q812yes—
122C2×Dic293C2×Q812yes—
123C23×C293—21no—
124C23×C293—21no—
125C23×C293—21no—
126C23×C293—21no—
127C23×C293C22×C412yes—
128C23×C293C2×Q812yes—
129C23×C293#4112yes—
130C23×C293C2411yescharacteristic
131C2×C4×C293—21no—
132C2×C4×C293—21no—
133C2×C4×C293—21no—
134C2×C4×C293—21no—
135C2×C4×C293—21no—
136C2×C4×C293—21no—
137C2×C4×C293—21no—
138C2×C4×C293—21no—
139C2×C4×C293—21no—
140C2×C4×C293—21no—
141C2×C4×C293—21no—
142C2×C4×C293—21no—
143C2×C4×C293—21no—
144C2×C4×C293—21no—
145C2×C4×C293—21no—
146C2×C4×C293—21no—
147C2×C4×C293—44no—
148C2×C4×C293C22×C412yes—
149C2×C4×C293C22×C412yes—
150C22×C293—22no—
151C22×C293—22no—
152C22×C293—22no—
153C22×C293—22no—
154C22×C293—22no—
155C22×C293—22no—
156C22×C293C4×Q814yes—
157C22×C293C23×C411yescharacteristic
158C22×C293C22×Q811yescharacteristic
159C22×C293C2×#4111yescharacteristic
160C22×C293H(5,𝔽2)11yescharacteristic
161C22×C293H(5,𝔽2)11yescharacteristic
162C22×C293H(5,𝔽2)11yescharacteristic
163C22×C2932−1+411yescharacteristic
164C4×C293—22no—
165C4×C293—22no—
166C4×C293—41no—
167C4×C293—41no—
168C4×C293—41no—
169C4×C293—41no—
170Dic293—22no—
171Dic293—22no—
172Dic293—22no—
173Dic293—22no—
174Dic293—22no—
175Dic293—22no—
176Dic293—41no—
177Dic293—41no—
178C2×C293—24no—
179C2×C293C2×C4×Q811yescharacteristic
180C2×C293#51811yescharacteristic
181C2×C293#52011yescharacteristic
182C2×C293#53811yescharacteristic
183C2×C293#53911yescharacteristic
184C2×C293#54311yescharacteristic
185C2×C293#54411yescharacteristic
186C293#232911yescharacteristic, complemented, Sylow
187#2329—2931nomaximal, Sylow
188C2×C22⋊Q8—2931no—
189#391—2931no—
190#391—2931no—
191#389—2931no—
192#389—2931no—
193#387—2931no—
194#387—2931no—
195#387—2931no—
196#387—2931no—
197#385—2931no—
198#385D29311yescharacteristic
199#380—2931no—
200C4×C22⋊C4—2931no—
201C4×C22⋊C4—2931no—
202C2×#95—2931no—
203C22×Q8—2931no—
204C23×C4—2931no—
205C23×C4—2931no—
206C23×C4C2×D29311yescharacteristic
207C22⋊Q8—2938no—
208C2×C4⋊C4—2931no—
209C2×C4⋊C4—2931no—
210C2×C4⋊C4—2931no—
211C2×C4⋊C4—2931no—
212C2×C4⋊C4—2931no—
213C2×C4⋊C4—2931no—
214C2×C4⋊C4—2931no—
215C2×C4⋊C4C2×D29311yescharacteristic
216C2×C4⋊C4C2×D29311yescharacteristic
217C2×C22⋊C4—2931no—
218C2×C22⋊C4—2931no—
219C2×C22⋊C4C2×D29311yescharacteristic
220C2×C22⋊C4C2×D29311yescharacteristic
221C2×C42—2931no—
222C2×C42—2931no—
223C2×C42—2931no—
224C2×C42—2931no—
225#95—2931no—
226#95—2931no—
227#95—2931no—
228#95—2931no—
229#95—2931no—
230#95—2931no—
231#95—2931no—
232#95—2931no—
233#95—2931no—
234#95—2931no—
235#95—2931no—
236#95—2931no—
237#95C2×D29311yescharacteristic
238#95C2×D29311yescharacteristic
239C24C22×D29311yescharacteristic
240C2×Q8—2934no—
241C2×Q8—11721no—
242C2×Q8—11721no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—2931no—
247C22×C4—2931no—
248C22×C4—2931no—
249C22×C4—2931no—
250C22×C4—2931no—
251C22×C4—2931no—
252C22×C4—2934no—
253C22×C4—2934no—
254C22×C4—2931no—
255C22×C4—2931no—
256C22×C4—11721no—
257C22×C4—11721no—
258C22×C4—11721no—
259C22×C4—11721no—
260C22×C4C22×D29311yescharacteristic
261C22×C4C22×D29311yescharacteristic
262C22×C4C22×D29311yescharacteristic
263C22×C4C22×D29311yescharacteristic
264C22×C4C22×D29311yescharacteristic
265C22×C4C22×D29311yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—2934no—
269C4⋊C4—2934no—
270C4⋊C4—2934no—
271C4⋊C4—11721no—
272C4⋊C4—11721no—
273C4⋊C4—11721no—
274C4⋊C4—11721no—
275C22⋊C4—22no—
276C22⋊C4—2934no—
277C22⋊C4—5862no—
278C22⋊C4C4×D29314yes—
279C42—11721no—
280C42—11721no—
281C42—11721no—
282C42—11721no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23C2×C4×D29312yes—
288C23Q8×D29312yes—
289C23#1798980112yes—
290C23C23×D29311yescharacteristic
291Q8—11724no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—2932no—
310C2×C4—2932no—
311C2×C4—2932no—
312C2×C4—2932no—
313C2×C4—2932no—
314C2×C4—2932no—
315C2×C4—5861no—
316C2×C4—5861no—
317C2×C4—5861no—
318C2×C4—5861no—
319C2×C4—5861no—
320C2×C4—5861no—
321C2×C4—5861no—
322C2×C4—5861no—
323C2×C4—5861no—
324C2×C4—5861no—
325C2×C4—5861no—
326C2×C4—5861no—
327C2×C4—5861no—
328C2×C4—5861no—
329C2×C4—5861no—
330C2×C4—5861no—
331C2×C4—5861no—
332C2×C4—5861no—
333C2×C4—11724no—
334C2×C4—11724no—
335C2×C4C2×C4×D29312yes—
336C2×C4C2×C4×D29312yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#2262758914yes—
344C22C22×C4×D29311yescharacteristic
345C22C2×Q8×D29311yescharacteristic
346C22C2×#1798980111yescharacteristic
347C22#2262766911yescharacteristic
348C22#2262766911yescharacteristic
349C22#2262767011yescharacteristic
350C22#2262767111yescharacteristic
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5862no—
358C4—5862no—
359C4—5862no—
360C4—5862no—
361C4—5862no—
362C4—5862no—
363C4—11721no—
364C4—11721no—
365C2—24no—
366C2C2×#2262758911yescharacteristic
367C2#3403660111yescharacteristic
368C2#3403660811yescharacteristic
369C2#3403671411yescharacteristic
370C2#3403672311yescharacteristic
371C2#3403675411yescharacteristic
372C2#3403675811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
Fitting#385×C293C2
socle, cosocleC23×C293C24
prime radicalC24×C293C23
hypercenter#385D293
hypocenterC293#2329
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #34036034C2semidirect
3. #34036033C2semidirect
4. #34036025C2semidirect
5. #34036024C2semidirect
6. #34036014C2semidirect
7. #34036011C2semidirect
8. #34036011C2semidirect
9. #34036010C2semidirect
19. #22627601C4semidirect
54. C2×Dic586C2×C4semidirect
60. C22×Dic293Q8semidirect
69. C4⋊Dic293C2×C4semidirect
74. #17989781C2×C4semidirect
75. #17989781C2×C4semidirect
186. C293#2329semidirect

Maximal subgroups

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

subgroupindex
C2×#226276012
#340360342
#340360332
#340360252
#340360242
#340360142
#340360112
#340360102
#340360022
#340359872
C22⋊C4×Dic2932
C2×#226275312
#385×C2932
#2329293