Finite Group Databasebeta

← #30598253

Subgroups — #30598253

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#21461676C211yesmaximal, characteristic
2#21461676C211yesmaximal, characteristic
3C2×#16840622C212yesmaximal, complemented
4C2×#16840621C212yesmaximal, complemented
5C2×#16840316C211yesmaximal, characteristic
6C2×#14519844C2212yes—
7C2×#14519844C2212yes—
8C2×#14519834—21no—
9C2×#14519834—21no—
10#16840628—21no—
11#16840628—21no—
12#16840622—24no—
13#16840621—24no—
14#16840316—24no—
15#16840316—24no—
16C2×#14519776C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×#14519767—21no—
19C2×#14519757C411yescharacteristic
20C2×#14519755—21no—
21C2×#14519753C411yescharacteristic
22C2×C4×Dic122—21no—
23C2×C4×Dic122C411yescharacteristic
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic
27C22×#92420—22no—
28C22×#92420—22no—
29#14519844—22no—
30#14519844—22no—
31#14519844—22no—
32#14519844—22no—
33#14519834—42no—
34#14519834—42no—
35C22×Dic122—21no—
36C22×Dic122C2×C411yescharacteristic
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40#14519776—22no—
41#14519776D414yes—
42#14519775—22no—
43#14519775D414yes—
44#14519767—28no—
45#14519757—22no—
46#14519757—22no—
47#14519755—28no—
48#14519753—22no—
49#14519753—22no—
50C4×Dic122—28no—
51C4×Dic122—22no—
52C4×Dic122—22no—
53#14519734—28no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C42×D61—22no—
68C42×D61—22no—
69C22×Q8×C61C2×C411yescharacteristic
70C4×Q8×C61—22no—
71C4×Q8×C61—22no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2×C411yescharacteristic
76C23×D61C22⋊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×C61C22⋊C411yescharacteristic
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61C22⋊C411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C22×Dic61C2×D411yescharacteristic
91C2×C4×D61—22no—
92C2×C4×D61—22no—
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—22no—
96C2×C4×D61—22no—
97C2×C4×D61—21no—
98C2×C4×D61—21no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61SD1612yes—
102C2×C4×D61Q1612yes—
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—
120C42×C61—24no—
121C42×C61—21no—
122C42×C61—21no—
123#11758198—24no—
124#11758198—24no—
125C4⋊Dic61—24no—
126C4⋊Dic61—21no—
127C4⋊Dic61C22⋊C412yes—
128#11758196—24no—
129#11758196—24no—
130#11758196—24no—
131#11758196—24no—
132#11758196—24no—
133C4×Dic61—24no—
134C4×Dic61—22no—
135C4×Dic61—22no—
136C4×Dic61C2×D412yes—
137C4×Dic61C2×D412yes—
138C23×C61C2×C22⋊C411yescharacteristic
139C22×D61—21no—
140C22×D61—21no—
141C22×D61—21no—
142C22×D61—21no—
143C22×D61—21no—
144C22×D61—21no—
145C22×D61#10312yes—
146Q8×C61—42no—
147Q8×C61—42no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61#10312yes—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×Dic61—21no—
165C2×Dic61—22no—
166C2×Dic61—22no—
167C2×Dic61—21no—
168C2×Dic61—21no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61#12012yes—
174C2×Dic61#12012yes—
175C2×Dic61C2×SD1611yescharacteristic
176C2×Dic61C2×Q1611yescharacteristic
177C2×Dic61#13512yes—
178C2×Dic61AGL(1,ℤ/8)11yescharacteristic
179C2×Dic61#13712yes—
180C2×Dic61#13711yescharacteristic
181C4×D61—24no—
182C4×D61—24no—
183C4×D61—42no—
184C4×D61—42no—
185C4×D61—42no—
186C4×D61—42no—
187Dic122—42no—
188Dic122—44no—
189Dic122—44no—
190Dic122—42no—
191#92420—82no—
192#92420—82no—
193#8343—611nomaximal, Sylow
194C22×C61#37911yescharacteristic
195C22×C61#41311yescharacteristic
196C22×C61#41311yescharacteristic
197C22×C61C2×#10311yescharacteristic
198C22×C61#41611yescharacteristic
199C22×C61#41711yescharacteristic
200C22×C61#41911yescharacteristic
201C2×D61—22no—
202C2×D61—22no—
203C2×D61—22no—
204C2×D61—41no—
205C2×D61—41no—
206C2×D61—41no—
207C2×D61—41no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—22no—
211C4×C61—42no—
212C4×C61—42no—
213C4×C61—41no—
214Dic61—42no—
215Dic61—41no—
216Dic61—42no—
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#154211yescharacteristic
227C2×C61#155011yescharacteristic
228C2×C61#155411yescharacteristic
229C2×C61#155911yescharacteristic
230C2×C61#178511yescharacteristic
231C2×C61#178711yescharacteristic
232C2×C61#178811yescharacteristic
233D61—42no—
234#533—1224no—
235#533—1224no—
236C2×C4⋊Q8—611no—
237C2×C4⋊Q8—611no—
238C2×#125—1221no—
239C2×C22⋊Q8—611no—
240C2×C22⋊Q8—611no—
241C2×C22⋊Q8—1221no—
242C2×C4×Q8—611no—
243C2×C4×Q8—1221no—
244C2×C4×Q8#692311yescharacteristic
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×C4—611no—
262C22×C8—1222no—
263C22×C8—1222no—
264C4⋊Q8—614no—
265C4⋊Q8—614no—
266C4⋊Q8—1222no—
267C4⋊Q8—1222no—
268#125—1228no—
269C22⋊Q8—1222no—
270C22⋊Q8—1228no—
271C22⋊Q8—1222no—
272C22⋊Q8—1222no—
273C22⋊Q8—1222no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—1222no—
277C4×Q8—1222no—
278C4×Q8—1228no—
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—1221no—
290C2×C22⋊C4—1221no—
291C2×C42—21no—
292C2×C42—611no—
293C2×C42—611no—
294C2×C42—611no—
295C2×C42—1222no—
296C2×C42—1222no—
297C2×C42—1221no—
298C2×C42C2×#692311yescharacteristic
299C4⋊C8—1222no—
300C4⋊C8—1222no—
301C4⋊C8—1222no—
302C4⋊C8—1222no—
303C22⋊C8—2442no—
304C22⋊C8—2442no—
305C24—611no—
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—611no—
321C22×C4—611no—
322C22×C4—612no—
323C22×C4—612no—
324C22×C4—611no—
325C22×C4—1222no—
326C22×C4—1222no—
327C22×C4—1221no—
328C22×C4—1222no—
329C22×C4—1222no—
330C22×C4—1221no—
331C22×C4—1221no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—1221no—
335C22×C4—2441no—
336C22×C4—2441no—
337C22×C4C22⋊#692311yescharacteristic
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—1224no—
358C22⋊C4—1224no—
359C42—24no—
360C42—21no—
361C42—21no—
362C42—612no—
363C42—612no—
364C42—1222no—
365C42—1222no—
366C42—1224no—
367C23—612no—
368C23—1221no—
369C23—1221no—
370C23—1221no—
371C23—1221no—
372C23—1221no—
373C23—1221no—
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—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
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—1224no—
403C2×C4—1224no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—1221no—
407C2×C4—1222no—
408C2×C4—1222no—
409C2×C4—2442no—
410C2×C4—2442no—
411C2×C4—2442no—
412C2×C4—2442no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4#1451984012yes—
418C2×C4C2×C22⋊#692311yescharacteristic
419C2×C4C2×C22⋊#692311yescharacteristic
420C8—4882no—
421C8—4882no—
422C22—1222no—
423C22—1222no—
424C22—1222no—
425C22—2441no—
426C22—2441no—
427C22—2441no—
428C22—2441no—
429C22#1684070311yescharacteristic
430C22#1684074811yescharacteristic
431C22#1684074811yescharacteristic
432C22C2×#1451984011yescharacteristic
433C22#1684075211yescharacteristic
434C22#1684075311yescharacteristic
435C22#1684075511yescharacteristic
436C4—21no—
437C4—21no—
438C4—22no—
439C4—42no—
440C4—41no—
441C4—42no—
442C4—612no—
443C4—612no—
444C4—612no—
445C4—612no—
446C4—2441no—
447C4—2442no—
448C4—2442no—
449C2—2442no—
450C2#2146132011yescharacteristic
451C2#2146133611yescharacteristic
452C2#2146134411yescharacteristic
453C2#2146135411yescharacteristic
454C2#2146165511yescharacteristic
455C2#2146165811yescharacteristic
456C2#2146165911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8343
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×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
#214616762
C2×#168406222
C2×#168406212
C2×#168403162
#834361