Finite Group Databasebeta

← #30598254

Subgroups — #30598254

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461675C211yesmaximal, characteristic
2#21461675C211yesmaximal, characteristic
3C2×#16840622C212yesmaximal, complemented
4C2×#16840621C212yesmaximal, complemented
5C2×#16840314C211yesmaximal, characteristic
6C2×#14519844C2212yes—
7C2×#14519844C2212yes—
8C2×C22⋊#92420—21no—
9C2×C22⋊#92420—21no—
10#16840628—21no—
11#16840628—21no—
12#16840622—24no—
13#16840621—24no—
14#16840314—24no—
15#16840314—24no—
16C2×#14519776C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×#14519767—21no—
19C2×#14519756C411yescharacteristic
20C2×#14519754—21no—
21C2×Q8×Dic61C411yescharacteristic
22C2×#14519742—21no—
23C2×#14519742C411yescharacteristic
24C2×#14519733—21no—
25C22×C4×Dic61C2211yescharacteristic
26C2×C22⋊Q8×C61C411yescharacteristic
27C22×#92420—22no—
28C22×#92420—22no—
29#14519844—22no—
30#14519844—22no—
31#14519844—22no—
32#14519844—22no—
33C22⋊#92420—42no—
34C22⋊#92420—42no—
35C22×Dic122—21no—
36C22×Dic122C2×C411yescharacteristic
37C23×Dic61D411yescharacteristic
38C23×Dic61D411yescharacteristic
39#14519776—22no—
40#14519776D414yes—
41#14519775—22no—
42#14519775D414yes—
43#14519767—28no—
44#14519756—22no—
45#14519756—22no—
46#14519754—28no—
47Q8×Dic61—22no—
48Q8×Dic61—22no—
49#14519742—28no—
50#14519742—22no—
51#14519742—22no—
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—41no—
79C2×Q8×C61—41no—
80C2×Q8×C61C22⋊C412yes—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
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—21no—
96C22×Dic61—21no—
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61SD1612yes—
100C22×Dic61Q1612yes—
101C22×Dic61C2×D411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C2×Dic122—21no—
104C2×Dic122—24no—
105C2×Dic122—42no—
106C2×Dic122—41no—
107C2×Dic122—41no—
108C2×Dic122—41no—
109C2×Dic122—41no—
110C2×Dic122C22⋊C412yes—
111C2×#92420—42no—
112C2×#92420—42no—
113C2×#92420—42no—
114C2×#92420—42no—
115C2×#92420—42no—
116C2×#92420—42no—
117C4⋊C4×C61—24no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61C22⋊C412yes—
120C22⋊C4×C61—24no—
121C22⋊Dic61—24no—
122C4⋊Dic61—24no—
123C4⋊Dic61—21no—
124C4⋊Dic61C22⋊C412yes—
125#11758196—24no—
126#11758196—24no—
127#11758196—24no—
128#11758196—24no—
129#11758196—24no—
130C4×Dic61—24no—
131C4×Dic61—24no—
132C4×Dic61—21no—
133C4×Dic61—21no—
134C4×Dic61—22no—
135C4×Dic61—22no—
136C4×Dic61C2×D412yes—
137C4×Dic61C2×D412yes—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61#10312yes—
145C23×C61C2×C22⋊C411yescharacteristic
146Q8×C61—42no—
147Q8×C61—42no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—24no—
166C2×Dic61—24no—
167C2×Dic61—21no—
168C2×Dic61—22no—
169C2×Dic61—22no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61#10312yes—
179C2×Dic61#12012yes—
180C2×Dic61#12012yes—
181C2×Dic61C2×SD1611yescharacteristic
182C2×Dic61C2×Q1611yescharacteristic
183C2×Dic61#13512yes—
184C2×Dic61AGL(1,ℤ/8)11yescharacteristic
185C2×Dic61#13712yes—
186C2×Dic61#13711yescharacteristic
187Dic122—42no—
188Dic122—44no—
189Dic122—44no—
190Dic122—42no—
191#92420—82no—
192#92420—82no—
193#8343—611nomaximal, Sylow
194C22×C61—22no—
195C22×C61—22no—
196C22×C61—22no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61#37911yescharacteristic
202C22×C61#41311yescharacteristic
203C22×C61#41311yescharacteristic
204C22×C61C2×#10311yescharacteristic
205C22×C61#41611yescharacteristic
206C22×C61#41711yescharacteristic
207C22×C61#41911yescharacteristic
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—41no—
211C4×C61—42no—
212C4×C61—42no—
213Dic61—22no—
214Dic61—42no—
215Dic61—42no—
216Dic61—41no—
217Dic61#44812yes—
218Dic61#44912yes—
219Dic61#45112yes—
220Dic61#45212yes—
221C2×#533—611no—
222#1797—611no—
223#1797—611no—
224C2×#333—612no—
225C2×#332—612no—
226C2×C61—42no—
227C2×C61#154211yescharacteristic
228C2×C61#155011yescharacteristic
229C2×C61#155411yescharacteristic
230C2×C61#155911yescharacteristic
231C2×C61#178511yescharacteristic
232C2×C61#178711yescharacteristic
233C2×C61#178811yescharacteristic
234#533—1224no—
235#533—1224no—
236C2×C4⋊Q8—611no—
237C2×C4⋊Q8—611no—
238C2×#125—1221no—
239C2×C22⋊Q8—611no—
240C2×C22⋊Q8—1221no—
241C2×C22⋊Q8#692311yescharacteristic
242C2×C4×Q8—611no—
243C2×C4×Q8—611no—
244C2×C4×Q8—1221no—
245C2×#117—1221no—
246C22×C42—611no—
247C2×C4⋊C8—612no—
248C2×C4⋊C8—612no—
249C2×C22⋊C8—1221no—
250C2×C22⋊C8—1221no—
251#336—1221no—
252#336—1221no—
253#333—1224no—
254#332—1224no—
255C61#834311yescharacteristic, complemented, Sylow
256C22×Q8—611no—
257C22×Q8—1221no—
258C22×Q8C2×#692311yescharacteristic
259C23×C4—611no—
260C23×C4—611no—
261C23×C4C2×#692311yescharacteristic
262C22×C8—1222no—
263C22×C8—1222no—
264C4⋊Q8—614no—
265C4⋊Q8—614no—
266C4⋊Q8—1222no—
267C4⋊Q8—1222no—
268#125—1228no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—1228no—
272C22⋊Q8—1222no—
273C22⋊Q8—1222no—
274C4×Q8—1222no—
275C4×Q8—1228no—
276C4×Q8—1222no—
277C4×Q8—1222no—
278C4×Q8—1222no—
279#117—1228no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—611no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4—1221no—
286C2×C4⋊C4—1221no—
287C2×C4⋊C4—1221no—
288C2×C4⋊C4C2×#692311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—1221no—
291C2×C42—611no—
292C2×C42—611no—
293C2×C42—611no—
294C2×C42—611no—
295C2×C42—1222no—
296C2×C42—1222no—
297C2×C42—1221no—
298C2×C42—1221no—
299C4⋊C8—1222no—
300C4⋊C8—1222no—
301C4⋊C8—1222no—
302C4⋊C8—1222no—
303C22⋊C8—2442no—
304C22⋊C8—2442no—
305C24C22⋊#692311yescharacteristic
306C2×Q8—21no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—612no—
310C2×Q8—1224no—
311C2×Q8—1221no—
312C2×Q8—2441no—
313C2×Q8—2441no—
314C2×Q8—2442no—
315C2×Q8—2441no—
316C2×Q8—2441no—
317C2×Q8C22⋊#692312yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—612no—
327C22×C4—612no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—1222no—
331C22×C4—1222no—
332C22×C4—1221no—
333C22×C4—1222no—
334C22×C4—1222no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—1221no—
338C22×C4C22⋊#692311yescharacteristic
339C22×C4C22×#692311yescharacteristic
340C2×C8—2442no—
341C2×C8—2442no—
342C2×C8—2442no—
343C2×C8—2442no—
344C2×C8—2442no—
345C2×C8—2442no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—612no—
349C4⋊C4—1224no—
350C4⋊C4—1224no—
351C4⋊C4—1224no—
352C4⋊C4—1224no—
353C4⋊C4—1224no—
354C4⋊C4—1224no—
355C4⋊C4—1221no—
356C4⋊C4C22⋊#692312yes—
357C22⋊C4—24no—
358C22⋊C4—1224no—
359C42—612no—
360C42—612no—
361C42—1222no—
362C42—1222no—
363C42—1224no—
364C42—1224no—
365C42—1221no—
366C42—1221no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23#1451984012yes—
374C23C2×C22⋊#692311yescharacteristic
375Q8—42no—
376Q8—42no—
377Q8—2442no—
378Q8—2442no—
379Q8—2444no—
380Q8—2444no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—611no—
395C2×C4—611no—
396C2×C4—612no—
397C2×C4—611no—
398C2×C4—611no—
399C2×C4—612no—
400C2×C4—612no—
401C2×C4—612no—
402C2×C4—612no—
403C2×C4—1224no—
404C2×C4—1224no—
405C2×C4—1221no—
406C2×C4—1221no—
407C2×C4—1221no—
408C2×C4—1221no—
409C2×C4—1221no—
410C2×C4—1222no—
411C2×C4—1222no—
412C2×C4—2442no—
413C2×C4—2442no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4C2×C22⋊#692311yescharacteristic
419C2×C4C2×C22⋊#692311yescharacteristic
420C8—4882no—
421C8—4882no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22#1684070311yescharacteristic
430C22#1684074911yescharacteristic
431C22#1684074911yescharacteristic
432C22C2×#1451984011yescharacteristic
433C22#1684075211yescharacteristic
434C22#1684075311yescharacteristic
435C22#1684075511yescharacteristic
436C4—21no—
437C4—21no—
438C4—41no—
439C4—42no—
440C4—42no—
441C4—612no—
442C4—612no—
443C4—612no—
444C4—612no—
445C4—1222no—
446C4—2442no—
447C4—2441no—
448C4—2442no—
449C2—42no—
450C2#2146132111yescharacteristic
451C2#2146133711yescharacteristic
452C2#2146134511yescharacteristic
453C2#2146135511yescharacteristic
454C2#2146165611yescharacteristic
455C2#2146165811yescharacteristic
456C2#2146165911yescharacteristic
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#8343
LCSG ▷ C22×C4×C61 ▷ C22×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×#16840622C2semidirect
4. C2×#16840621C2semidirect
255. C61#8343semidirect

Maximal subgroups

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

subgroupindex
#214616752
C2×#168406222
C2×#168406212
C2×#168403142
#834361