Finite Group Databasebeta

← #30603130

Subgroups — #30603130

Order 15616.

1334 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C2×#16840755C212yesmaximal
2C2×D61⋊#101C211yesmaximal, characteristic
3#21462293C211yesmaximal, characteristic
4C2×#16840633C212yesmaximal
5C2×C4⋊Q8×D61C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8#16840755—24no—
9#16840755—24no—
10C2×#14519840—24no—
11D61⋊#101—24no—
12C2×#14519838—22no—
13(C2×#6923)⋊Q8—24no—
14#16840705—22no—
15#16840633—24no—
16C22×Q8×D61C2211yescharacteristic
17C22×Q8×D61C2211yescharacteristic
18C2×#14519777C411yescharacteristic, complemented
19C2×#14519776—22no—
20C2×#14519775C411yescharacteristic, complemented
21C2×#14519775C411yescharacteristic, complemented
22C4⋊Q8×D61—24no—
23C4⋊Q8×D61—24no—
24C2×C4⋊C4×D61—22no—
25C2×C42×D61C2211yescharacteristic
26C2×C4⋊Q8×C61C411yescharacteristic, complemented
27C2×#11758213—22no—
28D61⋊M16—22no—
29D61⋊M16—22no—
30D61⋊M16—22no—
31D61⋊M16D414yes—
32C2×#11758217—22no—
33C2×D61⋊C8—22no—
34#14519856—22no—
35#14519856—22no—
36#14519856—22no—
37#14519856—22no—
38C2×C4⋊#6923D414yes—
39C2×C22⋊#6923—22no—
40C2×C4×#6923—22no—
41#14519840—216no—
42#14519838—28no—
43(C2×#6923)⋊C4—28no—
44C22×Dic122—21no—
45C22×Dic122C2×C411yescharacteristic
46C22×Dic122C2×C411yescharacteristic
47C22×Dic122D412yes—
48C2×Q8×D61—22no—
49C2×Q8×D61—22no—
50C2×Q8×D61—22no—
51C2×Q8×D61—22no—
52C2×Q8×D61—22no—
53C2×Q8×D61—22no—
54C2×Q8×D61—22no—
55C2×Q8×D61—22no—
56C2×Q8×D61—22no—
57C2×Q8×D61—22no—
58C2×Q8×D61—22no—
59C2×Q8×D61D414yes—
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61D412yes—
63C22×C4×D61C2311yescharacteristic
64#14519777—22no—
65#14519777—22no—
66#14519776—216no—
67#14519775—22no—
68#14519775—22no—
69#14519775—22no—
70#14519775—22no—
71C2×C4⋊Dic61—22no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C4⋊C4×D61—216no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61C2×C411yescharacteristic
77C42×D61—22no—
78C42×D61D414yes—
79C22×Q8×C61C2×C411yescharacteristic
80C22×Q8×C61C2×C411yescharacteristic
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61—22no—
83C2×C4⋊C4×C61—22no—
84C2×C42×C61C2×C411yescharacteristic
85C23×D61C2×D411yescharacteristic
86C22×#6923—24no—
87C2×Q8×C61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61—21no—
90C2×Q8×C61—22no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61—22no—
94C2×Q8×C61C22⋊C412yes—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61C22⋊C412yes—
98C22×C4×C61C22×C411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61C22⋊C412yes—
102C22×Dic61C2×D411yescharacteristic
103Q8×D61—24no—
104Q8×D61—24no—
105Q8×D61—24no—
106Q8×D61—24no—
107Q8×D61—42no—
108Q8×D61—42no—
109Q8×D61—42no—
110Q8×D61—42no—
111Q8×D61—42no—
112Q8×D61—42no—
113Q8×D61—42no—
114Q8×D61—42no—
115Q8×D61—42no—
116Q8×D61—42no—
117Q8×D61—42no—
118Q8×D61—42no—
119C2×C4×D61—28no—
120C2×C4×D61—24no—
121C2×C4×D61—24no—
122C2×C4×D61—22no—
123C2×C4×D61—24no—
124C2×C4×D61—24no—
125C2×C4×D61—22no—
126C2×C4×D61—41no—
127C2×C4×D61—42no—
128C2×C4×D61—41no—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2×D412yes—
131C2×C4×D61C2×D412yes—
132C2×C4×D61C2×D412yes—
133C2×C4×D61#4112yes—
134C2×C4×D61#4112yes—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—24no—
144C2×Dic122—24no—
145C2×Dic122—24no—
146C2×Dic122—22no—
147C2×Dic122—41no—
148C2×Dic122—42no—
149C2×Dic122—41no—
150C2×Dic122C22⋊C412yes—
151C22⋊#6923—42no—
152C22⋊#6923—42no—
153#11758217—24no—
154#11758217—42no—
155C2×#92420—42no—
156C4⋊#6923—24no—
157C4⋊#6923—24no—
158C4×#6923—42no—
159C4×#6923—42no—
160#11758213—24no—
161#11758213—42no—
162D61⋊C8—24no—
163D61⋊C8—42no—
164C4⋊C4×C61—28no—
165C42×C61—21no—
166C42×C61C22⋊C412yes—
167C4⋊Dic61—28no—
168#11758196—28no—
169#11758196—28no—
170C4×Dic61—21no—
171C4×Dic61—24no—
172C4×Dic61C22⋊C412yes—
173C23×C61C2×C22⋊C411yescharacteristic
174C22×D61—24no—
175C22×D61#12012yes—
176C22×D61#12011yescharacteristic
177C22×D61#12112yes—
178C22×D61#12311yescharacteristic
179C2×#6923—44no—
180C2×#6923—48no—
181Q8×C61—24no—
182Q8×C61—42no—
183Q8×C61—42no—
184Q8×C61—42no—
185C2×C4×C61—24no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61C4×D412yes—
196C2×C4×C61C4×D412yes—
197C2×Dic61—24no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61#12011yescharacteristic
206C2×Dic61#12012yes—
207C2×Dic61#12112yes—
208C2×Dic61#12311yescharacteristic
209C4×D61—216no—
210C4×D61—44no—
211C4×D61—44no—
212C4×D61—44no—
213C4×D61—44no—
214C4×D61#12012yes—
215C4×D61#12112yes—
216C4×D61#12112yes—
217C4×D61#12112yes—
218C4×D61#12112yes—
219C4×D61#12312yes—
220C4×D61#12412yes—
221C4×D61#12712yes—
222Dic122—24no—
223Dic122—42no—
224Dic122—42no—
225Dic122—42no—
226Dic122—44no—
227Dic122—44no—
228Dic122—44no—
229Dic122—44no—
230#92420—44no—
231#10978—611nomaximal, Sylow
232C22×C61—22no—
233C22×C61#37911yescharacteristic
234C22×C61#38611yescharacteristic
235C22×C61#38611yescharacteristic
236C2×D61—22no—
237C2×D61—24no—
238C2×D61—24no—
239C2×D61#39212yes—
240C2×D61#39412yes—
241C2×D61#45614yes—
242#6923—84no—
243C4×C61—24no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61#38612yes—
247C4×C61#39012yes—
248Dic61—24no—
249Dic61—42no—
250Dic61—42no—
251Dic61#39212yes—
252Dic61#39412yes—
253C22×C4⋊Q8—611no—
254C2×#419—612no—
255C22×#101—611no—
256#2310—611no—
257C2×#339—612no—
258C2×C61—21no—
259C2×C61—21no—
260C2×C61#143811yescharacteristic
261C2×C61#188612yes—
262D61—22no—
263D61#200514yes—
264C23×Q8—611no—
265C23×Q8—611no—
266C22×M16—612no—
267C2×C4⋊Q8—611no—
268C2×C4⋊Q8—611no—
269C2×C4⋊Q8—611no—
270C2×C4⋊Q8—1224no—
271C2×C4⋊Q8—1222no—
272C2×C4⋊Q8—1224no—
273C2×C4⋊Q8#692311yescharacteristic, complemented
274C2×#117—612no—
275C22×C4⋊C4—1222no—
276C22×C42—611no—
277#419—1224no—
278#419—1224no—
279C2×#103—1224no—
280C2×#101—1224no—
281C2×#101—1222no—
282#391—1224no—
283#381—1222no—
284#339—1224no—
285C61#1097811yescharacteristic, complemented, Sylow
286C22×Q8—612no—
287C22×Q8—611no—
288C22×Q8—614no—
289C22×Q8—611no—
290C22×Q8—1221no—
291C22×Q8—1222no—
292C22×Q8—1222no—
293C22×Q8—1222no—
294C22×Q8—1222no—
295C22×Q8—1222no—
296C22×Q8—1222no—
297C22×Q8—1222no—
298C22×Q8—1222no—
299C22×Q8—1222no—
300C22×Q8—1222no—
301C22×Q8—1222no—
302C22×Q8C2×#692311yescharacteristic
303C22×Q8C2×#692311yescharacteristic
304C23×C4—611no—
305C23×C4—612no—
306C23×C4—1221no—
307C23×C4—1221no—
308C2×M16—614no—
309C2×M16—1222no—
310C2×M16—1222no—
311C2×M16—1222no—
312C2×M16—1222no—
313C2×M16—1222no—
314C22×C8—1222no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—1222no—
318C4⋊Q8—1222no—
319C4⋊Q8—1222no—
320C4⋊Q8—12216no—
321C4⋊Q8—1222no—
322C4⋊Q8—1222no—
323C4⋊Q8—1222no—
324#117—1222no—
325#117—1222no—
326#117—1222no—
327#117—1222no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—614no—
330C2×C4⋊C4—12216no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1222no—
334C2×C22⋊C4—1222no—
335C2×C42—611no—
336C2×C42—614no—
337C2×C42—1222no—
338C2×C42—1221no—
339C2×C42—1222no—
340C2×C42C2×#692311yescharacteristic
341#103—12216no—
342#101—1228no—
343#95—1228no—
344C24—611no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—612no—
353C2×Q8—1222no—
354C2×Q8—1224no—
355C2×Q8—1224no—
356C2×Q8—1222no—
357C2×Q8—1224no—
358C2×Q8—1224no—
359C2×Q8—1224no—
360C2×Q8—1224no—
361C2×Q8—1224no—
362C2×Q8—1221no—
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8—1222no—
366C2×Q8—1222no—
367C2×Q8—1222no—
368C2×Q8—1222no—
369C2×Q8—2441no—
370C2×Q8—2441no—
371C2×Q8—2442no—
372C2×Q8—2442no—
373C2×Q8—2442no—
374C2×Q8—2442no—
375C2×Q8—2442no—
376C2×Q8—2442no—
377C2×Q8—2442no—
378C2×Q8—2442no—
379C2×Q8—2442no—
380C2×Q8—2442no—
381C2×Q8—2442no—
382C2×Q8—2442no—
383C2×Q8—2442no—
384C2×Q8C22⋊#692312yes—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—611no—
388C22×C4—612no—
389C22×C4—612no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—612no—
393C22×C4—612no—
394C22×C4—612no—
395C22×C4—1224no—
396C22×C4—1228no—
397C22×C4—1222no—
398C22×C4—1224no—
399C22×C4—1224no—
400C22×C4—1224no—
401C22×C4—1222no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1224no—
405C22×C4—2442no—
406C22×C4—2441no—
407C22×C4—2441no—
408C22×C4C22⋊#692312yes—
409C22×C4C22×#692311yescharacteristic
410M16—1224no—
411M16—1224no—
412M16—2442no—
413M16—2442no—
414C2×C8—1224no—
415C2×C8—2442no—
416C2×C8—2442no—
417C4⋊C4—28no—
418C4⋊C4—1228no—
419C4⋊C4—1228no—
420C4⋊C4—1228no—
421C4⋊C4—1224no—
422C4⋊C4—1224no—
423C22⋊C4—2442no—
424C22⋊C4—2442no—
425C42—21no—
426C42—612no—
427C42—1224no—
428C42—1221no—
429C42—2442no—
430C42—2442no—
431C42C22⋊#692312yes—
432C23—611no—
433C23—612no—
434C23—611no—
435C23—612no—
436C23—1224no—
437C23C2×C22⋊#692311yescharacteristic
438Q8—24no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—1224no—
443Q8—2444no—
444Q8—2444no—
445Q8—2444no—
446Q8—2444no—
447Q8—2442no—
448Q8—2442no—
449Q8—2442no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—611no—
459C2×C4—611no—
460C2×C4—612no—
461C2×C4—612no—
462C2×C4—612no—
463C2×C4—612no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—612no—
467C2×C4—612no—
468C2×C4—612no—
469C2×C4—612no—
470C2×C4—12216no—
471C2×C4—1224no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—2444no—
478C2×C4—2444no—
479C2×C4—2444no—
480C2×C4—2444no—
481C2×C4—2444no—
482C2×C4—2448no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4C2×C22⋊#692311yescharacteristic
486C2×C4C2×C22⋊#692311yescharacteristic
487C2×C4D4×#692312yes—
488C2×C4D4×#692312yes—
489C8—2444no—
490C22—22no—
491C22—612no—
492C22—614no—
493C22—612no—
494C22—1224no—
495C22—1224no—
496C22—1222no—
497C22#1684070311yescharacteristic
498C22#1684071011yescharacteristic
499C22#1684071011yescharacteristic
500C4—24no—
501C4—42no—
502C4—42no—
503C4—612no—
504C4—612no—
505C4—1224no—
506C4—4884no—
507C4—2442no—
508C4—2442no—
509C4#1684071012yes—
510C4#1684071412yes—
511C2—21no—
512C2—21no—
513C2—614no—
514C2—1222no—
515C2#2146100111yescharacteristic
516C2#2146182512yes—
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#10978
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
18. C2×#14519777C4semidirect
20. C2×#14519775C4semidirect
21. C2×#14519775C4semidirect
26. C2×C4⋊Q8×C61C4semidirect
273. C2×C4⋊Q8#6923semidirect
285. C61#10978semidirect

Maximal subgroups

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

subgroupindex
C2×#168407552
C2×D61⋊#1012
#214622932
C2×#168406332
C2×C4⋊Q8×D612
#1097861