Finite Group Databasebeta

← #30598168

Subgroups — #30598168

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461673C211yesmaximal, characteristic
2C4×C22⋊#92420C211yesmaximal, characteristic
3C2×#16840612C212yesmaximal, complemented
4C2×Q8⋊#92420C212yesmaximal, complemented
5C2×#16840314C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×C22⋊#92420—21no—
8C2×C22⋊#92420—21no—
9C2×C4×#92420C2212yes—
10#16840628—21no—
11#16840628—21no—
12#16840612—24no—
13Q8⋊#92420—24no—
14#16840314—24no—
15#16840314—24no—
16C2×#14519776C411yescharacteristic
17C2×#14519775C411yescharacteristic
18C2×#14519767—21no—
19C2×#14519756C411yescharacteristic
20C2×#14519754—21no—
21C2×Q8×Dic61C2211yescharacteristic
22C2×#14519742—21no—
23C2×#14519742C2211yescharacteristic
24C2×#14519733—21no—
25C22×C4×Dic61C2211yescharacteristic
26C2×C22⋊Q8×C61C411yescharacteristic
27C22×#92420—22no—
28C22×#92420—22no—
29C4⋊#92420—22no—
30C4⋊#92420—22no—
31C22⋊#92420—24no—
32C22⋊#92420—42no—
33C4×#92420—22no—
34C4×#92420—22no—
35C22×Dic122—21no—
36C22×Dic122C2×C411yescharacteristic
37C23×Dic61D411yescharacteristic
38C23×Dic61D411yescharacteristic
39#14519776—22no—
40#14519776—22no—
41#14519775—22no—
42#14519775—22no—
43#14519767—28no—
44#14519756—22no—
45#14519756—22no—
46#14519754—28no—
47Q8×Dic61—22no—
48Q8×Dic61D414yes—
49#14519742—28no—
50#14519742—22no—
51#14519742D414yes—
52#14519733—28no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×C22⋊Dic61—21no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—22no—
64C2×C4×Dic61—22no—
65C2×C4×Dic61C2×C411yescharacteristic
66C2×C4×Dic61D411yescharacteristic
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C22×Q8×C61C2×C411yescharacteristic
70C23×C4×C61C2×C411yescharacteristic
71C22⋊Q8×C61—22no—
72C22⋊Q8×C61—22no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61C2×C411yescharacteristic
75C2×C22⋊C4×C61—21no—
76C24×C61C22⋊C411yescharacteristic
77C2×Q8×C61—21no—
78C2×Q8×C61—22no—
79C2×Q8×C61—22no—
80C2×Q8×C61C22⋊C412yes—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22×C411yescharacteristic
89C22×Dic61—22no—
90C22×Dic61—22no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—22no—
94C22×Dic61—22no—
95C22×Dic61—41no—
96C22×Dic61C22⋊C411yescharacteristic
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61SD1612yes—
99C22×Dic61Q1612yes—
100C22×Dic61C2×D411yescharacteristic
101C22×Dic61C2×D411yescharacteristic
102C2×Dic122—22no—
103C2×Dic122—22no—
104C2×Dic122—24no—
105C2×Dic122—21no—
106C2×Dic122—42no—
107C2×Dic122—42no—
108C2×Dic122C22⋊C412yes—
109C2×#92420—24no—
110C2×#92420—42no—
111C2×#92420—42no—
112C2×#92420—42no—
113C2×#92420—42no—
114C2×#92420—42no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61C22⋊C412yes—
118C22⋊C4×C61—24no—
119C22⋊Dic61—24no—
120C4⋊Dic61—21no—
121C4⋊Dic61—24no—
122C4⋊Dic61C22⋊C412yes—
123#11758196—24no—
124#11758196—24no—
125#11758196—24no—
126#11758196—24no—
127#11758196—24no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130C4×Dic61—22no—
131C4×Dic61—22no—
132C4×Dic61—24no—
133C4×Dic61—24no—
134C4×Dic61C2×D412yes—
135C4×Dic61C2×D412yes—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61#10312yes—
143C23×C61C2×C22⋊C411yescharacteristic
144Q8×C61—24no—
145Q8×C61—42no—
146C2×C4×C61—21no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61C2×C22⋊C411yescharacteristic
158C2×C4×C61C2×C22⋊C411yescharacteristic
159C2×C4×C61#13112yes—
160C2×C4×C61#13112yes—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—24no—
168C2×Dic61—24no—
169C2×Dic61—22no—
170C2×Dic61—22no—
171C2×Dic61—44no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61#10312yes—
177C2×Dic61#10412yes—
178C2×Dic61#12012yes—
179C2×Dic61#12012yes—
180C2×Dic61C2×SD1611yescharacteristic
181C2×Dic61C2×Q1611yescharacteristic
182C2×Dic61#13712yes—
183Dic122—24no—
184Dic122—48no—
185Dic122—42no—
186#92420—44no—
187#92420—82no—
188#8314—611nomaximal, Sylow
189C22×C61—22no—
190C22×C61—22no—
191C22×C61—22no—
192C22×C61—41no—
193C22×C61—41no—
194C22×C61—41no—
195C22×C61—41no—
196C22×C61#37911yescharacteristic
197C22×C61#40811yescharacteristic
198C22×C61#40811yescharacteristic
199C22×C61C2×#10311yescharacteristic
200C22×C61#41911yescharacteristic
201C22×C61C2×#10411yescharacteristic
202C22×C61#42111yescharacteristic
203C4×C61—41no—
204C4×C61—42no—
205C4×C61—42no—
206C4×C61#40812yes—
207C4×C61#40812yes—
208Dic61—22no—
209Dic61—22no—
210Dic61—22no—
211Dic61—42no—
212Dic61—42no—
213Dic61#44812yes—
214Dic61#45112yes—
215C2×#533—611no—
216#1797—611no—
217C4×C22⋊C8—611no—
218C2×Q8⋊C8—612no—
219C2×Q8⋊C8—612no—
220C2×C61—42no—
221C2×C61#148111yescharacteristic
222C2×C61#148511yescharacteristic
223C2×C61#149311yescharacteristic
224C2×C61#149711yescharacteristic
225C2×C61#178011yescharacteristic
226C2×C61#178711yescharacteristic
227C2×C61#178911yescharacteristic
228#533—1224no—
229#533—1224no—
230C2×C4⋊Q8—611no—
231C2×C4⋊Q8—611no—
232C2×#125—1221no—
233C2×C22⋊Q8—611no—
234C2×C22⋊Q8—1221no—
235C2×C22⋊Q8#692311yescharacteristic
236C2×C4×Q8—611no—
237C2×C4×Q8—611no—
238C2×C4×Q8—1221no—
239C2×#117—1221no—
240C22×C42—611no—
241C2×C4⋊C8—612no—
242C2×C22⋊C8—1221no—
243C2×C22⋊C8—1221no—
244C2×C4×C8—612no—
245#336—1221no—
246#336—1221no—
247Q8⋊C8—1224no—
248Q8⋊C8—1224no—
249C61#831411yescharacteristic, complemented, Sylow
250C22×Q8—611no—
251C22×Q8—1221no—
252C22×Q8C2×#692311yescharacteristic
253C23×C4—611no—
254C23×C4—611no—
255C23×C4C2×#692311yescharacteristic
256C22×C8—1222no—
257C22×C8—1222no—
258C4⋊Q8—1222no—
259C4⋊Q8—1222no—
260C4⋊Q8—1222no—
261C4⋊Q8—1222no—
262#125—1228no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—1228no—
266C22⋊Q8—1222no—
267C22⋊Q8—1222no—
268C4×Q8—614no—
269C4×Q8—614no—
270C4×Q8—1222no—
271C4×Q8—1222no—
272C4×Q8—1228no—
273#117—1228no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4—1221no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4C2×#692311yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—1221no—
285C2×C42—611no—
286C2×C42—611no—
287C2×C42—611no—
288C2×C42—611no—
289C2×C42—1222no—
290C2×C42—1222no—
291C2×C42—1221no—
292C2×C42—1221no—
293C4⋊C8—1222no—
294C4⋊C8—1222no—
295C22⋊C8—1224no—
296C22⋊C8—2442no—
297C4×C8—1222no—
298C4×C8—1222no—
299C24C22⋊#692311yescharacteristic
300C2×Q8—21no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—612no—
304C2×Q8—1224no—
305C2×Q8—1221no—
306C2×Q8—1222no—
307C2×Q8—1222no—
308C2×Q8—2442no—
309C2×Q8—2442no—
310C2×Q8C22⋊#692312yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—611no—
318C22×C4—612no—
319C22×C4—612no—
320C22×C4—611no—
321C22×C4—611no—
322C22×C4—611no—
323C22×C4—1222no—
324C22×C4—1222no—
325C22×C4—1222no—
326C22×C4—1222no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—2441no—
330C22×C4C22⋊#692311yescharacteristic
331C22×C4C22×#692311yescharacteristic
332C2×C8—1224no—
333C2×C8—2442no—
334C2×C8—2442no—
335C2×C8—2442no—
336C2×C8—2442no—
337C2×C8—2442no—
338C4⋊C4—24no—
339C4⋊C4—21no—
340C4⋊C4—612no—
341C4⋊C4—1224no—
342C4⋊C4—1224no—
343C4⋊C4—1224no—
344C4⋊C4—1224no—
345C4⋊C4—1224no—
346C4⋊C4—1224no—
347C4⋊C4—1221no—
348C4⋊C4C22⋊#692312yes—
349C22⋊C4—24no—
350C22⋊C4—1224no—
351C42—612no—
352C42—612no—
353C42—1222no—
354C42—1222no—
355C42—1224no—
356C42—1224no—
357C42—1221no—
358C42—1221no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23#1451984012yes—
366C23C2×C22⋊#692311yescharacteristic
367Q8—24no—
368Q8—42no—
369Q8—1224no—
370Q8—2448no—
371Q8—2442no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—611no—
384C2×C4—611no—
385C2×C4—612no—
386C2×C4—612no—
387C2×C4—612no—
388C2×C4—612no—
389C2×C4—612no—
390C2×C4—1224no—
391C2×C4—1224no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1222no—
399C2×C4—1222no—
400C2×C4—2444no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4C2×C22⋊#692311yescharacteristic
406C2×C4C2×C22⋊#692311yescharacteristic
407C2×C4#1451986412yes—
408C2×C4#1451986412yes—
409C8—4882no—
410C8—2444no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22#1684070311yescharacteristic
419C22#1684073511yescharacteristic
420C22#1684073511yescharacteristic
421C22C2×#1451984011yescharacteristic
422C22#1684075511yescharacteristic
423C22D61⋊#10411yescharacteristic
424C22#1684075911yescharacteristic
425C4—41no—
426C4—42no—
427C4—42no—
428C4—612no—
429C4—612no—
430C4—1222no—
431C4—1222no—
432C4—1222no—
433C4—2442no—
434C4—2442no—
435C4#1684073512yes—
436C4#1684073612yes—
437C2—42no—
438C2#2146112711yescharacteristic
439C2#2146114111yescharacteristic
440C2#2146116711yescharacteristic
441C2#2146118111yescharacteristic
442C2#2146164311yescharacteristic
443C2#2146165811yescharacteristic
444C2#2146166111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#8314
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16840612C2semidirect
4. C2×Q8⋊#92420C2semidirect
249. C61#8314semidirect

Maximal subgroups

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

subgroupindex
#214616732
C4×C22⋊#924202
C2×#168406122
C2×Q8⋊#924202
C2×#168403142
#831461