Finite Group Databasebeta

← #30607725

Subgroups — #30607725

Order 15616.

676 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C22×#14519838C211yesmaximal, characteristic
2C2×#16840672C212yesmaximal, complemented
3#21461653C211yesmaximal, characteristic
4#21460917C212yesmaximal, complemented
5#21449464C211yesmaximal, characteristic
6C22×#11758217—21no—
7C2×#14519838—24no—
8C2×#14519838—21no—
9C2×#14519838—21no—
10C2×#14519838C2212yes—
11C2×#14519838C2212yes—
12#16840732—21no—
13#16840672—24no—
14C23×Dic122C2211yescharacteristic
15C2×#14519775C411yescharacteristic
16C2×#14519775C2211yescharacteristic
17#16839682—21no—
18#16839682—21no—
19#16839667C411yescharacteristic
20#16839667C2211yescharacteristic
21#16839634—21no—
22#16839634—21no—
23C4×C22⋊Dic61C411yescharacteristic
24C2×C22⋊Q8×C61C411yescharacteristic
25C2×#11758217—24no—
26C2×#11758217—22no—
27C2×#11758217—22no—
28C2×#11758217—41no—
29C2×#11758217—42no—
30C22×#92420—41no—
31#14519838—24no—
32#14519838—22no—
33#14519838—22no—
34#14519838—22no—
35#14519838—42no—
36#14519838D414yes—
37C22⋊#92420—41no—
38C22⋊#92420—41no—
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—42no—
42C22×Dic122—41no—
43C22×Dic122—41no—
44C22×Dic122C2×C411yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×Dic122D411yescharacteristic
47C22×Dic122C2311yescharacteristic
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50#14519775—22no—
51#14519775—22no—
52#14519775—42no—
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×C22⋊Dic61—21no—
58C2×C4×Dic61C2×C411yescharacteristic
59C2×C4×Dic61C2×C411yescharacteristic
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65C22×Q8×C61C2×C411yescharacteristic
66C23×C4×C61C2×C411yescharacteristic
67C22⋊Q8×C61—41no—
68C22⋊Q8×C61—41no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61C2×C411yescharacteristic
71C2×C22⋊C4×C61—21no—
72C24×C61C22⋊C411yescharacteristic
73C2×Q8×C61—22no—
74C2×Q8×C61—41no—
75C2×Q8×C61—41no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—22no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—41no—
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22×C411yescharacteristic
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—22no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—22no—
94C22×Dic61—41no—
95C22×Dic61C22⋊C411yescharacteristic
96C22×Dic61C22⋊C411yescharacteristic
97C22×Dic61C2×D411yescharacteristic
98C22×Dic61C2×D411yescharacteristic
99C2×Dic122—22no—
100C2×Dic122—22no—
101C2×Dic122—22no—
102C2×Dic122—21no—
103C2×Dic122—22no—
104C2×Dic122—21no—
105C2×Dic122—41no—
106C2×Dic122—41no—
107C2×Dic122—41no—
108C2×Dic122—41no—
109C2×Dic122—41no—
110C2×Dic122—41no—
111C2×Dic122—44no—
112C2×Dic122—44no—
113C2×Dic122—44no—
114C2×Dic122—44no—
115C2×Dic122C2×D412yes—
116C2×Dic122#4112yes—
117#11758217—24no—
118#11758217—44no—
119#11758217—42no—
120#11758217—42no—
121#11758217—42no—
122C2×#92420—44no—
123C2×#92420—42no—
124C2×#92420—42no—
125C4⋊C4×C61—42no—
126C4⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C22⋊C4×C61—41no—
129C22⋊Dic61—24no—
130#11758196—42no—
131#11758196—42no—
132#11758196—41no—
133#11758196—42no—
134#11758196—41no—
135C4×Dic61—22no—
136C4×Dic61—41no—
137C23×C61—22no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—41no—
141C23×C61#9911yescharacteristic
142C23×C61#9911yescharacteristic
143C23×C61C2×C22⋊C411yescharacteristic
144Q8×C61—82no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61C4×D412yes—
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—22no—
169C2×Dic61—22no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61#9912yes—
180C2×Dic61#12011yescharacteristic
181C2×Dic61#12112yes—
182C2×Dic61#12311yescharacteristic
183Dic122—42no—
184Dic122—42no—
185Dic122—44no—
186Dic122—44no—
187Dic122—42no—
188Dic122—42no—
189#92420—44no—
190#92420—82no—
191#13094—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—42no—
200C22×C61—42no—
201C22×C61#35612yescomplemented
202C22×C61#38611yescharacteristic
203C22×C61C2×#9911yescharacteristic
204C22×C61C2×#9911yescharacteristic
205C4×C61—22no—
206C4×C61—41no—
207C4×C61—81no—
208C4×C61—81no—
209Dic61—21no—
210Dic61—22no—
211Dic61—22no—
212Dic61—21no—
213Dic61—42no—
214Dic61—42no—
215Dic61—41no—
216C22×#101—611no—
217#2434—611no—
218C2×#356—612no—
219#1784—611no—
220#1394—612no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—22no—
224C2×C61—41no—
225C2×C61#189811yescharacteristic
226C2×C61C2×#35611yescharacteristic
227C2×C61#213311yescharacteristic
228C23×Q8—611no—
229C22×M16—1221no—
230C2×C4⋊Q8—611no—
231C2×C4⋊Q8—611no—
232C2×C22⋊Q8#692311yescharacteristic
233C2×#101—612no—
234C2×#101—612no—
235C2×#101—1224no—
236C2×#101—1221no—
237C2×#101—1221no—
238#407—1221no—
239#395—1221no—
240#395—1221no—
241#391—611no—
242#391—611no—
243#385—1221no—
244#385—1221no—
245C4×C22⋊C4—611no—
246#356—1224no—
247C61#1309411yescharacteristic, complemented, Sylow
248C22×Q8—611no—
249C22×Q8—611no—
250C22×Q8—611no—
251C22×Q8—611no—
252C22×Q8—1222no—
253C22×Q8—1222no—
254C22×Q8—2441no—
255C22×Q8—2441no—
256C22×Q8—2442no—
257C22×Q8C2×#692311yescharacteristic
258C23×C4—611no—
259C23×C4—611no—
260C23×C4C2×#692311yescharacteristic
261C2×M16—1222no—
262C2×M16—1224no—
263C2×M16—1222no—
264C2×M16—2442no—
265C2×M16—2441no—
266C22×C8—2441no—
267C4⋊Q8—1222no—
268C4⋊Q8—1222no—
269C4⋊Q8—2442no—
270C22⋊Q8—41no—
271C22⋊Q8—41no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4C2×#692311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—1221no—
280C2×C42—611no—
281C2×C42—611no—
282#101—614no—
283#101—1222no—
284#101—1224no—
285#101—1222no—
286#101—1222no—
287#101—2442no—
288C22⋊C8—2441no—
289C22⋊C8—2441no—
290#95—1221no—
291#95—1221no—
292#95—1221no—
293#95—1221no—
294#95—1221no—
295C24C22⋊#692311yescharacteristic
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—612no—
300C2×Q8—612no—
301C2×Q8—1221no—
302C2×Q8—1222no—
303C2×Q8—1222no—
304C2×Q8—1222no—
305C2×Q8—1221no—
306C2×Q8—1222no—
307C2×Q8—2441no—
308C2×Q8—2441no—
309C2×Q8—2441no—
310C2×Q8—2441no—
311C2×Q8—2441no—
312C2×Q8—2441no—
313C2×Q8—2444no—
314C2×Q8—2444no—
315C2×Q8—2444no—
316C2×Q8—2444no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—611no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—1221no—
328C22×C4—1222no—
329C22×C4—1221no—
330C22×C4—1222no—
331C22×C4—1221no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—1221no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—2441no—
338C22×C4C22⋊#692311yescharacteristic
339C22×C4C22×#692311yescharacteristic
340M16—1224no—
341M16—2442no—
342M16—2442no—
343M16—2444no—
344M16—2442no—
345C2×C8—2442no—
346C2×C8—2444no—
347C2×C8—2442no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—2441no—
351C4⋊C4—2442no—
352C4⋊C4—2441no—
353C4⋊C4—2442no—
354C4⋊C4—2442no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—1224no—
358C42—1222no—
359C42—2441no—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—41no—
364C23#1451983511yescharacteristic
365C23#1451983511yescharacteristic
366C23C2×C22⋊#692311yescharacteristic
367Q8—82no—
368Q8—2442no—
369Q8—2442no—
370Q8—2444no—
371Q8—2444no—
372Q8—2442no—
373Q8—2442no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—611no—
387C2×C4—612no—
388C2×C4—611no—
389C2×C4—612no—
390C2×C4—1222no—
391C2×C4—1222no—
392C2×C4—1222no—
393C2×C4—1222no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—1222no—
401C2×C4—1222no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—2442no—
406C2×C4—2442no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4D4×#692312yes—
412C8—4882no—
413C8—2444no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22#1684067212yescomplemented
424C22#1684071011yescharacteristic
425C22C2×#1451983511yescharacteristic
426C22D61⋊#9911yescharacteristic
427C4—22no—
428C4—41no—
429C4—81no—
430C4—81no—
431C4—1222no—
432C4—1221no—
433C4—1222no—
434C4—1221no—
435C4—2442no—
436C4—2442no—
437C4—2441no—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—41no—
442C2#2146184011yescharacteristic
443C2C2×#1684067211yescharacteristic
444C2#2146211411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC22×Dic122C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#13094
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16840672C2semidirect
4. #21460917C2semidirect
201. C22×C61#356semidirect
247. C61#13094semidirect
423. C22#16840672semidirect

Maximal subgroups

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

subgroupindex
C22×#145198382
C2×#168406722
#214616532
#214609172
#214494642
#1309461