Finite Group Databasebeta

← #29576063

Subgroups — #29576063

Order 15616.

612 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451626C211yesmaximal, characteristic
2#21448416C211yesmaximal, characteristic
3#21444391C211yesmaximal, characteristic, complemented
4C2×#16839485C211yesmaximal, characteristic
5#21443687C211yesmaximal, characteristic, complemented
6#21443686C211yesmaximal, characteristic, complemented
7C61×#1520C211yesmaximal, characteristic, complemented
8C22×#11758198C2211yescharacteristic
9C2×C22⋊#92418C2211yescharacteristic
10#16839697C412yescomplemented
11#16839684—22no—
12#16839673C412yescomplemented
13#16839654—22no—
14#16839652—22no—
15#16839634—21no—
16#16839622C2211yescharacteristic
17#16839622C2211yescharacteristic
18#16839617—21no—
19#16839617—21no—
20C22⋊C4×Dic61—21no—
21#16839499—22no—
22#16839485—22no—
23C2×C22⋊Q8×C61C2211yescharacteristic
24C2×C22⋊C8×C61C2211yescharacteristic
25C61×#380C2211yescharacteristic
26C61×#328—22no—
27C24×D61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C22×#92418—21no—
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61—21no—
32C2×#11758196—22no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×#11758198C2×C412yes—
44C2×#11758198D412yes—
45C2×C4×Dic61—22no—
46C22⋊#92418—21no—
47C22⋊#92418—21no—
48#14519684—22no—
49#14519684—22no—
50#14519684C2×C412yes—
51#14519684D412yes—
52C22×Q8×C61C2×C411yescharacteristic
53C23×C4×C61C2311yescharacteristic
54C22×C8×C61—21no—
55C22⋊Q8×C61—21no—
56C22⋊Q8×C61—21no—
57C22⋊Q8×C61—21no—
58C22⋊Q8×C61—21no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C22⋊C8×C61—21no—
65C22⋊C8×C61—21no—
66#95×C61D412yes—
67C24×C61C2×D411yescharacteristic
68C23×D61—21no—
69C23×D61—21no—
70C23×D61—21no—
71C23×D61—21no—
72C23×D61—21no—
73C23×D61—21no—
74C23×D61—81no—
75C23×D61—81no—
76C23×D61C22⋊C412yes—
77C2×Q8×C61—21no—
78C2×Q8×C61—21no—
79C2×Q8×C61—81no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—22no—
83C22×C4×C61—81no—
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—22no—
93C22×Dic61—22no—
94C22×Dic61—81no—
95C22×Dic61C22⋊C412yes—
96C2×C8×C61—41no—
97C2×C8×C61—81no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—81no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C22⋊Dic61—41no—
108C22⋊Dic61—41no—
109C22⋊Dic61—41no—
110C22⋊Dic61—41no—
111#11758198—22no—
112#11758198—22no—
113#11758198—22no—
114#11758198—22no—
115#11758198—82no—
116#11758198—82no—
117#11758198—82no—
118#11758198—82no—
119C4⋊Dic61—81no—
120C4⋊Dic61—81no—
121#11758196—82no—
122C4×Dic61—82no—
123C2×#92418—41no—
124C2×#92418—81no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61C2×C22⋊C411yescharacteristic
128C23×C61#12011yescharacteristic
129C23×C61#12311yescharacteristic
130C23×C61#13511yescharacteristic
131C23×C61#13511yescharacteristic
132C23×C61#13711yescharacteristic
133C23×C61#13711yescharacteristic
134C22×D61—22no—
135C22×D61—22no—
136C22×D61—22no—
137C22×D61—42no—
138C22×D61—42no—
139C22×D61—42no—
140C22×D61—42no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—81no—
146C22×D61—81no—
147C22×D61—82no—
148C22×D61—82no—
149C22×D61—82no—
150C22×D61—82no—
151Q8×C61—81no—
152Q8×C61—81no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61C4×D411yescharacteristic
167C2×C4×C61C4×D411yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×Dic61—22no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—81no—
180C8×C61—81no—
181#92418—81no—
182#9201—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61#38611yescharacteristic
194C22×C61#41611yescharacteristic
195C22×C61#41911yescharacteristic
196C22×C61#45211yescharacteristic
197C22×C61#45211yescharacteristic
198C22×C61#48511yescharacteristic
199C22×C61#48511yescharacteristic
200C2×D61—44no—
201C2×D61—42no—
202C2×D61—42no—
203C2×D61—42no—
204C2×D61—82no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—82no—
208C2×D61—82no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—82no—
214C4×C61—81no—
215C4×C61—81no—
216Dic61—42no—
217Dic61—82no—
218Dic61—82no—
219#2626—611no—
220#2312—611no—
221#1520—611no—
222#1520D6111yescharacteristic, complemented
223C2×#323—611no—
224#1289—611no—
225#1289—611no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—21no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61#187311yescharacteristic
233C2×C61#191111yescharacteristic
234C2×C61#191311yescharacteristic
235D61—82no—
236C2×C22⋊Q8C2×D6111yescharacteristic
237C22×C22⋊C4—611no—
238C2×C22⋊C8—611no—
239C2×C22⋊C8C2×D6111yescharacteristic
240#399—612no—
241#396—1222no—
242#393—612no—
243#388—1222no—
244#388—1222no—
245#385—1221no—
246#380—611no—
247#380—611no—
248#380C2×D6111yescharacteristic
249#379—1221no—
250#379—1221no—
251C4×C22⋊C4—1221no—
252#328—22no—
253#328—1222no—
254#323—1222no—
255C61#920111yescharacteristic, complemented, Sylow
256C25—611no—
257C22×Q8C4×D6111yescharacteristic
258C23×C4—611no—
259C23×C4C22×D6111yescharacteristic
260C22×C8—21no—
261C22×C8—1221no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4C4×D6111yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—612no—
275C2×C22⋊C4—612no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1222no—
281C2×C22⋊C4—1222no—
282C2×C22⋊C4—1222no—
283C2×C22⋊C4—1222no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C42—1222no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291#95—612no—
292#95—612no—
293#95—1222no—
294#95—1222no—
295#95C22⋊D6112yes—
296C24—612no—
297C24—1221no—
298C24—1221no—
299C24—1221no—
300C24—1221no—
301C24—1221no—
302C24—1221no—
303C24—4881no—
304C24—4881no—
305C24D4×D6111yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—81no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—81no—
313C22×C4—612no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1222no—
317C22×C4—1222no—
318C22×C4—4881no—
319C22×C4C2×C4×D6111yescharacteristic
320C22×C4#1175822211yescharacteristic
321C22×C4D4×D6111yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4#1175822411yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325C2×C8—41no—
326C2×C8—81no—
327C2×C8—4881no—
328C2×C8—2441no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—4882no—
335C4⋊C4—4881no—
336C4⋊C4—4881no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—1222no—
342C22⋊C4—1222no—
343C22⋊C4—1222no—
344C22⋊C4—1222no—
345C22⋊C4—4882no—
346C22⋊C4—4882no—
347C22⋊C4—4882no—
348C22⋊C4—4882no—
349C22⋊C4—2441no—
350C22⋊C4—2441no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C42—4882no—
354C23—41no—
355C23—41no—
356C23—1222no—
357C23—1222no—
358C23—1222no—
359C23—4881no—
360C23—4881no—
361C23—4882no—
362C23—4882no—
363C23—4882no—
364C23—4882no—
365C23—2442no—
366C23—2442no—
367C23—2442no—
368C23—2442no—
369C23—2441no—
370C23—2441no—
371C23—2441no—
372C23—2441no—
373C23C22⋊C4×D6111yescharacteristic
374C23#1451974511yescharacteristic
375C23#1451975911yescharacteristic
376C23#1451979811yescharacteristic
377C23#1451980011yescharacteristic
378C23#1451980611yescharacteristic
379C23#1451980911yescharacteristic
380Q8—81no—
381Q8—81no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—1222no—
396C2×C4—4881no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4—2442no—
401C2×C4—2442no—
402C2×C4—2442no—
403C2×C4—2442no—
404C2×C4—2442no—
405C2×C4#1451973711yescharacteristic
406C2×C4C4×C22⋊D6111yescharacteristic
407C2×C4#1451974811yescharacteristic
408C2×C4#1451975011yescharacteristic
409C8—81no—
410C8—4881no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—4882no—
422C22—4882no—
423C22—4882no—
424C22—4882no—
425C22—4882no—
426C22—2444no—
427C22—2442no—
428C22—2442no—
429C22—2442no—
430C22#1683963811yescharacteristic
431C22#1683975711yescharacteristic
432C22#1683976911yescharacteristic
433C22#1683992111yescharacteristic
434C22#1683992311yescharacteristic
435C22#1684010111yescharacteristic
436C22#1684010211yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—4882no—
445C4—4882no—
446C4—2442no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—4882no—
454C2#2144612811yescharacteristic
455C2#2144634811yescharacteristic
456C2#2144635811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1520C2
socleC22×C61#386
prime radicalC24×D61D4
cosocleC23×C4×C61C23
hypercenter#1520D61
hypocenterC61#9201
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
3. #21444391C2semidirect
5. #21443687C2semidirect
6. #21443686C2semidirect
7. C61×#1520C2semidirect
10. #16839697C4semidirect
12. #16839673C4semidirect
222. #1520D61semidirect
255. C61#9201semidirect

Maximal subgroups

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

subgroupindex
#214516262
#214484162
#214443912
C2×#168394852
#214436872
#214436862
C61×#15202
#920161