Finite Group Databasebeta

← #30598396

Subgroups — #30598396

Order 15616.

966 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 118 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#21461683C211yesmaximal, characteristic
2#21461332C212yesmaximal, complemented
3#21461149C212yesmaximal, complemented
4#21461058C211yesmaximal, characteristic
5C2×#16840425C211yesmaximal, characteristic
6#16840779C2212yes—
7#16840765C2212yes—
8C2×#14519837—21no—
9#16840642—21no—
10#16840622—24no—
11#16840612—24no—
12#16840606—21no—
13#16840605—21no—
14C23×Dic122C411yescharacteristic
15C22×Q8×D61C411yescharacteristic
16#16840425—24no—
17#16840425—24no—
18C2×#14519776—21no—
19C2×#14519775C2211yescharacteristic
20C2×#14519757C411yescharacteristic
21C2×#14519755—21no—
22C2×#14519754—21no—
23C2×#14519742C2211yescharacteristic
24C2×C4×C22⋊D61C2211yescharacteristic
25C2×#14519737—21no—
26C2×C22⋊Q8×C61C411yescharacteristic
27C2×#11758217—22no—
28#14519844—22no—
29#14519844—22no—
30C4⋊#92420—22no—
31#14519837—42no—
32C22⋊#92420—22no—
33C4×#92420—22no—
34C22×Dic122—21no—
35C22×Dic122—21no—
36C22×Dic122—22no—
37C22×Dic122—22no—
38C22×Dic122—44no—
39C22×Dic122C2×C411yescharacteristic
40C22×Dic122C2×C411yescharacteristic
41C2×Q8×D61—22no—
42C2×Q8×D61—22no—
43C2×Q8×D61—44no—
44C22×C22⋊D61D411yescharacteristic
45C23×Dic61—21no—
46C22×C4×D61—21no—
47C22×C4×D61C2×C411yescharacteristic
48#14519776—28no—
49#14519775—22no—
50#14519775D414yes—
51#14519757—22no—
52#14519757—22no—
53#14519755—28no—
54#14519754—28no—
55#14519742—22no—
56#14519742D414yes—
57C4×C22⋊D61—22no—
58C4×C22⋊D61—22no—
59#14519737—28no—
60C2×C4⋊Dic61—21no—
61C2×#11758196—21no—
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758196D411yescharacteristic
65C2×C22⋊Dic61D411yescharacteristic
66C2×#11758198—21no—
67C2×#11758198D411yescharacteristic
68C2×C4×Dic61—21no—
69C2×C4×Dic61C2311yescharacteristic
70C22×Q8×C61C2×C411yescharacteristic
71C23×C4×C61C2×C411yescharacteristic
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—22no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C24×C61C22⋊C411yescharacteristic
78C23×D61C22⋊C411yescharacteristic
79C2×Q8×C61—21no—
80C2×Q8×C61—42no—
81C2×Q8×C61C22⋊C412yes—
82C22×C4×C61—21no—
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—22no—
90C2×C22⋊D61—22no—
91C2×C22⋊D61SD1612yes—
92C2×C22⋊D61Q1612yes—
93C22×Dic61—21no—
94C22×Dic61—24no—
95C22×Dic61—21no—
96C22×Dic61—44no—
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C2×D411yescharacteristic
99C22×Dic61C2×D411yescharacteristic
100Q8×D61—48no—
101Q8×D61—44no—
102Q8×D61—44no—
103C2×C4×D61—24no—
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×C4×D61—44no—
107C2×C4×D61—42no—
108C2×Dic122—21no—
109C2×Dic122—24no—
110C2×Dic122—24no—
111C2×Dic122—24no—
112C2×Dic122—21no—
113C2×Dic122—48no—
114C2×Dic122—44no—
115C2×Dic122—44no—
116C2×Dic122—44no—
117C2×Dic122—44no—
118C2×Dic122—42no—
119C2×Dic122C22⋊C412yes—
120C2×Dic122C22⋊C412yes—
121#11758217—42no—
122#11758217—42no—
123C2×#92420—42no—
124C2×#92420—42no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61C22⋊C412yes—
128C22⋊C4×C61—24no—
129C22⋊Dic61—22no—
130#11758198—24no—
131#11758198—22no—
132C4⋊Dic61—24no—
133#11758196—24no—
134#11758196—24no—
135#11758196—24no—
136#11758196—22no—
137C4×Dic61—24no—
138C4×Dic61C2×D412yes—
139C4×Dic61C2×D412yes—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61#10312yes—
145C23×C61C2×C22⋊C411yescharacteristic
146C22×D61—21no—
147C22×D61—21no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61#10312yes—
151Q8×C61—42no—
152Q8×C61—42no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—21no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C22⋊D61—24no—
165C22⋊D61—24no—
166C2×Dic61—24no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—21no—
170C2×Dic61—41no—
171C2×Dic61—48no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61#12012yes—
175C2×Dic61#12012yes—
176C2×Dic61C2×SD1611yescharacteristic
177C2×Dic61C2×Q1611yescharacteristic
178C2×Dic61#13712yes—
179C4×D61—48no—
180C4×D61—42no—
181C4×D61—42no—
182Dic122—24no—
183Dic122—22no—
184Dic122—48no—
185Dic122—48no—
186Dic122—42no—
187Dic122—42no—
188Dic122#10014yescomplemented
189#92420—82no—
190#92420—82no—
191#8391—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—22no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61#37911yescharacteristic
200C22×C61C2×#10311yescharacteristic
201C22×C61#41911yescharacteristic
202C2×D61—22no—
203C2×D61—42no—
204C2×D61—41no—
205C2×D61—41no—
206C2×D61—42no—
207C4×C61—21no—
208C4×C61—42no—
209C4×C61—41no—
210C4×C61—42no—
211C4×C61C2×#10012yes—
212Dic61—22no—
213Dic61—24no—
214Dic61—41no—
215Dic61—42no—
216Dic61#44812yes—
217Dic61#45112yes—
218C2×D4×Q8—611no—
219#1800—611no—
220#1548—612no—
221#1487—612no—
222#1460—611no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61—42no—
226C2×C61#178311yescharacteristic
227C2×C61#178711yescharacteristic
228C2×C61#179611yescharacteristic
229D61—42no—
230C23×Q8—611no—
231C23×Q8—611no—
232D4×Q8—1224no—
233D4×Q8—1224no—
234C2×C4⋊Q8—611no—
235C2×C4⋊Q8—1221no—
236C2×C22⋊Q8—611no—
237C2×C22⋊Q8—1221no—
238C2×C22⋊Q8—1221no—
239C2×C22⋊Q8#692311yescharacteristic
240C2×C4×Q8—611no—
241C2×C4×D4—611no—
242C2×C4×D4—1221no—
243#431—612no—
244#423—612no—
245C2×#100—1221no—
246#343—1221no—
247#333—1224no—
248Q8⋊C8—1224no—
249#323—1221no—
250#323—1221no—
251C61#839111yescharacteristic, complemented, Sylow
252C22×Q8—611no—
253C22×Q8—611no—
254C22×Q8—1221no—
255C22×Q8—1222no—
256C22×Q8—1222no—
257C22×Q8—1222no—
258C22×Q8—1222no—
259C22×Q8—1221no—
260C22×Q8—2444no—
261C22×Q8—2444no—
262C22×Q8C2×#692311yescharacteristic
263C22×D4—611no—
264C23×C4—611no—
265C23×C4—1221no—
266C23×C4—1221no—
267C23×C4C2×#692311yescharacteristic
268C2×M16—1222no—
269C4⋊Q8—614no—
270C4⋊Q8—1222no—
271C4⋊Q8—1228no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C22⋊Q8—1228no—
275C22⋊Q8—1222no—
276C22⋊Q8—1228no—
277C22⋊Q8—1222no—
278C4×Q8—614no—
279C4×Q8—1222no—
280C4×D4—1222no—
281C4×D4—1222no—
282C4×D4—1228no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—611no—
285C2×C4⋊C4—1221no—
286C2×C4⋊C4—1221no—
287C2×C4⋊C4—1221no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4C2×#692311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—611no—
292C2×C22⋊C4—611no—
293C2×C22⋊C4—1221no—
294C2×C42—611no—
295C2×C42—1221no—
296C4⋊C8—1222no—
297C4⋊C8—1222no—
298C4⋊C8—1222no—
299#100—2442no—
300C22⋊C8—1222no—
301C4×C8—1222no—
302C24—611no—
303C24C22⋊#692311yescharacteristic
304C2×Q8—21no—
305C2×Q8—42no—
306C2×Q8—612no—
307C2×Q8—612no—
308C2×Q8—1224no—
309C2×Q8—1224no—
310C2×Q8—1224no—
311C2×Q8—1221no—
312C2×Q8—1221no—
313C2×Q8—2442no—
314C2×Q8—2444no—
315C2×Q8—2444no—
316C2×Q8—2444no—
317C2×Q8—2448no—
318C2×Q8—2444no—
319C2×Q8—2444no—
320C2×Q8—2448no—
321C2×Q8—2444no—
322C2×Q8C22⋊#692312yes—
323C2×D4—612no—
324C2×D4—612no—
325C2×D4—1222no—
326C2×D4—1222no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—611no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—1224no—
336C22×C4—1221no—
337C22×C4—1224no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—2442no—
342C22×C4—2444no—
343C22×C4—2444no—
344C22×C4C22⋊#692311yescharacteristic
345C22×C4C22×#692311yescharacteristic
346M16—2442no—
347M16—2442no—
348C2×C8—2442no—
349C2×C8—2442no—
350C4⋊C4—24no—
351C4⋊C4—21no—
352C4⋊C4—1222no—
353C4⋊C4—1224no—
354C4⋊C4—1224no—
355C4⋊C4—1224no—
356C4⋊C4—1224no—
357C4⋊C4C22⋊#692312yes—
358C22⋊C4—24no—
359C22⋊C4—1222no—
360C22⋊C4—1224no—
361C22⋊C4—1222no—
362C42—612no—
363C42—612no—
364C42—1224no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—612no—
370C23—1221no—
371C23—1221no—
372C23—2441no—
373C23—2441no—
374C23#1451984012yes—
375C23C2×C22⋊#692311yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—614no—
379Q8—1224no—
380Q8—1222no—
381Q8—2442no—
382Q8—2448no—
383Q8—2448no—
384Q8—2442no—
385D4—1224no—
386D4—1224no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—611no—
397C2×C4—611no—
398C2×C4—612no—
399C2×C4—612no—
400C2×C4—612no—
401C2×C4—1224no—
402C2×C4—1222no—
403C2×C4—1221no—
404C2×C4—1222no—
405C2×C4—2442no—
406C2×C4—2448no—
407C2×C4—2441no—
408C2×C4—2448no—
409C2×C4—2442no—
410C2×C4—2442no—
411C2×C4—2441no—
412C2×C4C2×C22⋊#692311yescharacteristic
413C2×C4C2×C22⋊#692311yescharacteristic
414C8—4882no—
415C8—4882no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—1222no—
424C22—2442no—
425C22—2442no—
426C22—2441no—
427C22—2441no—
428C22#1684070311yescharacteristic
429C22C2×#1451984011yescharacteristic
430C22#1684075511yescharacteristic
431C4—21no—
432C4—41no—
433C4—42no—
434C4—42no—
435C4—612no—
436C4—612no—
437C4—1224no—
438C4—1222no—
439C4—2442no—
440C4—2441no—
441C4D61⋊#10012yes—
442C2—21no—
443C2—21no—
444C2—42no—
445C2—2442no—
446C2#2146165211yescharacteristic
447C2#2146165811yescharacteristic
448C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61#379
prime radicalC22×C22⋊D61D4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#8391
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #21461332C2semidirect
3. #21461149C2semidirect
188. Dic122#100semidirect
251. C61#8391semidirect

Maximal subgroups

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

subgroupindex
#214616832
#214613322
#214611492
#214610582
C2×#168404252
#839161