Finite Group Databasebeta

← #37581972

Subgroups — #37581972

Order 24448.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259343C211yesmaximal, characteristic, complemented
2C2×#20259341C211yesmaximal, characteristic, complemented
3#27133212C211yesmaximal, characteristic, complemented
4C4×#15690286C211yesmaximal, characteristic, complemented
5#27132886C211yesmaximal, characteristic, complemented
6#27132882C211yesmaximal, characteristic, complemented
7#27132878C211yesmaximal, characteristic, complemented
8#27132872C211yesmaximal, characteristic, complemented
9#27132866C211yesmaximal, characteristic, complemented
10#27132862C211yesmaximal, characteristic, complemented
11#27132856C211yesmaximal, characteristic, complemented
12C4⋊C8×D191C211yesmaximal, characteristic, complemented
13#27132658C211yesmaximal, characteristic, complemented
14#27132656C211yesmaximal, characteristic, complemented
15#463×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×#15690286C2211yescharacteristic
19#20259343—24no—
20#20259341—24no—
21C2×Dic764C2211yescharacteristic
22C2×#15690277C2211yescharacteristic
23C2×#15690276C2211yescharacteristic
24C2×#15690275C2211yescharacteristic
25C2×#15690275C2211yescharacteristic
26C2×#15690272C2211yescharacteristic
27C2×C8×D191C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259320C2211yescharacteristic
30#20259318C2211yescharacteristic
31#20259310C2211yescharacteristic
32#20259295—21no—
33#20259295—21no—
34C4×Dic382C2211yescharacteristic
35#20259287C2211yescharacteristic
36D4×Dic191C2211yescharacteristic
37C4×C22⋊D191—21no—
38(C22⋊D191)⋊C4—21no—
39(C22⋊D191)⋊C4—21no—
40(C22⋊D191)⋊C4—21no—
41#20259280C2211yescharacteristic
42#20259278—21no—
43C2×C4×Dic191—21no—
44C42×D191C2211yescharacteristic
45C4⋊(C191⋊C8)C2211yescharacteristic
46Dic191⋊C8C2211yescharacteristic
47Dic191⋊C8C2211yescharacteristic
48#20259247C2211yescharacteristic
49#20259246C2211yescharacteristic
50#20259245C2211yescharacteristic
51#20259244C2211yescharacteristic
52#20259243C2211yescharacteristic
53#20259242C2211yescharacteristic
54C2×SD16×C191C2211yescharacteristic
55C2×D8×C191C2211yescharacteristic
56#124×C191C2211yescharacteristic
57C4×D4×C191C2211yescharacteristic
58C4⋊C8×C191C2211yescharacteristic
59#103×C191C2211yescharacteristic
60#102×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—22no—
63#15690286—22no—
64#15690286—22no—
65#15690286—22no—
66#15690286D414yescomplemented
67C2×Dic382C2311yescharacteristic
68C2×Dic382C2311yescharacteristic
69C2×D764—21no—
70C2×D764C2311yescharacteristic
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C22×Dic191—21no—
76C22×Dic191—21no—
77C22×Dic191—21no—
78C2×C4×D191—21no—
79C2×C4×D191D411yescharacteristic
80C2×C4×D191D411yescharacteristic
81C2×C4×D191C2311yescharacteristic
82Dic764—22no—
83#15690277—22no—
84#15690276—22no—
85#15690275—22no—
86#15690275—22no—
87#15690272—22no—
88C2×C191⋊C8C2311yescharacteristic
89C2×C191⋊C8C2311yescharacteristic
90C8×D191—22no—
91C8×D191—22no—
92C4⋊Dic191C2311yescharacteristic
93Dic191⋊C4—21no—
94Dic191⋊C4—21no—
95Dic191⋊C4—21no—
96Dic191⋊C4—21no—
97C22⋊Dic191—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4—21no—
101C4×Dic191—21no—
102C4×Dic191—21no—
103C4×Dic191—21no—
104C4×Dic191D411yescharacteristic
105C4×Dic191D411yescharacteristic
106C4×Dic191C2311yescharacteristic
107C2×Q8×C191C2311yescharacteristic
108C2×D4×C191C2311yescharacteristic
109C2×D4×C191C2311yescharacteristic
110C22×C4×C191—21no—
111SD16×C191—22no—
112D8×C191—22no—
113C2×C8×C191C2311yescharacteristic
114C2×C8×C191C2311yescharacteristic
115C4⋊C4×C191C2311yescharacteristic
116C22⋊C4×C191—21no—
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C42×C191C2311yescharacteristic
120C22×D191—21no—
121C22×D191C2×D411yescharacteristic
122Dic382—21no—
123Dic382—21no—
124Dic382—21no—
125Dic382C2×D412yes—
126D764—21no—
127D764—21no—
128D764—42no—
129C22⋊D191—24no—
130C22⋊D191—42no—
131C22⋊D191—42no—
132C22⋊D191—42no—
133C2×Dic191—21no—
134C2×Dic191—24no—
135C2×Dic191—21no—
136C2×Dic191—21no—
137C2×Dic191—42no—
138C2×Dic191—42no—
139C2×Dic191C2×D411yescharacteristic
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C4×D191—22no—
143C4×D191—41no—
144C4×D191—42no—
145C4×D191C2×D412yes—
146C4×D191#4112yes—
147C191⋊C8—21no—
148C191⋊C8—21no—
149C23×C191—21no—
150C23×C191—21no—
151Q8×C191—21no—
152Q8×C191—21no—
153D4×C191—21no—
154D4×C191—21no—
155D4×C191—21no—
156D4×C191C2×D412yes—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—42no—
160C2×C4×C191C2×D411yescharacteristic
161C2×C4×C191C2×D411yescharacteristic
162C2×C4×C191C2411yescharacteristic
163C8×C191—21no—
164C8×C191—21no—
165C22×C191—22no—
166C22×C191—41no—
167C22×C191—41no—
168C22×C191—41no—
169C22×C191—41no—
170C22×C191—41no—
171C22×C191C22×D411yescharacteristic
172C2×D191—21no—
173C2×D191—21no—
174C2×D191—41no—
175C2×D191—41no—
176C2×D191—41no—
177C2×D191#12112yes—
178C4×C191—21no—
179C4×C191—21no—
180C4×C191—41no—
181C4×C191—41no—
182C4×C191C22×D411yescharacteristic
183C4×C191C2×#4111yescharacteristic
184Dic191—22no—
185Dic191—21no—
186Dic191—41no—
187Dic191—41no—
188Dic191#12112yes—
189Dic191#13512yescomplemented
190C2×C191—42no—
191C2×C191—81no—
192C2×C191C2×#12111yescharacteristic
193C2×C191C2×#13511yescharacteristic
194C2×C191#57511yescharacteristic
195D191—22no—
196D191—81no—
197C191#303511yescharacteristic, complemented, Sylow
198#3035—1911nomaximal, Sylow
199C2×#135—1911no—
200C2×#135—1911no—
201#532—1911no—
202C4×#41—1911no—
203#464—1911no—
204#464—1911no—
205#463—1911no—
206#463D19111yescharacteristic, complemented
207#462—1911no—
208#461—1911no—
209#460—1911no—
210#459—1911no—
211C2×C4⋊C8—1911no—
212#416—1911no—
213#416—1911no—
214C2×#41—1911no—
215C2×#41—1911no—
216C2×#41—1911no—
217#135—3824no—
218#135—3824no—
219C2×Q16—1911no—
220C2×Q16—1911no—
221C2×SD16—1911no—
222C2×SD16—1911no—
223C2×SD16—1911no—
224C2×SD16C2×D19111yescharacteristic
225C2×D8—1911no—
226C2×D8C2×D19111yescharacteristic
227C22×C8—1911no—
228C22×C8—1911no—
229C4⋊Q8—1911no—
230#127—1911no—
231#124—1911no—
232#124C2×D19111yescharacteristic
233#121—3821no—
234#121—3821no—
235C4×Q8—1911no—
236C4×Q8—1911no—
237C4×D4—3821no—
238C4×D4—3821no—
239C4×D4—3821no—
240C4×D4—3821no—
241C4×D4—1911no—
242C4×D4C2×D19111yescharacteristic
243#117—3821no—
244#117—1911no—
245C2×C42—3821no—
246C2×C42—1911no—
247C4⋊C8—1911no—
248C4⋊C8—1911no—
249C4⋊C8—1911no—
250C4⋊C8C2×D19111yescharacteristic
251#103—1911no—
252#103—1911no—
253#103—1911no—
254#103C2×D19111yescharacteristic
255#102—1911no—
256#102—1911no—
257#102—1911no—
258#102C2×D19111yescharacteristic
259#41—3822no—
260#41—3822no—
261#41—3822no—
262#41—3822no—
263#41—3822no—
264#41—1914no—
265C2×Q8—1911no—
266C2×Q8—1911no—
267C2×Q8C22×D19111yescharacteristic
268C2×D4—3821no—
269C2×D4—3821no—
270C2×D4—3821no—
271C2×D4—3821no—
272C2×D4—3821no—
273C2×D4—1911no—
274C2×D4C22×D19111yescharacteristic
275C2×D4C22×D19111yescharacteristic
276C22×C4—21no—
277C22×C4—3821no—
278C22×C4—3821no—
279C22×C4—3821no—
280C22×C4—3821no—
281C22×C4—1911no—
282C22×C4—1911no—
283C22×C4—1911no—
284Q16—3822no—
285Q16—3822no—
286SD16—22no—
287SD16—3822no—
288SD16—3822no—
289SD16—3822no—
290D8—22no—
291D8—3822no—
292C2×C8—3822no—
293C2×C8—3822no—
294C2×C8—1911no—
295C2×C8—1911no—
296C2×C8C22×D19111yescharacteristic
297C2×C8C22×D19111yescharacteristic
298C4⋊C4—3821no—
299C4⋊C4—3821no—
300C4⋊C4—3821no—
301C4⋊C4—3821no—
302C4⋊C4—1911no—
303C4⋊C4C22×D19111yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—3821no—
308C22⋊C4—3821no—
309C22⋊C4—3821no—
310C22⋊C4—3821no—
311C42—3821no—
312C42—3821no—
313C42—3821no—
314C42—1911no—
315C42—1911no—
316C42—1911no—
317C42C22×D19111yescharacteristic
318C23—21no—
319C23—21no—
320C23—3821no—
321C23—1911no—
322Q8—21no—
323Q8—21no—
324Q8—3821no—
325Q8—3821no—
326Q8—3821no—
327Q8—1912no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—3824no—
332D4—3821no—
333D4—3821no—
334D4—7642no—
335D4—7642no—
336D4—7642no—
337D4—7642no—
338D4D4×D19112yes—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—3822no—
343C2×C4—3821no—
344C2×C4—3824no—
345C2×C4—3821no—
346C2×C4—3821no—
347C2×C4—1911no—
348C2×C4—1911no—
349C2×C4—1911no—
350C2×C4—1912no—
351C2×C4—1912no—
352C2×C4—7641no—
353C2×C4—7642no—
354C2×C4—7642no—
355C2×C4—7642no—
356C2×C4D4×D19111yescharacteristic
357C2×C4D4×D19111yescharacteristic
358C2×C4C23×D19111yescharacteristic
359C8—21no—
360C8—21no—
361C8—3821no—
362C8—3821no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—3821no—
370C22—3821no—
371C22—1912no—
372C22—7641no—
373C22—7641no—
374C22—7641no—
375C22C2×D4×D19111yescharacteristic
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—3821no—
381C4—3822no—
382C4—1912no—
383C4—1912no—
384C4—7641no—
385C4—7641no—
386C4C2×D4×D19111yescharacteristic
387C4#41×D19111yescharacteristic
388C2—42no—
389C2—81no—
390C2—3822no—
391C2—15281no—
392C2#121×D19111yescharacteristic
393C2#135×D19111yescharacteristic
394C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#463×C191C2
socleC22×C191C22×D4
hypercenter#463D191
hypocenterC191#3035
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#20259343C2semidirect
2. C2×#20259341C2semidirect
3. #27133212C2semidirect
4. C4×#15690286C2semidirect
5. #27132886C2semidirect
6. #27132882C2semidirect
7. #27132878C2semidirect
8. #27132872C2semidirect
9. #27132866C2semidirect
10. #27132862C2semidirect
11. #27132856C2semidirect
12. C4⋊C8×D191C2semidirect
13. #27132658C2semidirect
14. #27132656C2semidirect
15. #463×C191C2semidirect
66. #15690286D4semidirect
189. Dic191#135semidirect
197. C191#3035semidirect
206. #463D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593432
C2×#202593412
#271332122
C4×#156902862
#271328862
#271328822
#271328782
#271328722
#271328662
#271328622
#271328562
C4⋊C8×D1912
#271326582
#271326562
#463×C1912
#3035191