Finite Group Databasebeta

← #49168239

Subgroups — #49168239

Order 37504.

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×#22627647C211yesmaximal, characteristic, complemented
2C2×#22627645C211yesmaximal, characteristic, complemented
3#34036678C211yesmaximal, characteristic, complemented
4C4×#17989803C211yesmaximal, characteristic, complemented
5#34036351C211yesmaximal, characteristic, complemented
6#34036345C211yesmaximal, characteristic, complemented
7#34036341C211yesmaximal, characteristic, complemented
8#34036337C211yesmaximal, characteristic, complemented
9#34036333C211yesmaximal, characteristic, complemented
10#34036327C211yesmaximal, characteristic, complemented
11#34036323C211yesmaximal, characteristic, complemented
12C4⋊C8×D293C211yesmaximal, characteristic, complemented
13#34036124C211yesmaximal, characteristic, complemented
14#34036122C211yesmaximal, characteristic, complemented
15#464×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989803C2211yescharacteristic
18C2×#17989801C2211yescharacteristic
19#22627647—24no—
20#22627645—24no—
21C2×#17989792C2211yescharacteristic
22C2×#17989791C2211yescharacteristic
23C2×#17989789C2211yescharacteristic
24C2×#17989789C2211yescharacteristic
25C2×D2344C2211yescharacteristic
26C2×#17989787C2211yescharacteristic
27C2×C8×D293C2211yescharacteristic
28C2×C8×D293C2211yescharacteristic
29#22627622C2211yescharacteristic
30#22627620C2211yescharacteristic
31#22627612C2211yescharacteristic
32#22627597—21no—
33#22627597—21no—
34Q8×Dic293C2211yescharacteristic
35C4×D1172—21no—
36C4×D1172C2211yescharacteristic
37#22627586—21no—
38#22627586C2211yescharacteristic
39#22627584—21no—
40#22627584—21no—
41#22627582—21no—
42#22627582C2211yescharacteristic
43C42×D293—21no—
44C42×D293C2211yescharacteristic
45C4⋊#14556873C2211yescharacteristic
46#22627553C2211yescharacteristic
47#22627553C2211yescharacteristic
48#22627549C2211yescharacteristic
49#22627548C2211yescharacteristic
50#22627547C2211yescharacteristic
51#22627546C2211yescharacteristic
52#22627545C2211yescharacteristic
53#22627544C2211yescharacteristic
54C2×Q16×C293C2211yescharacteristic
55C2×SD16×C293C2211yescharacteristic
56#124×C293C2211yescharacteristic
57C4×Q8×C293C2211yescharacteristic
58C4⋊C8×C293C2211yescharacteristic
59#103×C293C2211yescharacteristic
60#102×C293C2211yescharacteristic
61#17989803—22no—
62#17989803—22no—
63#17989803—22no—
64#17989803D414yescomplemented
65#17989801—22no—
66#17989801—22no—
67C2×Dic586C2311yescharacteristic
68C2×D1172—21no—
69C2×D1172—21no—
70C2×D1172C2311yescharacteristic
71C2×D1172C2311yescharacteristic
72C2×C22⋊D293—21no—
73C2×C22⋊D293—21no—
74C2×C22⋊D293—21no—
75C22×Dic293—21no—
76C2×C4×D293—21no—
77C2×C4×D293—21no—
78C2×C4×D293—21no—
79C2×C4×D293—21no—
80C2×C4×D293D411yescharacteristic
81C2×C4×D293D411yescharacteristic
82C2×C4×D293C2311yescharacteristic
83#17989792—22no—
84#17989791—22no—
85#17989789—22no—
86#17989789—22no—
87D2344—22no—
88#17989787—22no—
89C2×#14556873C2311yescharacteristic
90C2×#14556873C2311yescharacteristic
91C8×D293—22no—
92C8×D293—22no—
93C4⋊Dic293—21no—
94C4⋊Dic293C2311yescharacteristic
95#17989781—21no—
96#17989781—21no—
97#17989779—21no—
98#17989779—21no—
99#17989779—21no—
100#17989779—21no—
101#17989779—21no—
102C4×Dic293—21no—
103C4×Dic293—21no—
104C4×Dic293D411yescharacteristic
105C4×Dic293D411yescharacteristic
106C4×Dic293C2311yescharacteristic
107C2×Q8×C293C2311yescharacteristic
108C2×Q8×C293C2311yescharacteristic
109C2×D4×C293C2311yescharacteristic
110Q16×C293—22no—
111SD16×C293—22no—
112C2×C8×C293C2311yescharacteristic
113C2×C8×C293C2311yescharacteristic
114C4⋊C4×C293—21no—
115C4⋊C4×C293C2311yescharacteristic
116C22⋊C4×C293—21no—
117C22⋊C4×C293—21no—
118C42×C293—21no—
119C42×C293C2311yescharacteristic
120C22×D293—21no—
121C22×D293—21no—
122C22×D293C2×D411yescharacteristic
123Dic586—21no—
124Dic586—21no—
125D1172—24no—
126D1172—21no—
127D1172—21no—
128D1172—21no—
129D1172—42no—
130D1172C2×D412yes—
131C22⋊D293—42no—
132C22⋊D293—42no—
133C22⋊D293—42no—
134C2×Dic293—21no—
135C2×Dic293—21no—
136C2×Dic293—42no—
137C2×Dic293C2×D411yescharacteristic
138C2×Dic293C2×D411yescharacteristic
139C2×Dic293C2×D411yescharacteristic
140C4×D293—24no—
141C4×D293—22no—
142C4×D293—41no—
143C4×D293—42no—
144C4×D293—42no—
145C4×D293—42no—
146C4×D293C2×D412yes—
147C4×D293#4112yes—
148#14556873—21no—
149#14556873—21no—
150C23×C293—21no—
151Q8×C293—21no—
152Q8×C293—21no—
153Q8×C293—21no—
154Q8×C293C2×D412yes—
155D4×C293—21no—
156D4×C293—21no—
157C2×C4×C293—21no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293C2×D411yescharacteristic
161C2×C4×C293C2×D411yescharacteristic
162C2×C4×C293C2411yescharacteristic
163C8×C293—21no—
164C8×C293—21no—
165C22×C293—41no—
166C22×C293—41no—
167C22×C293—41no—
168C22×C293C22×D411yescharacteristic
169C2×D293—21no—
170C2×D293—21no—
171C2×D293—22no—
172C2×D293—41no—
173C2×D293—41no—
174C2×D293—41no—
175C2×D293—41no—
176C2×D293—41no—
177C2×D293#12112yes—
178C4×C293—22no—
179C4×C293—21no—
180C4×C293—21no—
181C4×C293—41no—
182C4×C293—41no—
183C4×C293C22×D411yescharacteristic
184C4×C293C2×#4111yescharacteristic
185Dic293—21no—
186Dic293—41no—
187Dic293—41no—
188Dic293#12112yes—
189Dic293#13512yescomplemented
190C2×C293—81no—
191C2×C293C2×#12111yescharacteristic
192C2×C293C2×#13511yescharacteristic
193C2×C293#57511yescharacteristic
194D293—22no—
195D293—42no—
196D293—81no—
197C293#303511yescharacteristic, complemented, Sylow
198#3035—2931nomaximal, Sylow
199C2×#135—2931no—
200C2×#135—2931no—
201#532—2931no—
202C4×#41—2931no—
203#464—2931no—
204#464D29311yescharacteristic, complemented
205#463—2931no—
206#463—2931no—
207#462—2931no—
208#461—2931no—
209#460—2931no—
210#459—2931no—
211C2×C4⋊C8—2931no—
212#416—2931no—
213#416—2931no—
214C2×#41—2931no—
215C2×#41—2931no—
216C2×#41—2931no—
217#135—5864no—
218#135—5864no—
219C2×Q16—2931no—
220C2×Q16C2×D29311yescharacteristic
221C2×SD16—2931no—
222C2×SD16—2931no—
223C2×SD16—2931no—
224C2×SD16C2×D29311yescharacteristic
225C2×D8—2931no—
226C2×D8—2931no—
227C22×C8—2931no—
228C22×C8—2931no—
229C4⋊Q8—2931no—
230#127—2931no—
231#124—2931no—
232#124C2×D29311yescharacteristic
233#121—5861no—
234#121—5861no—
235C4×Q8—2931no—
236C4×Q8C2×D29311yescharacteristic
237C4×D4—2931no—
238C4×D4—2931no—
239C4×D4—5861no—
240C4×D4—5861no—
241C4×D4—5861no—
242C4×D4—5861no—
243#117—2931no—
244#117—5861no—
245C2×C42—2931no—
246C2×C42—5861no—
247C4⋊C8—2931no—
248C4⋊C8—2931no—
249C4⋊C8—2931no—
250C4⋊C8C2×D29311yescharacteristic
251#103—2931no—
252#103—2931no—
253#103—2931no—
254#103C2×D29311yescharacteristic
255#102—2931no—
256#102—2931no—
257#102—2931no—
258#102C2×D29311yescharacteristic
259#41—2934no—
260#41—5862no—
261#41—5862no—
262#41—5862no—
263#41—5862no—
264#41—5862no—
265C2×Q8—2931no—
266C2×Q8C22×D29311yescharacteristic
267C2×Q8C22×D29311yescharacteristic
268C2×D4—2931no—
269C2×D4—2931no—
270C2×D4—5861no—
271C2×D4—5861no—
272C2×D4—5861no—
273C2×D4—5861no—
274C2×D4—5861no—
275C2×D4C22×D29311yescharacteristic
276C22×C4—2931no—
277C22×C4—2931no—
278C22×C4—2931no—
279C22×C4—5861no—
280C22×C4—5861no—
281C22×C4—5861no—
282C22×C4—5861no—
283C22×C4—5861no—
284Q16—22no—
285Q16—5862no—
286SD16—22no—
287SD16—5862no—
288SD16—5862no—
289SD16—5862no—
290D8—5862no—
291D8—5862no—
292C2×C8—2931no—
293C2×C8—2931no—
294C2×C8—5862no—
295C2×C8—5862no—
296C2×C8C22×D29311yescharacteristic
297C2×C8C22×D29311yescharacteristic
298C4⋊C4—21no—
299C4⋊C4—2931no—
300C4⋊C4—5861no—
301C4⋊C4—5861no—
302C4⋊C4—5861no—
303C4⋊C4C22×D29311yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—5861no—
307C22⋊C4—5861no—
308C22⋊C4—5861no—
309C22⋊C4—5861no—
310C22⋊C4—5861no—
311C42—21no—
312C42—2931no—
313C42—2931no—
314C42—2931no—
315C42—5861no—
316C42—5861no—
317C42C22×D29311yescharacteristic
318C23—21no—
319C23—2931no—
320C23—5861no—
321C23—5861no—
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—5861no—
326Q8—5861no—
327Q8D4×D29312yes—
328D4—21no—
329D4—21no—
330D4—2932no—
331D4—5864no—
332D4—5861no—
333D4—5861no—
334D4—5861no—
335D4—11722no—
336D4—11722no—
337D4—11722no—
338D4—11722no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2931no—
343C2×C4—2931no—
344C2×C4—2932no—
345C2×C4—2932no—
346C2×C4—2931no—
347C2×C4—5862no—
348C2×C4—5864no—
349C2×C4—5861no—
350C2×C4—5861no—
351C2×C4—11721no—
352C2×C4—11722no—
353C2×C4—11722no—
354C2×C4—11722no—
355C2×C4—11722no—
356C2×C4D4×D29311yescharacteristic
357C2×C4D4×D29311yescharacteristic
358C2×C4C23×D29311yescharacteristic
359C8—21no—
360C8—21no—
361C8—5861no—
362C8—5861no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—2932no—
367C22—5861no—
368C22—5861no—
369C22—5862no—
370C22—11721no—
371C22—11721no—
372C22—11721no—
373C22—11721no—
374C22—11721no—
375C22C2×D4×D29311yescharacteristic
376C4—22no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—2932no—
382C4—2932no—
383C4—5861no—
384C4—11721no—
385C4—11721no—
386C4C2×D4×D29311yescharacteristic
387C4#41×D29311yescharacteristic
388C2—81no—
389C2—23441no—
390C2—5862no—
391C2—11722no—
392C2#121×D29311yescharacteristic
393C2#135×D29311yescharacteristic
394C2#3403692811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#464×C293C2
socleC22×C293C22×D4
hypercenter#464D293
hypocenterC293#3035
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22627647C2semidirect
2. C2×#22627645C2semidirect
3. #34036678C2semidirect
4. C4×#17989803C2semidirect
5. #34036351C2semidirect
6. #34036345C2semidirect
7. #34036341C2semidirect
8. #34036337C2semidirect
9. #34036333C2semidirect
10. #34036327C2semidirect
11. #34036323C2semidirect
12. C4⋊C8×D293C2semidirect
13. #34036124C2semidirect
14. #34036122C2semidirect
15. #464×C293C2semidirect
64. #17989803D4semidirect
189. Dic293#135semidirect
197. C293#3035semidirect
204. #464D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276472
C2×#226276452
#340366782
C4×#179898032
#340363512
#340363452
#340363412
#340363372
#340363332
#340363272
#340363232
C4⋊C8×D2932
#340361242
#340361222
#464×C2932
#3035293