Finite Group Databasebeta

← #30606261

Subgroups — #30606261

Order 15616.

838 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 152 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×#16840748C211yesmaximal, characteristic
2C2×#16840741C211yesmaximal, characteristic
3#21460904C214yesmaximal, complemented
4#21448307C211yesmaximal, characteristic
5C2×D61⋊M16—21no—
6C2×#14519856—21no—
7#16840748—22no—
8#16840748C414yescomplemented
9C2×#14519838—21no—
10C2×#14519837C2212yes—
11#16840741—22no—
12#16840741C414yescomplemented
13C2×#14519836—21no—
14C2×#14519835C2212yes—
15C22×#11758222C2211yescharacteristic
16C22×C4×Dic61C411yescharacteristic, complemented
17#16839667—21no—
18#16839663—21no—
19#16839639C412yescomplemented
20#16839639C2212yes—
21C4×#11758198—21no—
22C4×#11758198—21no—
23C2×#117×C61C411yescharacteristic, complemented
24C2×#11758213—41no—
25D61⋊M16—24no—
26D61⋊M16—42no—
27C2×#11758217—22no—
28C2×D61⋊C8—41no—
29#14519856—24no—
30#14519856—42no—
31C2×C4⋊#6923—41no—
32C2×C22⋊#6923—22no—
33C2×C4×#6923—41no—
34#14519838—24no—
35#14519837—22no—
36#14519837C2×C414yes—
37#14519836—24no—
38#14519835—22no—
39#14519835C2×C414yes—
40C2×#11758222—41no—
41C2×#11758222—41no—
42C2×#11758222—42no—
43C2×#11758222C2×C412yes—
44C2×#11758222C2×C412yes—
45C2×#11758222D412yes—
46C2×#11758222Q812yes—
47C22×Dic122D411yescharacteristic
48C22×D244D411yescharacteristic
49C22×C22⋊D61C2×C411yescharacteristic
50C22×C22⋊D61C2311yescharacteristic
51C23×Dic61C2×C412yes—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C2×#11758198—22no—
55C2×#11758198—22no—
56C2×C4×Dic61—21no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61—41no—
59C2×C4×Dic61—41no—
60#14519684—22no—
61#14519684—22no—
62C23×C4×C61C2×C411yescharacteristic
63#117×C61—42no—
64C2×C4⋊C4×C61—21no—
65C2×C22⋊C4×C61C2×C412yes—
66C2×C42×C61—21no—
67C24×C61C22×C411yescharacteristic
68C23×D61C22⋊C411yescharacteristic
69C23×D61C2×D411yescharacteristic
70C22×#6923—42no—
71C22×C4×C61—22no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—22no—
75C22×C4×C61C4212yes—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C4⋊C412yes—
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—41no—
82C2×C22⋊D61—41no—
83C2×C22⋊D61C4212yes—
84C2×C22⋊D61C4⋊C412yes—
85C2×C22⋊D61C22×C411yescharacteristic
86C2×C22⋊D61C22×C411yescharacteristic
87C2×C22⋊D61C2×D411yescharacteristic
88C2×C22⋊D61C2×Q811yescharacteristic
89C22×Dic61—22no—
90C22×Dic61—44no—
91C22×Dic61—42no—
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C2×D411yescharacteristic
94#11758222—42no—
95#11758222—42no—
96#11758222—42no—
97#11758222—42no—
98#11758222—42no—
99#11758222—42no—
100#11758222—42no—
101#11758222—42no—
102C2×D244—41no—
103C2×D244—41no—
104C2×D244C22⋊C412yes—
105C2×D244C4⋊C412yes—
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—22no—
109C2×C4×D61—21no—
110C2×C4×D61—22no—
111C2×C4×D61—21no—
112C2×C4×D61C22⋊C412yes—
113C2×C4×D61C22⋊C412yes—
114C2×C4×D61C4⋊C412yes—
115C2×C4×D61C4⋊C412yes—
116C2×Dic122—41no—
117C2×Dic122—41no—
118C2×Dic122C22⋊C412yes—
119C2×Dic122C4⋊C412yes—
120C22⋊#6923—24no—
121C22⋊#6923—42no—
122#11758217—24no—
123#11758217—42no—
124C2×#92420—42no—
125C4⋊#6923—44no—
126C4×#6923—44no—
127#11758213—44no—
128D61⋊C8—44no—
129C4⋊C4×C61—42no—
130C22⋊C4×C61—42no—
131C42×C61—42no—
132#11758198—44no—
133#11758198—44no—
134C4×Dic61—42no—
135C4×Dic61—41no—
136C4×Dic61—41no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—42no—
140C23×C61C2×C4211yescharacteristic
141C23×C61C2×C22⋊C411yescharacteristic
142C23×C61C2×C4⋊C411yescharacteristic
143C22×D61—21no—
144C22×D61—21no—
145C22×D61—21no—
146C22×D61—21no—
147C22×D61—42no—
148C22×D61—42no—
149C22×D61#9512yes—
150C22×D61C2×C22⋊C411yescharacteristic
151C22×D61C2×C4⋊C411yescharacteristic
152C2×#6923—44no—
153C2×#6923—84no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61#9512yes—
164C2×C4×C61#9512yes—
165C2×C4×C61#9912yes—
166C2×C4×C61#9911yescharacteristic
167C2×C4×C61#9911yescharacteristic
168C22⋊D61—42no—
169C22⋊D61—42no—
170C22⋊D61—41no—
171C22⋊D61—41no—
172C22⋊D61—41no—
173C22⋊D61—41no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—24no—
179C2×Dic61—42no—
180C2×Dic61—44no—
181C2×Dic61—42no—
182C2×Dic61—44no—
183C2×Dic61#9512yes—
184C2×Dic61C2×C22⋊C411yescharacteristic
185C2×Dic61C2×C4⋊C411yescharacteristic
186D244—42no—
187D244—42no—
188C4×D61—22no—
189C4×D61—22no—
190C4×D61—22no—
191C4×D61—22no—
192C4×D61—22no—
193C4×D61—22no—
194C4×D61—22no—
195C4×D61—22no—
196Dic122—42no—
197Dic122—42no—
198#92420—44no—
199#12573—611nomaximal, Sylow
200C22×C61—22no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61C2×#9511yescharacteristic
208C22×C61C2×#9911yescharacteristic
209C22×C61C2×#9911yescharacteristic
210C2×D61—21no—
211C2×D61—22no—
212C2×D61—21no—
213C2×D61—42no—
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—41no—
217C2×D61—41no—
218C2×D61—81no—
219C2×D61—81no—
220#6923—84no—
221C4×C61—22no—
222C4×C61—82no—
223C4×C61#34212yes—
224C4×C61#34212yes—
225Dic61—21no—
226Dic61—22no—
227Dic61—21no—
228Dic61—44no—
229C2×#413—611no—
230C2×#410—611no—
231#2297—611no—
232#1390—614no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—41no—
236C2×C61—41no—
237C2×C61C2×#34211yescharacteristic
238C2×C61#212012yes—
239D61—41no—
240D61—42no—
241D61—41no—
242C22×#41—611no—
243C22×M16—1221no—
244C2×#117—1221no—
245C2×#117#692311yescharacteristic, complemented
246C22×C42—611no—
247#413—614no—
248#413—1222no—
249C2×#101—1221no—
250C2×#100—612no—
251#410—614no—
252#410—1222no—
253C2×#99—612no—
254C2×#99—1221no—
255#391—1221no—
256#390—1221no—
257#386—612no—
258#386—612no—
259C4×C22⋊C4—1221no—
260C4×C22⋊C4—1221no—
261C61#1257311yescharacteristic, complemented, Sylow
262C2×#41—612no—
263C2×#41—612no—
264C2×#41—612no—
265C2×#41—612no—
266C2×#41—2442no—
267C2×#41—2441no—
268C2×#41—2441no—
269C22×Q8—611no—
270C22×D4—611no—
271C22×D4—611no—
272C22×D4—611no—
273C23×C4—611no—
274C23×C4—611no—
275C23×C4—612no—
276C23×C4C2×#692311yescharacteristic
277C2×M16—1222no—
278C2×M16—1224no—
279C2×M16—2442no—
280C2×M16—2441no—
281C22×C8—2441no—
282#117—42no—
283#117—1224no—
284#117—2442no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—2441no—
287C2×C22⋊C4—1222no—
288C2×C22⋊C4—1222no—
289C2×C22⋊C4—1222no—
290C2×C22⋊C4C2×#692312yes—
291C2×C42—21no—
292C2×C42—1221no—
293C2×C42—1221no—
294C2×C42—2441no—
295C2×C42—2441no—
296C2×C42—2441no—
297#101—1224no—
298#100—614no—
299#100—1222no—
300#99—614no—
301#99—1222no—
302#99—1224no—
303#95—1222no—
304#95—1222no—
305C24—611no—
306C24—611no—
307C24C22×#692311yescharacteristic
308#41—2442no—
309#41—2442no—
310#41—2442no—
311#41—2442no—
312#41—2442no—
313#41—2442no—
314#41—2442no—
315#41—2442no—
316C2×Q8—612no—
317C2×Q8—612no—
318C2×Q8—2441no—
319C2×Q8—2441no—
320C2×D4—611no—
321C2×D4—611no—
322C2×D4—612no—
323C2×D4—611no—
324C2×D4—612no—
325C2×D4—611no—
326C2×D4—612no—
327C2×D4—612no—
328C2×D4—2441no—
329C2×D4—2441no—
330C2×D4—2441no—
331C2×D4—2441no—
332C2×D4—2441no—
333C2×D4—2441no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—612no—
341C22×C4—612no—
342C22×C4—612no—
343C22×C4—612no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—1222no—
351C22×C4—2444no—
352C22×C4—2442no—
353C22×C4—2442no—
354C22×C4C4×#692312yes—
355C22×C4C4⋊#692312yes—
356C22×C4C22⋊#692311yescharacteristic
357C22×C4C22⋊#692311yescharacteristic
358M16—1224no—
359M16—2444no—
360M16—2442no—
361C2×C8—2444no—
362C2×C8—2442no—
363C4⋊C4—42no—
364C4⋊C4—2444no—
365C22⋊C4—42no—
366C22⋊C4—1224no—
367C22⋊C4—2444no—
368C22⋊C4—2444no—
369C22⋊C4—2442no—
370C42—42no—
371C42—2442no—
372C42—2444no—
373C42—2441no—
374C42—2441no—
375C23—21no—
376C23—21no—
377C23—42no—
378C23—612no—
379C23—611no—
380C23—611no—
381C23—1221no—
382C23—1221no—
383C23—1221no—
384C23—1221no—
385C23—2442no—
386C23—2442no—
387C23C2×C4×#692311yescharacteristic
388C23C2×C22⋊#692311yescharacteristic
389C23C2×C4⋊#692311yescharacteristic
390Q8—2442no—
391Q8—2442no—
392D4—2442no—
393D4—2442no—
394D4—2441no—
395D4—2441no—
396D4—2441no—
397D4—2442no—
398D4—2442no—
399D4—2441no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—612no—
410C2×C4—611no—
411C2×C4—611no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1222no—
417C2×C4—1222no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—1224no—
425C2×C4—4884no—
426C2×C4—2444no—
427C2×C4—2442no—
428C2×C4—2444no—
429C2×C4—2442no—
430C2×C4—2444no—
431C2×C4(C2×#6923)⋊C412yes—
432C2×C4(C2×#6923)⋊C412yes—
433C2×C4#1451983512yes—
434C2×C4#1451983611yescharacteristic
435C2×C4#1451983611yescharacteristic
436C8—2444no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—42no—
443C22—42no—
444C22—1221no—
445C22—1222no—
446C22—1221no—
447C22—4881no—
448C22—4881no—
449C22—2441no—
450C22—2442no—
451C22—2441no—
452C22—2442no—
453C22—2442no—
454C22C2×(C2×#6923)⋊C411yescharacteristic
455C22C2×#1451983511yescharacteristic
456C22C2×#1451983611yescharacteristic
457C4—22no—
458C4—82no—
459C4—1221no—
460C4—1222no—
461C4—1221no—
462C4—4884no—
463C4—2444no—
464C4#1684064012yes—
465C4#1684064012yes—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—2441no—
471C2—2441no—
472C2—2442no—
473C2C2×#1684064011yescharacteristic
474C2#2146207212yes—
subgroupG/H
centerC2×C4#14519836
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC22×C22⋊D61C23
hypercenterC2×#117#6923
hypocenterC61#12573
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21460904C2semidirect
8. #16840748C4semidirect
12. #16840741C4semidirect
16. C22×C4×Dic61C4semidirect
19. #16839639C4semidirect
23. C2×#117×C61C4semidirect
245. C2×#117#6923semidirect
261. C61#12573semidirect

Maximal subgroups

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

subgroupindex
C2×#168407482
C2×#168407412
#214609042
#214483072
#1257361