Finite Group Databasebeta

← #16822190

Subgroups — #16822190

Order 3712.

696 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×#11745301C211yesmaximal, characteristic
2C2×C4×C22⋊D29C211yesmaximal, characteristic, complemented
3#14271054C211yesmaximal, characteristic, complemented
4#14271051C211yesmaximal, characteristic, complemented
5#14271047C211yesmaximal, characteristic, complemented
6#14271044C211yesmaximal, characteristic, complemented
7#14271043C211yesmaximal, characteristic, complemented
8#14271042C211yesmaximal, characteristic, complemented
9#14271040C211yesmaximal, characteristic, complemented
10#14271039C211yesmaximal, characteristic, complemented
11#14271036C211yesmaximal, characteristic, complemented
12#14271034C211yesmaximal, characteristic, complemented
13#14271021C211yesmaximal, characteristic, complemented
14C2×#11745179C211yesmaximal, characteristic, complemented
15C2×C4×D4×C29C211yesmaximal, characteristic, complemented
16C22×D4×C29C2211yescharacteristic
17C23×C4×C29C2211yescharacteristic
18C23×C4×C29C2211yescharacteristic
19C22×C22⋊D29C2211yescharacteristic
20C22×C22⋊D29C2211yescharacteristic
21C23×Dic29C2211yescharacteristic
22C23×Dic29C2211yescharacteristic
23C22×C4×D29C2211yescharacteristic
24C4×D4×C29—24no—
25C2×C4⋊C4×C29C2211yescharacteristic
26C2×C22⋊C4×C29C2211yescharacteristic
27C2×C22⋊C4×C29C2211yescharacteristic
28C2×C42×C29C2211yescharacteristic
29C2×C22⋊Dic29C2211yescharacteristic
30C2×C22⋊Dic29C2211yescharacteristic
31C2×C22⋊Dic29C2211yescharacteristic
32C2×C22⋊Dic29C2211yescharacteristic
33#11745301C418yescomplemented
34C4×C22⋊D29—24no—
35C2×#90781C2211yescharacteristic
36C2×#90781C2211yescharacteristic
37C2×#90781C2211yescharacteristic
38C2×#90781C2211yescharacteristic
39C2×#90779C2211yescharacteristic
40C2×#90779C2211yescharacteristic
41C2×#90779C2211yescharacteristic
42C2×#90779C2211yescharacteristic
43C2×C4×Dic29C2211yescharacteristic
44#11745179C2211yescharacteristic
45#11745179C2211yescharacteristic
46#11745179C2211yescharacteristic
47#11745179C2211yescharacteristic
48#11745179C2211yescharacteristic
49#11745179C2211yescharacteristic
50#11745179C2211yescharacteristic
51#11745179C2211yescharacteristic
52#11745179C2211yescharacteristic
53#11745179C2211yescharacteristic
54C24×C29C2311yescharacteristic
55C24×C29C2311yescharacteristic
56C23×D29C2311yescharacteristic
57C2×D4×C29—22no—
58C2×D4×C29—22no—
59C2×D4×C29C2×C414yescomplemented
60C22×C4×C29—22no—
61C22×C4×C29—41no—
62C22×C4×C29—41no—
63C22×C4×C29—41no—
64C22×C4×C29—41no—
65C22×C4×C29D414yescomplemented
66C22×C4×C29C2311yescharacteristic
67C22×C4×C29C2311yescharacteristic
68C22×C4×C29C2311yescharacteristic
69C22×C4×C29C2311yescharacteristic
70C22×C4×C29C2311yescharacteristic
71C2×C22⋊D29—22no—
72C2×C22⋊D29—22no—
73C2×C22⋊D29—41no—
74C2×C22⋊D29—41no—
75C2×C22⋊D29C2×C414yescomplemented
76C2×C22⋊D29C2×C414yescomplemented
77C22×Dic29—22no—
78C22×Dic29—41no—
79C22×Dic29—41no—
80C22×Dic29—41no—
81C22×Dic29—41no—
82C22×Dic29C2×C414yescomplemented
83C22×Dic29C2311yescharacteristic
84C22×Dic29C2311yescharacteristic
85C22×Dic29C2311yescharacteristic
86C22×Dic29C2311yescharacteristic
87C22×Dic29C2311yescharacteristic
88C22×Dic29C2311yescharacteristic
89C22×Dic29C2311yescharacteristic
90C2×C4×D29—22no—
91C2×C4×D29—41no—
92C2×C4×D29—41no—
93C4⋊C4×C29—22no—
94C22⋊C4×C29—22no—
95C22⋊C4×C29—22no—
96C42×C29—22no—
97C22⋊Dic29—22no—
98C22⋊Dic29—41no—
99C22⋊Dic29—41no—
100C22⋊Dic29C2×C414yescomplemented
101#90781—22no—
102#90781—41no—
103#90781—41no—
104#90781C2×C414yescomplemented
105#90779—22no—
106#90779—41no—
107#90779—41no—
108#90779C2×C414yescomplemented
109C4×Dic29—22no—
110C23×C29—21no—
111C23×C29—21no—
112C23×C29—21no—
113C23×C29—21no—
114C23×C29—21no—
115C23×C29—21no—
116C23×C29—21no—
117C23×C29—21no—
118C23×C29—21no—
119C23×C29—21no—
120C23×C29C22×C412yes—
121C23×C29C22×C412yes—
122C23×C29C2×D412yes—
123C23×C29#4112yes—
124C23×C29C2411yescharacteristic
125C22×D29—21no—
126C22×D29—21no—
127C22×D29—21no—
128C22×D29—21no—
129C22×D29—21no—
130C22×D29—21no—
131C22×D29C22×C412yes—
132D4×C29—28no—
133C2×C4×C29—21no—
134C2×C4×C29—21no—
135C2×C4×C29—21no—
136C2×C4×C29—21no—
137C2×C4×C29—21no—
138C2×C4×C29—21no—
139C2×C4×C29—21no—
140C2×C4×C29—21no—
141C2×C4×C29—21no—
142C2×C4×C29—21no—
143C2×C4×C29—21no—
144C2×C4×C29—21no—
145C2×C4×C29—44no—
146C2×C4×C29—44no—
147C2×C4×C29C22×C412yes—
148C2×C4×C29C2×D412yes—
149C2×C4×C29C2×D412yes—
150C22⋊D29—28no—
151C22⋊D29—44no—
152C2×Dic29—21no—
153C2×Dic29—21no—
154C2×Dic29—21no—
155C2×Dic29—21no—
156C2×Dic29—21no—
157C2×Dic29—21no—
158C2×Dic29—21no—
159C2×Dic29—21no—
160C2×Dic29—21no—
161C2×Dic29—21no—
162C2×Dic29—21no—
163C2×Dic29—21no—
164C2×Dic29—21no—
165C2×Dic29—21no—
166C2×Dic29—21no—
167C2×Dic29—21no—
168C2×Dic29—21no—
169C2×Dic29—21no—
170C2×Dic29—44no—
171C2×Dic29—44no—
172C2×Dic29C22×C412yes—
173C2×Dic29C22×C412yes—
174C2×Dic29C22×C412yes—
175C4×D29—44no—
176#2325—291nomaximal, Sylow
177C22×C29—22no—
178C22×C29—22no—
179C22×C29—22no—
180C22×C29—22no—
181C22×C29—22no—
182C22×C29—22no—
183C22×C29—22no—
184C22×C29—22no—
185C22×C29—22no—
186C22×C29—41no—
187C22×C29—41no—
188C22×C29—41no—
189C22×C29—41no—
190C22×C29C4×D414yes—
191C22×C29C23×C411yescharacteristic
192C22×C29C22×D411yescharacteristic
193C22×C29C2×#4111yescharacteristic
194C22×C29H(5,𝔽2)11yescharacteristic
195C22×C29H(5,𝔽2)11yescharacteristic
196C22×C29H(5,𝔽2)11yescharacteristic
197C22×C292−1+411yescharacteristic
198C2×D29—22no—
199C2×D29—22no—
200C2×D29—22no—
201C2×D29—41no—
202C2×D29—41no—
203C2×D29—41no—
204C2×D29—41no—
205C4×C29—22no—
206C4×C29—22no—
207C4×C29—22no—
208C4×C29—41no—
209C4×C29—41no—
210Dic29—22no—
211Dic29—22no—
212Dic29—22no—
213Dic29—41no—
214Dic29—41no—
215Dic29—41no—
216Dic29—41no—
217C2×#121—291no—
218C2×C4×D4—291no—
219C2×C4×D4D2911yescharacteristic, complemented
220#388—291no—
221#388—291no—
222#387—291no—
223#387—291no—
224#386—291no—
225#386—291no—
226#386—291no—
227#386—291no—
228#385—291no—
229#385—291no—
230#380—291no—
231C2×#95—291no—
232C2×C29—24no—
233C2×C29—42no—
234C2×C29C2×C4×D411yescharacteristic
235C2×C29#51811yescharacteristic
236C2×C29#52011yescharacteristic
237C2×C29#53411yescharacteristic
238C2×C29#53711yescharacteristic
239C2×C29#53811yescharacteristic
240C2×C29#53911yescharacteristic
241D29—42no—
242C22×D4—291no—
243C22×D4—291no—
244C22×D4C2×D2911yescharacteristic
245C23×C4—291no—
246C23×C4—291no—
247C23×C4—291no—
248C23×C4C2×D2911yescharacteristic
249C23×C4C2×D2911yescharacteristic
250#121—298no—
251C4×D4—24no—
252C4×D4—584no—
253C2×C4⋊C4—291no—
254C2×C4⋊C4—291no—
255C2×C4⋊C4—291no—
256C2×C4⋊C4—291no—
257C2×C4⋊C4C2×D2911yescharacteristic
258C2×C22⋊C4—291no—
259C2×C22⋊C4—291no—
260C2×C22⋊C4—291no—
261C2×C22⋊C4—291no—
262C2×C22⋊C4—291no—
263C2×C22⋊C4—291no—
264C2×C22⋊C4—291no—
265C2×C22⋊C4—291no—
266C2×C22⋊C4C2×D2911yescharacteristic
267C2×C22⋊C4C2×D2911yescharacteristic
268C2×C42—291no—
269C2×C42C2×D2911yescharacteristic
270#95—291no—
271#95—291no—
272#95—291no—
273#95—291no—
274#95—291no—
275#95—291no—
276#95—291no—
277#95—291no—
278#95—291no—
279#95—291no—
280C29#232511yescharacteristic, complemented, Sylow
281C24—291no—
282C24C22×D2911yescharacteristic
283C24C22×D2911yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—294no—
287C2×D4—294no—
288C2×D4—582no—
289C2×D4—582no—
290C2×D4—1161no—
291C2×D4—1161no—
292C2×D4C4×D2914yescomplemented
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—291no—
299C22×C4—291no—
300C22×C4—291no—
301C22×C4—294no—
302C22×C4—291no—
303C22×C4—291no—
304C22×C4—291no—
305C22×C4—291no—
306C22×C4—582no—
307C22×C4—582no—
308C22×C4—1161no—
309C22×C4—1161no—
310C22×C4—1161no—
311C22×C4—1161no—
312C22×C4—1161no—
313C22×C4—1161no—
314C22×C4C22⋊D2914yescomplemented
315C22×C4C22×D2911yescharacteristic
316C22×C4C22×D2911yescharacteristic
317C22×C4C22×D2911yescharacteristic
318C22×C4C22×D2911yescharacteristic
319C22×C4C22×D2911yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—294no—
322C4⋊C4—582no—
323C4⋊C4—1161no—
324C4⋊C4—1161no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—294no—
328C22⋊C4—294no—
329C22⋊C4—582no—
330C22⋊C4—582no—
331C22⋊C4—1161no—
332C22⋊C4—1161no—
333C22⋊C4—1161no—
334C22⋊C4—1161no—
335C42—22no—
336C42—582no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—292no—
348C23—581no—
349C23—581no—
350C23—581no—
351C23—581no—
352C23—581no—
353C23—581no—
354C23C2×C4×D2912yes—
355C23C2×C4×D2912yes—
356C23#9080512yes—
357C23C2×C22⋊D2912yes—
358C23C23×D2911yescharacteristic
359D4—28no—
360D4—588no—
361D4—1164no—
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—44no—
375C2×C4—44no—
376C2×C4—292no—
377C2×C4—292no—
378C2×C4—292no—
379C2×C4—581no—
380C2×C4—581no—
381C2×C4—581no—
382C2×C4—581no—
383C2×C4—581no—
384C2×C4—581no—
385C2×C4—581no—
386C2×C4—581no—
387C2×C4—581no—
388C2×C4—581no—
389C2×C4—581no—
390C2×C4—581no—
391C2×C4—581no—
392C2×C4—581no—
393C2×C4—581no—
394C2×C4—581no—
395C2×C4—581no—
396C2×C4—581no—
397C2×C4—1164no—
398C2×C4—1164no—
399C2×C4—1164no—
400C2×C4C2×C4×D2912yes—
401C2×C4C2×C22⋊D2912yes—
402C2×C4C2×C22⋊D2912yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—582no—
417C22—582no—
418C22—582no—
419C22—1161no—
420C22—1161no—
421C22—1161no—
422C22—1161no—
423C22C4×C22⋊D2914yes—
424C22C22×C4×D2911yescharacteristic
425C22C2×#9080511yescharacteristic
426C22#1174535911yescharacteristic
427C22#1174535911yescharacteristic
428C22#1174536411yescharacteristic
429C22#1174536511yescharacteristic
430C22C22×C22⋊D2911yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—582no—
437C4—582no—
438C4—582no—
439C4—1161no—
440C4—1161no—
441C4—1161no—
442C4—1161no—
443C2—24no—
444C2—42no—
445C2—1162no—
446C2C2×C4×C22⋊D2911yescharacteristic
447C2#1427163511yescharacteristic
448C2#1427164211yescharacteristic
449C2#1427172211yescharacteristic
450C2#1427174111yescharacteristic
451C2#1427174611yescharacteristic
452C2#1427175511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutatorC22×C29C23×C4
FittingC2×C4×D4×C29C2
socle, cosocleC23×C29C24
prime radicalC2×#11745301C2
hypercenterC2×C4×D4D29
hypocenterC29#2325
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D29C2semidirect
3. #14271054C2semidirect
4. #14271051C2semidirect
5. #14271047C2semidirect
6. #14271044C2semidirect
7. #14271043C2semidirect
8. #14271042C2semidirect
9. #14271040C2semidirect
10. #14271039C2semidirect
11. #14271036C2semidirect
12. #14271034C2semidirect
13. #14271021C2semidirect
14. C2×#11745179C2semidirect
15. C2×C4×D4×C29C2semidirect
33. #11745301C4semidirect
59. C2×D4×C29C2×C4semidirect
65. C22×C4×C29D4semidirect
75. C2×C22⋊D29C2×C4semidirect
76. C2×C22⋊D29C2×C4semidirect
82. C22×Dic29C2×C4semidirect
100. C22⋊Dic29C2×C4semidirect
104. #90781C2×C4semidirect
108. #90779C2×C4semidirect
219. C2×C4×D4D29semidirect
280. C29#2325semidirect
292. C2×D4C4×D29semidirect
314. C22×C4C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117453012
C2×C4×C22⋊D292
#142710542
#142710512
#142710472
#142710442
#142710432
#142710422
#142710402
#142710392
#142710362
#142710342
#142710212
C2×#117451792
C2×C4×D4×C292
#232529