Finite Group Databasebeta

← #49161836

Subgroups — #49161836

Order 37504.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627597C211yesmaximal, characteristic
2C2×C4×D1172C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D293C211yesmaximal, characteristic, complemented
4#34036022C211yesmaximal, characteristic, complemented
5#34036020C211yesmaximal, characteristic, complemented
6#34036016C211yesmaximal, characteristic, complemented
7#34036013C211yesmaximal, characteristic, complemented
8#34036008C211yesmaximal, characteristic, complemented
9#34036007C211yesmaximal, characteristic, complemented
10#34036005C211yesmaximal, characteristic, complemented
11#34036005C211yesmaximal, characteristic, complemented
12C4×#17989779C211yesmaximal, characteristic, complemented
13#34035976C211yesmaximal, characteristic, complemented
14#95×D293C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C293C211yesmaximal, characteristic, complemented
16C22×D1172C2211yescharacteristic
17C22×C22⋊D293C2211yescharacteristic
18C22×C22⋊D293C2211yescharacteristic
19C22×C4×D293C2211yescharacteristic
20C22×C4×D293C2211yescharacteristic
21C22×C4×D293C2211yescharacteristic
22C22×C4×D293C2211yescharacteristic
23#22627597C418yescomplemented
24C4×D1172—24no—
25C4×C22⋊D293—24no—
26C2×C4⋊Dic293C2211yescharacteristic
27C2×#17989781C2211yescharacteristic
28C2×#17989781C2211yescharacteristic
29C2×C22⋊Dic293C2211yescharacteristic
30C2×C22⋊Dic293C2211yescharacteristic
31C2×#17989779C2211yescharacteristic
32C2×#17989779C2211yescharacteristic
33C2×#17989779C2211yescharacteristic
34C2×#17989779C2211yescharacteristic
35C2×#17989779C2211yescharacteristic
36C2×#17989779C2211yescharacteristic
37C2×C4×Dic293C2211yescharacteristic
38C2×C4×Dic293C2211yescharacteristic
39#22627531C2211yescharacteristic
40#22627531C2211yescharacteristic
41#22627531C2211yescharacteristic
42#22627531C2211yescharacteristic
43#22627531C2211yescharacteristic
44#22627531C2211yescharacteristic
45#22627531C2211yescharacteristic
46#22627531C2211yescharacteristic
47C23×C4×C293C2211yescharacteristic
48C2×C22⋊C4×C293C2211yescharacteristic
49C2×C22⋊C4×C293C2211yescharacteristic
50C2×C42×C293C2211yescharacteristic
51C2×C42×C293C2211yescharacteristic
52#95×C293C2211yescharacteristic
53#95×C293C2211yescharacteristic
54C23×D293C2311yescharacteristic
55C23×D293C2311yescharacteristic
56C2×D1172—22no—
57C2×D1172—22no—
58C2×D1172C2×C414yescomplemented
59C2×C22⋊D293—22no—
60C2×C22⋊D293—22no—
61C2×C22⋊D293—41no—
62C2×C22⋊D293—41no—
63C2×C22⋊D293C2×C414yescomplemented
64C2×C22⋊D293C2×C414yescomplemented
65C22×Dic293C2311yescharacteristic
66C22×Dic293C2311yescharacteristic
67C22×Dic293C2311yescharacteristic
68C22×Dic293C2311yescharacteristic
69C22×Dic293C2311yescharacteristic
70C22×Dic293C2311yescharacteristic
71C2×C4×D293—22no—
72C2×C4×D293—22no—
73C2×C4×D293—22no—
74C2×C4×D293—22no—
75C2×C4×D293—22no—
76C2×C4×D293—22no—
77C2×C4×D293—41no—
78C2×C4×D293—41no—
79C2×C4×D293—41no—
80C2×C4×D293—41no—
81C2×C4×D293—41no—
82C2×C4×D293C2×C414yescomplemented
83C4⋊Dic293—22no—
84#17989781—22no—
85#17989781C2×C414yescomplemented
86C22⋊Dic293—22no—
87C22⋊Dic293—41no—
88#17989779—22no—
89#17989779—22no—
90#17989779—22no—
91#17989779—41no—
92#17989779C2×C414yescomplemented
93#17989779D414yescomplemented
94C4×Dic293—22no—
95C4×Dic293—41no—
96C24×C293C2311yescharacteristic
97C22×C4×C293—22no—
98C22×C4×C293—22no—
99C22×C4×C293—41no—
100C22×C4×C293C2311yescharacteristic
101C22×C4×C293C2311yescharacteristic
102C22×C4×C293C2311yescharacteristic
103C22×C4×C293C2311yescharacteristic
104C22×C4×C293C2311yescharacteristic
105C22×C4×C293C2311yescharacteristic
106C22⋊C4×C293—22no—
107C22⋊C4×C293C2×C414yescomplemented
108C42×C293—22no—
109C42×C293—41no—
110C22×D293—21no—
111C22×D293—21no—
112C22×D293—21no—
113C22×D293—21no—
114C22×D293—21no—
115C22×D293—21no—
116C22×D293—21no—
117C22×D293—21no—
118C22×D293—21no—
119C22×D293—21no—
120C22×D293C22×C412yes—
121C22×D293C22×C412yes—
122C22×D293C2×D412yes—
123C22×D293#4112yes—
124D1172—28no—
125C22⋊D293—28no—
126C22⋊D293—44no—
127C2×Dic293—21no—
128C2×Dic293—21no—
129C2×Dic293—21no—
130C2×Dic293—21no—
131C2×Dic293—21no—
132C2×Dic293—21no—
133C2×Dic293—21no—
134C2×Dic293—21no—
135C2×Dic293—21no—
136C2×Dic293—21no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C2×Dic293—21no—
140C2×Dic293—21no—
141C2×Dic293—21no—
142C2×Dic293C22×C412yes—
143C2×Dic293C22×C412yes—
144C2×Dic293C2×D412yes—
145C4×D293—44no—
146C4×D293—44no—
147C4×D293—44no—
148C4×D293—44no—
149C23×C293—21no—
150C23×C293—21no—
151C23×C293—21no—
152C23×C293—21no—
153C23×C293—21no—
154C23×C293—21no—
155C23×C293C22×C412yes—
156C23×C293C2411yescharacteristic
157C2×C4×C293—21no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—21no—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293—21no—
168C2×C4×C293—21no—
169C2×C4×C293—21no—
170C2×C4×C293—44no—
171C2×C4×C293C22×C412yes—
172C2×C4×C293C22×C412yes—
173C2×C4×C293C2×D412yes—
174C2×C4×C293#4112yes—
175C2×C4×C293#4112yes—
176C22×C293—22no—
177C22×C293—22no—
178C22×C293—22no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293—41no—
182C22×C293—41no—
183C22×C293C23×C411yescharacteristic
184C22×C293C22×D411yescharacteristic
185C22×C293C2×#4111yescharacteristic
186C22×C293C2×#4111yescharacteristic
187C22×C293C2×#4111yescharacteristic
188C22×C293H(5,𝔽2)11yescharacteristic
189C22×C293H(5,𝔽2)11yescharacteristic
190C2×D293—22no—
191C2×D293—22no—
192C2×D293—22no—
193C2×D293—22no—
194C2×D293—22no—
195C2×D293—22no—
196C2×D293—22no—
197C2×D293—22no—
198C2×D293—22no—
199C2×D293—41no—
200C2×D293—41no—
201C2×D293—41no—
202C2×D293—41no—
203C2×D293C4×D414yes—
204C4×C293—22no—
205C4×C293—22no—
206C4×C293—22no—
207C4×C293—22no—
208C4×C293—22no—
209C4×C293—41no—
210Dic293—22no—
211Dic293—22no—
212Dic293—22no—
213Dic293—41no—
214Dic293—41no—
215Dic293—41no—
216C2×C293—42no—
217C2×C293C2×C4×D411yescharacteristic
218C2×C293C4×#4111yescharacteristic
219C2×C293#51811yescharacteristic
220C2×C293#54611yescharacteristic
221C2×C293#54611yescharacteristic
222C2×C293#55311yescharacteristic
223C2×C293#55311yescharacteristic
224D293—24no—
225D293—42no—
226C293#236411yescharacteristic, complemented, Sylow
227#2364—2931nomaximal, Sylow
228C2×#121—2931no—
229C2×C4×D4—2931no—
230C2×C4×D4—2931no—
231#388—2931no—
232#388—2931no—
233#387—2931no—
234#387—2931no—
235#386—2931no—
236#386—2931no—
237#386—2931no—
238#386—2931no—
239C4×C22⋊C4—2931no—
240C4×C22⋊C4—2931no—
241C4×C22⋊C4D29311yescharacteristic, complemented
242C2×#95—2931no—
243C22×D4—2931no—
244C22×D4—2931no—
245C22×D4—2931no—
246C23×C4—2931no—
247C23×C4—2931no—
248C23×C4—2931no—
249C23×C4—2931no—
250C23×C4C2×D29311yescharacteristic
251#121—2938no—
252C4×D4—5864no—
253C4×D4—5864no—
254C2×C4⋊C4—2931no—
255C2×C4⋊C4—2931no—
256C2×C4⋊C4—2931no—
257C2×C22⋊C4—2931no—
258C2×C22⋊C4—2931no—
259C2×C22⋊C4—2931no—
260C2×C22⋊C4—2931no—
261C2×C22⋊C4—2931no—
262C2×C22⋊C4—2931no—
263C2×C22⋊C4—2931no—
264C2×C22⋊C4—2931no—
265C2×C22⋊C4C2×D29311yescharacteristic
266C2×C22⋊C4C2×D29311yescharacteristic
267C2×C42—2931no—
268C2×C42—2931no—
269C2×C42C2×D29311yescharacteristic
270C2×C42C2×D29311yescharacteristic
271#95—2931no—
272#95—2931no—
273#95—2931no—
274#95—2931no—
275#95—2931no—
276#95—2931no—
277#95—2931no—
278#95—2931no—
279#95C2×D29311yescharacteristic
280#95C2×D29311yescharacteristic
281C24—2931no—
282C24—2931no—
283C24C22×D29311yescharacteristic
284C2×D4—2934no—
285C2×D4—2934no—
286C2×D4—2934no—
287C2×D4—5862no—
288C2×D4—5862no—
289C2×D4—5862no—
290C2×D4—5862no—
291C2×D4—11721no—
292C2×D4—11721no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—2931no—
297C22×C4—2931no—
298C22×C4—2931no—
299C22×C4—2934no—
300C22×C4—2931no—
301C22×C4—2931no—
302C22×C4—2931no—
303C22×C4—5862no—
304C22×C4—5862no—
305C22×C4—5862no—
306C22×C4—5862no—
307C22×C4—5862no—
308C22×C4—5862no—
309C22×C4—11721no—
310C22×C4—11721no—
311C22×C4—11721no—
312C22×C4—11721no—
313C22×C4—11721no—
314C22×C4C22×D29311yescharacteristic
315C22×C4C22×D29311yescharacteristic
316C22×C4C22×D29311yescharacteristic
317C22×C4C22×D29311yescharacteristic
318C22×C4C22×D29311yescharacteristic
319C22×C4C22×D29311yescharacteristic
320C4⋊C4—2934no—
321C4⋊C4—5862no—
322C4⋊C4—5862no—
323C22⋊C4—22no—
324C22⋊C4—2934no—
325C22⋊C4—2934no—
326C22⋊C4—5862no—
327C22⋊C4—5862no—
328C22⋊C4—5862no—
329C22⋊C4—5862no—
330C22⋊C4—11721no—
331C22⋊C4—11721no—
332C22⋊C4C4×D29314yescomplemented
333C42—22no—
334C42—41no—
335C42—5862no—
336C42—11721no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—2932no—
344C23—2932no—
345C23—2932no—
346C23—2932no—
347C23—5861no—
348C23—5861no—
349C23—5861no—
350C23—5861no—
351C23—5861no—
352C23—5861no—
353C23—5861no—
354C23—5861no—
355C23—5861no—
356C23—5861no—
357C23C2×C4×D29312yes—
358C23C23×D29311yescharacteristic
359D4—5868no—
360D4—5868no—
361D4—11724no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—2932no—
377C2×C4—2932no—
378C2×C4—2932no—
379C2×C4—5861no—
380C2×C4—5861no—
381C2×C4—5861no—
382C2×C4—5861no—
383C2×C4—5861no—
384C2×C4—5861no—
385C2×C4—5861no—
386C2×C4—5861no—
387C2×C4—5861no—
388C2×C4—5861no—
389C2×C4—5861no—
390C2×C4—5861no—
391C2×C4—5861no—
392C2×C4—5861no—
393C2×C4—5861no—
394C2×C4—11724no—
395C2×C4—11724no—
396C2×C4—11724no—
397C2×C4—11724no—
398C2×C4C2×C4×D29312yes—
399C2×C4C2×C4×D29312yes—
400C2×C4D4×D29312yes—
401C2×C4#1798980212yes—
402C2×C4#1798980212yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—2934no—
411C22—5862no—
412C22—5862no—
413C22—5862no—
414C22—5862no—
415C22—5862no—
416C22—5862no—
417C22—5862no—
418C22—5862no—
419C22—5862no—
420C22—11721no—
421C22—11721no—
422C22—11721no—
423C22—11721no—
424C22C22×C4×D29311yescharacteristic
425C22C2×D4×D29311yescharacteristic
426C22#41×D29311yescharacteristic
427C22C2×#1798980211yescharacteristic
428C22C2×#1798980211yescharacteristic
429C22#2262766911yescharacteristic
430C22#2262767011yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—41no—
437C4—5862no—
438C4—5862no—
439C4—5862no—
440C4—11721no—
441C4—11721no—
442C4—11721no—
443C2—42no—
444C2—5864no—
445C2—11722no—
446C2C4×D4×D29311yescharacteristic
447C2C4×#1798980211yescharacteristic
448C2#3403660111yescharacteristic
449C2#3403676911yescharacteristic
450C2#3403677511yescharacteristic
451C2#3403682311yescharacteristic
452C2#3403682911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
FittingC4×C22⋊C4×C293C2
socle, cosocleC23×C293C24
prime radicalC2×#22627597C2
hypercenterC4×C22⋊C4D293
hypocenterC293#2364
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D1172C2semidirect
3. C2×C4×C22⋊D293C2semidirect
4. #34036022C2semidirect
5. #34036020C2semidirect
6. #34036016C2semidirect
7. #34036013C2semidirect
8. #34036008C2semidirect
9. #34036007C2semidirect
10. #34036005C2semidirect
11. #34036005C2semidirect
12. C4×#17989779C2semidirect
13. #34035976C2semidirect
14. #95×D293C2semidirect
15. C4×C22⋊C4×C293C2semidirect
23. #22627597C4semidirect
58. C2×D1172C2×C4semidirect
63. C2×C22⋊D293C2×C4semidirect
64. C2×C22⋊D293C2×C4semidirect
82. C2×C4×D293C2×C4semidirect
85. #17989781C2×C4semidirect
92. #17989779C2×C4semidirect
93. #17989779D4semidirect
107. C22⋊C4×C293C2×C4semidirect
226. C293#2364semidirect
241. C4×C22⋊C4D293semidirect
332. C22⋊C4C4×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226275972
C2×C4×D11722
C2×C4×C22⋊D2932
#340360222
#340360202
#340360162
#340360132
#340360082
#340360072
#340360052
C4×#179897792
#340359762
#95×D2932
C4×C22⋊C4×C2932
#2364293