Finite Group Databasebeta

← #30598341

Subgroups — #30598341

Order 15616.

814 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C4×#14519837C211yesmaximal, characteristic, complemented
2#21461283C212yesmaximal, complemented
3#21461158C212yesmaximal, complemented
4#21461078C211yesmaximal, characteristic, complemented
5C2×#16840459C211yesmaximal, characteristic
6#16840783C2212yes—
7C2×#14519837—21no—
8C4×#11758217C2212yes—
9#16840642—21no—
10#16840619—24no—
11D4⋊#92420—24no—
12#16840606—21no—
13#16840605—21no—
14#16840459—24no—
15#16840459—24no—
16C2×#14519770—21no—
17C2×#14519768C2211yescharacteristic
18C2×#14519758—21no—
19C2×#14519748—21no—
20C2×#14519748C411yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C2×C4×C22⋊D61C2211yescharacteristic
23C2×#14519737—21no—
24C2×#14519736C411yescharacteristic
25C2×C4⋊C4×D61C411yescharacteristic
26C2×#121×C61C411yescharacteristic
27C2×#11758217—22no—
28C4⋊#92420—22no—
29#14519837—24no—
30C22⋊#92420—22no—
31#14519832—22no—
32#14519832—22no—
33C4×#92420—22no—
34C22×C22⋊D61—21no—
35C22×C22⋊D61D411yescharacteristic
36C23×Dic61—21no—
37C22×C4×D61—21no—
38C22×C4×D61C2×C411yescharacteristic
39#14519770—28no—
40#14519768—22no—
41#14519768D414yes—
42#14519758—28no—
43#14519748—28no—
44#14519748—22no—
45#14519748—22no—
46D4×Dic61—22no—
47D4×Dic61D414yes—
48C4×C22⋊D61—22no—
49C4×C22⋊D61—22no—
50#14519737—28no—
51#14519736—22no—
52#14519736—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196D411yescharacteristic
59C4⋊C4×D61—22no—
60C4⋊C4×D61—22no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—21no—
64C2×C22⋊Dic61D411yescharacteristic
65C2×#11758198D411yescharacteristic
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2311yescharacteristic
68C22×D4×C61—21no—
69C22×D4×C61C2×C411yescharacteristic
70C23×C4×C61C2×C411yescharacteristic
71#121×C61—22no—
72#121×C61—22no—
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C24×C61—21no—
76C24×C61C22⋊C411yescharacteristic
77C23×D61C22⋊C411yescharacteristic
78C2×D4×C61—24no—
79C2×D4×C61—21no—
80C2×D4×C61—42no—
81C2×D4×C61—82no—
82C2×D4×C61C22⋊C412yes—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—42no—
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22×C411yescharacteristic
89C2×C22⋊D61—24no—
90C2×C22⋊D61—22no—
91C2×C22⋊D61—22no—
92C2×C22⋊D61—41no—
93C2×C22⋊D61—42no—
94C2×C22⋊D61—42no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—24no—
98C22×Dic61—82no—
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C2×D411yescharacteristic
101C22×Dic61C2×D411yescharacteristic
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—24no—
105C2×C4×D61—42no—
106C2×C4×D61—82no—
107#11758217—24no—
108#11758217—42no—
109C2×#92420—42no—
110C2×#92420—42no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61C22⋊C412yes—
113C22⋊C4×C61—24no—
114C22⋊Dic61—24no—
115C22⋊Dic61—24no—
116C22⋊Dic61—24no—
117C22⋊Dic61—22no—
118#11758198—22no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—24no—
122C4⋊Dic61C22⋊C412yes—
123C4⋊Dic61C22⋊C412yes—
124#11758196—24no—
125#11758196—24no—
126#11758196—22no—
127C4×Dic61—24no—
128C4×Dic61C2×D412yes—
129C4×Dic61C2×D412yes—
130C23×C61—21no—
131C23×C61—22no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61C2×C22⋊C411yescharacteristic
141C22×D61—21no—
142C22×D61—21no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61#10412yes—
146D4×C61—24no—
147D4×C61—42no—
148D4×C61—84no—
149C2×C4×C61—22no—
150C2×C4×C61—21no—
151C2×C4×C61—22no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—41no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61C2×C22⋊C411yescharacteristic
158C22⋊D61—44no—
159C22⋊D61—48no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61—21no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—84no—
171C2×Dic61#12012yes—
172C2×Dic61#12012yes—
173C4×D61—42no—
174C4×D61—42no—
175C4×D61—84no—
176#92420—44no—
177#92420—82no—
178#8373—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—42no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—42no—
185C22×C61—41no—
186C22×C61—42no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—82no—
190C22×C61—82no—
191C22×C61#37911yescharacteristic
192C22×C61C2×#10411yescharacteristic
193C22×C61#42111yescharacteristic
194C2×D61—22no—
195C2×D61—42no—
196C2×D61—41no—
197C2×D61—41no—
198C2×D61—42no—
199C4×C61—21no—
200C4×C61—42no—
201C4×C61—41no—
202C4×C61#41312yes—
203Dic61—22no—
204Dic61—22no—
205Dic61—42no—
206Dic61—41no—
207Dic61—42no—
208C2×#553—611no—
209C4×#100—611no—
210#1528—612no—
211#1490—612no—
212#1466—611no—
213C2×C61—21no—
214C2×C61—21no—
215C2×C61—81no—
216C2×C61—82no—
217C2×C61#178511yescharacteristic
218C2×C61#178911yescharacteristic
219C2×C61#179511yescharacteristic
220D61—42no—
221#553—1224no—
222#553—1224no—
223C2×#126—1221no—
224C2×#125—611no—
225C2×#123—1221no—
226C2×#121—611no—
227C2×#121—1221no—
228C2×#121#692311yescharacteristic
229C2×C4×D4—611no—
230C2×C4×D4—611no—
231C2×C4×D4—1221no—
232C2×#117—611no—
233C22×C4⋊C4—611no—
234#432—612no—
235C2×#100—1221no—
236C4×M16—612no—
237#343—1221no—
238#330—1224no—
239D4⋊C8—1224no—
240#323—1221no—
241#323—1221no—
242C61#837311yescharacteristic, complemented, Sylow
243C22×D4—21no—
244C22×D4—611no—
245C22×D4—1221no—
246C22×D4C2×#692311yescharacteristic
247C23×C4—611no—
248C23×C4—1221no—
249C23×C4—1221no—
250C23×C4C2×#692311yescharacteristic
251C2×M16—1222no—
252#126—1228no—
253#125—614no—
254#125—1222no—
255#123—1228no—
256#121—22no—
257#121—22no—
258#121—1222no—
259#121—1228no—
260#121—1222no—
261C4×D4—614no—
262C4×D4—1222no—
263C4×D4—1222no—
264C4×D4—1222no—
265C4×D4—1228no—
266#117—1222no—
267#117—1222no—
268C2×C4⋊C4—611no—
269C2×C4⋊C4—611no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—1222no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1222no—
276C2×C4⋊C4C2×#692311yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—611no—
279C2×C22⋊C4—611no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C42—611no—
284C2×C42—1221no—
285C4⋊C8—1222no—
286#100—1224no—
287C22⋊C8—1222no—
288#97—1222no—
289#97—1222no—
290C4×C8—1222no—
291C24—21no—
292C24—611no—
293C24C22⋊#692311yescharacteristic
294C2×D4—24no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—82no—
298C2×D4—1222no—
299C2×D4—1222no—
300C2×D4—1224no—
301C2×D4—2441no—
302C2×D4—2442no—
303C2×D4—2442no—
304C2×D4C22⋊#692312yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—1224no—
313C22×C4—1224no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—4882no—
319C22×C4—4882no—
320C22×C4—2442no—
321C22×C4C22⋊#692311yescharacteristic
322C22×C4C22×#692311yescharacteristic
323M16—1224no—
324M16—2442no—
325C2×C8—2442no—
326C2×C8—2442no—
327C4⋊C4—21no—
328C4⋊C4—612no—
329C4⋊C4—612no—
330C4⋊C4—1222no—
331C4⋊C4—1224no—
332C4⋊C4—1224no—
333C4⋊C4—1224no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4C22⋊#692312yes—
337C22⋊C4—24no—
338C22⋊C4—1222no—
339C22⋊C4—1222no—
340C22⋊C4—1224no—
341C22⋊C4—1224no—
342C22⋊C4—1224no—
343C42—612no—
344C42—612no—
345C42—1224no—
346C23—22no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—612no—
357C23—1221no—
358C23—1221no—
359C23—2441no—
360C23—2441no—
361C23C2×C22⋊#692311yescharacteristic
362D4—24no—
363D4—42no—
364D4—84no—
365D4—2444no—
366D4—2448no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—612no—
375C2×C4—612no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4—1221no—
381C2×C4—4884no—
382C2×C4—4884no—
383C2×C4—2442no—
384C2×C4—2441no—
385C2×C4—2442no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4C2×C22⋊#692311yescharacteristic
391C2×C4C2×C22⋊#692311yescharacteristic
392C8—4882no—
393C8—2444no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22—1222no—
407C22—2442no—
408C22—2442no—
409C22—2441no—
410C22—2441no—
411C22#1684070311yescharacteristic
412C22D61⋊#10411yescharacteristic
413C22#1684075911yescharacteristic
414C4—21no—
415C4—41no—
416C4—42no—
417C4—1222no—
418C4—1222no—
419C4—2441no—
420C4—2442no—
421C4—2442no—
422C4#1684074912yes—
423C2—21no—
424C2—21no—
425C2—82no—
426C2—81no—
427C2—2442no—
428C2#2146165611yescharacteristic
429C2#2146166111yescharacteristic
430C2#2146166911yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#121×C61C4
socleC22×C61#379
prime radicalC2×#16840459C2
cosocleC2×C4×Dic61C23
hypercenterC2×#121#6923
hypocenterC61#8373
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C4×#14519837C2semidirect
2. #21461283C2semidirect
3. #21461158C2semidirect
4. #21461078C2semidirect
242. C61#8373semidirect

Maximal subgroups

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

subgroupindex
C4×#145198372
#214612832
#214611582
#214610782
C2×#168404592
#837361