Finite Group Databasebeta

← #49161952

Subgroups — #49161952

Order 37504.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627601C211yesmaximal, characteristic
2#34036035C211yesmaximal, characteristic, complemented
3#34036033C211yesmaximal, characteristic, complemented
4#34036026C211yesmaximal, characteristic, complemented
5#34036025C211yesmaximal, characteristic, complemented
6#34036023C211yesmaximal, characteristic
7#34036023C211yesmaximal, characteristic
8#34036022C211yesmaximal, characteristic
9#34036014C211yesmaximal, characteristic, complemented
10#34036013C211yesmaximal, characteristic, complemented
11#34036011C211yesmaximal, characteristic, complemented
12#34036010C211yesmaximal, characteristic, complemented
13#34035990C211yesmaximal, characteristic
14#34035987C211yesmaximal, characteristic
15#388×C293C211yesmaximal, characteristic
16C22×Dic586C2211yescharacteristic
17C23×Dic293C2211yescharacteristic
18#22627601C418yescomplemented
19C2×C4⋊Dic293C2211yescharacteristic
20C2×C4⋊Dic293C2211yescharacteristic
21C2×#17989781C2211yescharacteristic
22C2×#17989781C2211yescharacteristic
23C2×#17989781C2211yescharacteristic
24C2×#17989781C2211yescharacteristic
25C2×#17989781C2211yescharacteristic
26C2×#17989781C2211yescharacteristic
27C2×C22⋊Dic293C2211yescharacteristic
28C2×C22⋊Dic293C2211yescharacteristic
29C2×C22⋊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
45C2×C4⋊C4×C293C2211yescharacteristic
46C2×C22⋊C4×C293C2211yescharacteristic
47C2×C22⋊C4×C293C2211yescharacteristic
48C2×C22⋊C4×C293C2211yescharacteristic
49C2×C42×C293C2211yescharacteristic
50#95×C293C2211yescharacteristic
51#95×C293C2211yescharacteristic
52C2×Dic586—41no—
53C2×Dic586—41no—
54C2×Dic586C2×C414yes—
55C22×Dic293—41no—
56C22×Dic293—41no—
57C22×Dic293C2×C414yes—
58C22×Dic293C2311yescharacteristic
59C22×Dic293C2311yescharacteristic
60C22×Dic293C2311yescharacteristic
61C22×Dic293C2311yescharacteristic
62C22×Dic293C2311yescharacteristic
63C22×Dic293C2311yescharacteristic
64C22×Dic293C2311yescharacteristic
65C22×Dic293C2311yescharacteristic
66C4⋊Dic293—41no—
67C4⋊Dic293C2×C414yes—
68#17989781—41no—
69#17989781—41no—
70#17989781—41no—
71#17989781—41no—
72#17989781C2×C414yes—
73#17989781C2×C414yes—
74C22⋊Dic293—41no—
75C22⋊Dic293—41no—
76C22⋊Dic293C2×C414yes—
77C4×Dic293—41no—
78C4×Dic293—41no—
79C4×Dic293—41no—
80C24×C293C2311yescharacteristic
81C22×C4×C293C2311yescharacteristic
82C22×C4×C293C2311yescharacteristic
83C22×C4×C293C2311yescharacteristic
84C22×C4×C293C2311yescharacteristic
85C22×C4×C293C2311yescharacteristic
86C22×C4×C293C2311yescharacteristic
87C4⋊C4×C293—41no—
88C22⋊C4×C293—41no—
89C22⋊C4×C293—41no—
90C22⋊C4×C293C2×C414yes—
91C42×C293—41no—
92Dic586—44no—
93C2×Dic293—21no—
94C2×Dic293—21no—
95C2×Dic293—21no—
96C2×Dic293—21no—
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—44no—
114C2×Dic293C22×C412yes—
115C2×Dic293C22×C412yes—
116C2×Dic293C22×C412yes—
117C2×Dic293C22×C412yes—
118C23×C293—21no—
119C23×C293—21no—
120C23×C293—21no—
121C23×C293—21no—
122C23×C293—21no—
123C23×C293—21no—
124C23×C293C22×C412yes—
125C23×C293C2411yescharacteristic
126C2×C4×C293—21no—
127C2×C4×C293—21no—
128C2×C4×C293—21no—
129C2×C4×C293—21no—
130C2×C4×C293—21no—
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×C293C22×C412yes—
143C2×C4×C293C22×C412yes—
144C22×C293—22no—
145C22×C293—22no—
146C22×C293—22no—
147C22×C293—41no—
148C22×C293—41no—
149C22×C293—41no—
150C22×C293—41no—
151C22×C293C23×C411yescharacteristic
152C22×C293H(5,𝔽2)11yescharacteristic
153C22×C293H(5,𝔽2)11yescharacteristic
154C22×C293H(5,𝔽2)11yescharacteristic
155C22×C2932−1+411yescharacteristic
156C22×C2932−1+411yescharacteristic
157C22×C2932−1+411yescharacteristic
158C4×C293—22no—
159C4×C293—22no—
160C4×C293—41no—
161C4×C293—41no—
162C4×C293—41no—
163C4×C293—41no—
164Dic293—22no—
165Dic293—22no—
166Dic293—22no—
167Dic293—22no—
168Dic293—41no—
169Dic293—41no—
170Dic293—41no—
171Dic293—41no—
172C2×C293—42no—
173C2×C293#51811yescharacteristic
174C2×C293#51911yescharacteristic
175C2×C293#52011yescharacteristic
176C2×C293#56211yescharacteristic
177C2×C293#56211yescharacteristic
178C2×C293#56311yescharacteristic
179C2×C293#56311yescharacteristic
180C293#237811yescharacteristic, complemented, Sylow
181#2378—2931nomaximal, Sylow
182C2×C22⋊Q8—2931no—
183#391—2931no—
184#391—2931no—
185#389—2931no—
186#389—2931no—
187#388—2931no—
188#388—2931no—
189#388—2931no—
190#388D29311yescharacteristic
191#387—2931no—
192#387—2931no—
193#387—2931no—
194#387—2931no—
195#381—2931no—
196#380—2931no—
197C22×Q8—2931no—
198C23×C4—2931no—
199C22⋊Q8—2938no—
200C2×C4⋊C4—2931no—
201C2×C4⋊C4—2931no—
202C2×C4⋊C4—2931no—
203C2×C4⋊C4—2931no—
204C2×C4⋊C4—2931no—
205C2×C4⋊C4—2931no—
206C2×C4⋊C4—2931no—
207C2×C4⋊C4—2931no—
208C2×C4⋊C4C2×D29311yescharacteristic
209C2×C22⋊C4—2931no—
210C2×C22⋊C4—2931no—
211C2×C22⋊C4—2931no—
212C2×C22⋊C4C2×D29311yescharacteristic
213C2×C22⋊C4C2×D29311yescharacteristic
214C2×C22⋊C4C2×D29311yescharacteristic
215C2×C42—2931no—
216C2×C42—2931no—
217C2×C42—2931no—
218C2×C42C2×D29311yescharacteristic
219#95—2931no—
220#95—2931no—
221#95—2931no—
222#95—2931no—
223#95—2931no—
224#95—2931no—
225#95—2931no—
226#95—2931no—
227#95—2931no—
228#95—2931no—
229#95—2931no—
230#95—2931no—
231#95C2×D29311yescharacteristic
232#95C2×D29311yescharacteristic
233C24C22×D29311yescharacteristic
234C2×Q8—2934no—
235C2×Q8—11721no—
236C2×Q8—11721no—
237C22×C4—2931no—
238C22×C4—2931no—
239C22×C4—2931no—
240C22×C4—2931no—
241C22×C4—2934no—
242C22×C4—2931no—
243C22×C4—2931no—
244C22×C4—2931no—
245C22×C4—2931no—
246C22×C4—11721no—
247C22×C4—11721no—
248C22×C4C22×D29311yescharacteristic
249C22×C4C22×D29311yescharacteristic
250C22×C4C22×D29311yescharacteristic
251C22×C4C22×D29311yescharacteristic
252C22×C4C22×D29311yescharacteristic
253C22×C4C22×D29311yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—2934no—
256C4⋊C4—2934no—
257C4⋊C4—2934no—
258C4⋊C4—11721no—
259C4⋊C4—11721no—
260C4⋊C4—11721no—
261C4⋊C4—11721no—
262C4⋊C4—11721no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—2934no—
266C22⋊C4—11721no—
267C22⋊C4—11721no—
268C22⋊C4C4×D29314yes—
269C42—41no—
270C42—11721no—
271C42—11721no—
272C42—11721no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×C4×D29312yes—
280C23C23×D29311yescharacteristic
281Q8—11724no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—2932no—
299C2×C4—2932no—
300C2×C4—2932no—
301C2×C4—2932no—
302C2×C4—5861no—
303C2×C4—5861no—
304C2×C4—5861no—
305C2×C4—5861no—
306C2×C4—5861no—
307C2×C4—5861no—
308C2×C4—5861no—
309C2×C4—5861no—
310C2×C4—5861no—
311C2×C4—5861no—
312C2×C4—5861no—
313C2×C4—5861no—
314C2×C4—5861no—
315C2×C4—5861no—
316C2×C4—5861no—
317C2×C4—5861no—
318C2×C4—5861no—
319C2×C4—5861no—
320C2×C4—5861no—
321C2×C4—5861no—
322C2×C4—11724no—
323C2×C4C2×C4×D29312yes—
324C2×C4C2×C4×D29312yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C4×D29311yescharacteristic
333C22#2262766911yescharacteristic
334C22#2262766911yescharacteristic
335C22#2262767011yescharacteristic
336C22#2262767111yescharacteristic
337C22#2262767111yescharacteristic
338C22#2262767111yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—5862no—
346C4—5862no—
347C4—5862no—
348C4—5862no—
349C4—11721no—
350C4—11721no—
351C4—11721no—
352C4—11721no—
353C2—42no—
354C2#3403660111yescharacteristic
355C2#3403660611yescharacteristic
356C2#3403660811yescharacteristic
357C2#3403687311yescharacteristic
358C2#3403687411yescharacteristic
359C2#3403688111yescharacteristic
360C2#3403688211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
Fitting#388×C293C2
socle, cosocleC23×C293C24
prime radicalC24×C293C23
hypercenter#388D293
hypocenterC293#2378
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #34036035C2semidirect
3. #34036033C2semidirect
4. #34036026C2semidirect
5. #34036025C2semidirect
9. #34036014C2semidirect
10. #34036013C2semidirect
11. #34036011C2semidirect
12. #34036010C2semidirect
18. #22627601C4semidirect
180. C293#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#226276012
#340360352
#340360332
#340360262
#340360252
#340360232
#340360222
#340360142
#340360132
#340360112
#340360102
#340359902
#340359872
#388×C2932
#2378293