Finite Group Databasebeta

← #16822525

Subgroups — #16822525

Order 3712.

688 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745246C211yesmaximal, characteristic
2C2×#11745243C211yesmaximal, characteristic, complemented
3C2×#11745234C211yesmaximal, characteristic, complemented
4#14271064C211yesmaximal, characteristic, complemented
5#14271063C211yesmaximal, characteristic, complemented
6#14271063C211yesmaximal, characteristic, complemented
7#14271058C211yesmaximal, characteristic, complemented
8#14271052C211yesmaximal, characteristic, complemented
9#14271041C211yesmaximal, characteristic, complemented
10#14271040C211yesmaximal, characteristic, complemented
11#14271014C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic29C211yesmaximal, characteristic, complemented
13C4×#90781C211yesmaximal, characteristic, complemented
14#14271010C211yesmaximal, characteristic, complemented
15C29×#388C211yesmaximal, characteristic, complemented
16C22×C22⋊D29C2211yescharacteristic
17C22×C22⋊D29C2211yescharacteristic
18C23×Dic29C2211yescharacteristic
19C22×D116C2211yescharacteristic
20C22×C4×D29C2211yescharacteristic
21C22×C4×D29C2211yescharacteristic
22C2×C4⋊C4×C29C2211yescharacteristic
23C2×C22⋊C4×C29C2211yescharacteristic
24C2×C22⋊C4×C29C2211yescharacteristic
25C2×C22⋊C4×C29C2211yescharacteristic
26C2×C42×C29C2211yescharacteristic
27C2×C22⋊Dic29C2211yescharacteristic
28C2×#90781C2211yescharacteristic
29C2×#90781C2211yescharacteristic
30C2×#90781C2211yescharacteristic
31C2×#90781C2211yescharacteristic
32C2×#90781C2211yescharacteristic
33C2×#90781C2211yescharacteristic
34C2×#90781C2211yescharacteristic
35C2×#90781C2211yescharacteristic
36C2×#90779C2211yescharacteristic
37C2×#90779C2211yescharacteristic
38C2×#90779C2211yescharacteristic
39C2×#90779C2211yescharacteristic
40C2×C4×Dic29C2211yescharacteristic
41C2×C4×Dic29C2211yescharacteristic
42C2×C4×Dic29C2211yescharacteristic
43C2×C4×Dic29C2211yescharacteristic
44C2×C4×Dic29C2211yescharacteristic
45#11745246C418yescomplemented
46#11745243—24no—
47#11745234—24no—
48C29×#95C2211yescharacteristic
49C29×#95C2211yescharacteristic
50#11745179C2211yescharacteristic
51#11745179C2211yescharacteristic
52#11745179C2211yescharacteristic
53#11745179C2211yescharacteristic
54C24×C29C2311yescharacteristic
55C23×D29C2311yescharacteristic
56C23×D29C2311yescharacteristic
57C22×C4×C29C2311yescharacteristic
58C22×C4×C29C2311yescharacteristic
59C22×C4×C29C2311yescharacteristic
60C22×C4×C29C2311yescharacteristic
61C22×C4×C29C2311yescharacteristic
62C22×C4×C29C2311yescharacteristic
63C2×C22⋊D29—22no—
64C2×C22⋊D29—22no—
65C2×C22⋊D29—41no—
66C2×C22⋊D29—41no—
67C2×C22⋊D29C2×C414yes—
68C2×C22⋊D29C2×C414yescomplemented
69C22×Dic29—22no—
70C22×Dic29—22no—
71C22×Dic29—41no—
72C22×Dic29C2311yescharacteristic
73C22×Dic29C2311yescharacteristic
74C22×Dic29C2311yescharacteristic
75C22×Dic29C2311yescharacteristic
76C22×Dic29C2311yescharacteristic
77C22×Dic29C2311yescharacteristic
78C2×D116—41no—
79C2×D116—41no—
80C2×D116C2×C414yescomplemented
81C2×C4×D29—22no—
82C2×C4×D29—22no—
83C2×C4×D29—22no—
84C2×C4×D29—22no—
85C2×C4×D29—41no—
86C2×C4×D29C2×C414yescomplemented
87C4⋊C4×C29—41no—
88C22⋊C4×C29—22no—
89C22⋊C4×C29—41no—
90C22⋊C4×C29C2×C414yescomplemented
91C42×C29—22no—
92C22⋊Dic29—22no—
93#90781—22no—
94#90781—22no—
95#90781—22no—
96#90781—22no—
97#90781—22no—
98#90781—22no—
99#90781—41no—
100#90781C2×C414yescomplemented
101#90779—22no—
102#90779—41no—
103#90779C2×C414yescomplemented
104#90779D414yescomplemented
105C4×Dic29—22no—
106C4×Dic29—22no—
107C4×Dic29—22no—
108C4×Dic29—41no—
109C4×Dic29—41no—
110C23×C29—21no—
111C23×C29—21no—
112C23×C29—21no—
113C23×C29—21no—
114C23×C29—21no—
115C23×C29—21no—
116C23×C29C22×C412yes—
117C23×C29C2411yescharacteristic
118C22×D29—21no—
119C22×D29—21no—
120C22×D29—21no—
121C22×D29—21no—
122C22×D29—21no—
123C22×D29—21no—
124C22×D29—21no—
125C22×D29—21no—
126C22×D29—21no—
127C22×D29—21no—
128C22×D29—21no—
129C22×D29—21no—
130C22×D29C22×C412yes—
131C22×D29C22×C412yes—
132C2×C4×C29—21no—
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—21no—
146C2×C4×C29—21no—
147C2×C4×C29C22×C412yes—
148C2×C4×C29C22×C412yes—
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—44no—
164C2×Dic29C22×C412yes—
165C2×Dic29C22×C412yes—
166C2×Dic29C2×D412yes—
167C2×Dic29C2×D412yes—
168C2×Dic29#4112yes—
169C2×Dic29#4112yes—
170C2×Dic29#4112yes—
171D116—44no—
172C4×D29—28no—
173C4×D29—44no—
174#2363—291nomaximal, Sylow
175C22×C29—22no—
176C22×C29—22no—
177C22×C29—22no—
178C22×C29—41no—
179C22×C29—41no—
180C22×C29—41no—
181C22×C29—41no—
182C22×C29C23×C411yescharacteristic
183C22×C29C22×D411yescharacteristic
184C22×C29C2×#4111yescharacteristic
185C22×C29C2×#4111yescharacteristic
186C22×C29C2×#4111yescharacteristic
187C22×C29H(5,𝔽2)11yescharacteristic
188C22×C29H(5,𝔽2)11yescharacteristic
189C2×D29—22no—
190C2×D29—22no—
191C2×D29—22no—
192C2×D29—22no—
193C2×D29—22no—
194C2×D29—22no—
195C2×D29—41no—
196C2×D29—41no—
197C2×D29—41no—
198C2×D29—41no—
199C2×D29—41no—
200C2×D29—41no—
201C2×D29—41no—
202C2×D29—41no—
203C4×C29—22no—
204C4×C29—22no—
205C4×C29—22no—
206C4×C29—41no—
207C4×C29—41no—
208C4×C29—41no—
209Dic29—22no—
210Dic29—22no—
211Dic29—22no—
212Dic29—22no—
213Dic29—41no—
214Dic29C4×D414yes—
215C2×#121—291no—
216C2×C4×D4—291no—
217C2×#117—291no—
218#390—291no—
219#390—291no—
220#390—291no—
221#389—291no—
222#388—291no—
223#388D2911yescharacteristic, complemented
224#386—291no—
225#386—291no—
226C4×C4⋊C4—291no—
227C4×C22⋊C4—291no—
228C4×C22⋊C4—291no—
229C4×C22⋊C4—291no—
230C2×C29—42no—
231C2×C29C2×C4×D411yescharacteristic
232C2×C29C4×#4111yescharacteristic
233C2×C29#51811yescharacteristic
234C2×C29#54611yescharacteristic
235C2×C29#55011yescharacteristic
236C2×C29#55311yescharacteristic
237C2×C29#55511yescharacteristic
238D29—42no—
239D29—42no—
240C22×D4—291no—
241C22×D4—291no—
242C22×D4—291no—
243C23×C4—291no—
244C23×C4—291no—
245C23×C4—291no—
246#121—298no—
247C4×D4—584no—
248#117—584no—
249C2×C4⋊C4—291no—
250C2×C4⋊C4—291no—
251C2×C4⋊C4—291no—
252C2×C4⋊C4—291no—
253C2×C4⋊C4C2×D2911yescharacteristic
254C2×C22⋊C4—291no—
255C2×C22⋊C4—291no—
256C2×C22⋊C4—291no—
257C2×C22⋊C4—291no—
258C2×C22⋊C4—291no—
259C2×C22⋊C4—291no—
260C2×C22⋊C4—291no—
261C2×C22⋊C4—291no—
262C2×C22⋊C4—291no—
263C2×C22⋊C4C2×D2911yescharacteristic
264C2×C22⋊C4C2×D2911yescharacteristic
265C2×C22⋊C4C2×D2911yescharacteristic
266C2×C42—291no—
267C2×C42—291no—
268C2×C42—291no—
269C2×C42—291no—
270C2×C42—291no—
271C2×C42C2×D2911yescharacteristic
272#95—291no—
273#95—291no—
274#95—291no—
275#95—291no—
276#95C2×D2911yescharacteristic
277#95C2×D2911yescharacteristic
278C29#236311yescharacteristic, complemented, Sylow
279C24—291no—
280C24—291no—
281C24C22×D2911yescharacteristic
282C2×D4—294no—
283C2×D4—294no—
284C2×D4—294no—
285C2×D4—582no—
286C2×D4—582no—
287C2×D4—1161no—
288C2×D4—1161no—
289C2×D4—1161no—
290C2×D4—1161no—
291C22×C4—291no—
292C22×C4—291no—
293C22×C4—291no—
294C22×C4—291no—
295C22×C4—291no—
296C22×C4—294no—
297C22×C4—291no—
298C22×C4—582no—
299C22×C4—582no—
300C22×C4—582no—
301C22×C4—582no—
302C22×C4—582no—
303C22×C4—582no—
304C22×C4—1161no—
305C22×C4—1161no—
306C22×C4C22×D2911yescharacteristic
307C22×C4C22×D2911yescharacteristic
308C22×C4C22×D2911yescharacteristic
309C22×C4C22×D2911yescharacteristic
310C22×C4C22×D2911yescharacteristic
311C22×C4C22×D2911yescharacteristic
312C4⋊C4—41no—
313C4⋊C4—294no—
314C4⋊C4—294no—
315C4⋊C4—582no—
316C4⋊C4—1161no—
317C22⋊C4—22no—
318C22⋊C4—41no—
319C22⋊C4—294no—
320C22⋊C4—582no—
321C22⋊C4—582no—
322C22⋊C4—582no—
323C22⋊C4—582no—
324C22⋊C4—582no—
325C22⋊C4—582no—
326C22⋊C4—582no—
327C22⋊C4—1161no—
328C22⋊C4C4×D2914yescomplemented
329C42—22no—
330C42—582no—
331C42—582no—
332C42—582no—
333C42—1161no—
334C42—1161no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—292no—
342C23—292no—
343C23—581no—
344C23—581no—
345C23—581no—
346C23—581no—
347C23—581no—
348C23—581no—
349C23—581no—
350C23—581no—
351C23—581no—
352C23—581no—
353C23—581no—
354C23—581no—
355C23C2×C4×D2912yes—
356C23C23×D2911yescharacteristic
357D4—588no—
358D4—1164no—
359D4—1164no—
360C2×C4—21no—
361C2×C4—21no—
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—292no—
376C2×C4—292no—
377C2×C4—292no—
378C2×C4—292no—
379C2×C4—292no—
380C2×C4—292no—
381C2×C4—292no—
382C2×C4—588no—
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—1164no—
395C2×C4—1164no—
396C2×C4C2×C4×D2912yes—
397C2×C4C2×C4×D2912yes—
398C2×C4D4×D2912yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—582no—
407C22—582no—
408C22—582no—
409C22—582no—
410C22—582no—
411C22—582no—
412C22—1161no—
413C22—1161no—
414C22—1161no—
415C22—1161no—
416C22—1161no—
417C22—1161no—
418C22—1161no—
419C22—1161no—
420C22C22×C4×D2911yescharacteristic
421C22C2×D4×D2911yescharacteristic
422C22#1174535911yescharacteristic
423C22#41×D2911yescharacteristic
424C22#41×D2911yescharacteristic
425C22#41×D2911yescharacteristic
426C22#1174536411yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—294no—
434C4—582no—
435C4—582no—
436C4—582no—
437C4—582no—
438C4—1161no—
439C2—42no—
440C2—1162no—
441C2—1162no—
442C2C4×D4×D2911yescharacteristic
443C2#1427163111yescharacteristic
444C2#1427163511yescharacteristic
445C2#1427180511yescharacteristic
446C2#1427183111yescharacteristic
447C2#1427186411yescharacteristic
448C2#1427187511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutatorC22×C29C23×C4
FittingC29×#388C2
socle, cosocleC23×C29C24
prime radicalC2×#11745246C2
hypercenter#388D29
hypocenterC29#2363
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#11745243C2semidirect
3. C2×#11745234C2semidirect
4. #14271064C2semidirect
5. #14271063C2semidirect
6. #14271063C2semidirect
7. #14271058C2semidirect
8. #14271052C2semidirect
9. #14271041C2semidirect
10. #14271040C2semidirect
11. #14271014C2semidirect
12. C22⋊C4×Dic29C2semidirect
13. C4×#90781C2semidirect
14. #14271010C2semidirect
15. C29×#388C2semidirect
45. #11745246C4semidirect
68. C2×C22⋊D29C2×C4semidirect
80. C2×D116C2×C4semidirect
86. C2×C4×D29C2×C4semidirect
90. C22⋊C4×C29C2×C4semidirect
100. #90781C2×C4semidirect
103. #90779C2×C4semidirect
104. #90779D4semidirect
223. #388D29semidirect
278. C29#2363semidirect
328. C22⋊C4C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452462
C2×#117452432
C2×#117452342
#142710642
#142710632
#142710582
#142710522
#142710412
#142710402
#142710142
C22⋊C4×Dic292
C4×#907812
#142710102
C29×#3882
#236329