Finite Group Databasebeta

← #30603541

Subgroups — #30603541

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462299C211yesmaximal, characteristic
2#21461507C212yesmaximal
3C2×#16840634C212yesmaximal
4#21460996C211yesmaximal, characteristic
5#343×D61C211yesmaximal, characteristic
6C2×#14519856C2212yes—
7C22×C4⋊#6923C2211yescharacteristic
8C22×C4×#6923C2211yescharacteristic
9#16840713—22no—
10#16840713—22no—
11#16840704—22no—
12C2×(C2×#6923)⋊C4—22no—
13#16840636—22no—
14#16840636—22no—
15#16840634—24no—
16C2×M16×D61C2212yes—
17C2×C4⋊C4×D61C2211yescharacteristic
18#16839531C411yescharacteristic, complemented
19#16839530C412yescomplemented
20C61×#343C411yescharacteristic, complemented
21C23×#6923—21no—
22C23×#6923—21no—
23#14519856—22no—
24#14519856—22no—
25#14519856—22no—
26#14519856—22no—
27C2×C4⋊#6923—22no—
28C2×C4⋊#6923—22no—
29C2×C4⋊#6923—21no—
30C2×C4⋊#6923—22no—
31C2×C4⋊#6923—22no—
32C2×C4⋊#6923D412yes—
33C2×C4⋊#6923D412yes—
34C2×C4⋊#6923Q812yes—
35C2×C22⋊#6923—22no—
36C2×C4×#6923—22no—
37C2×C4×#6923—22no—
38C2×C4×#6923—22no—
39C2×C4×#6923—21no—
40C2×C4×#6923D412yes—
41(C2×#6923)⋊C4—24no—
42(C2×#6923)⋊C4—44no—
43C22×C4×D61D412yes—
44C22×C4×D61C2311yescharacteristic
45C2×#11758194C2×C412yes—
46C2×#11758189—22no—
47M16×D61—22no—
48M16×D61—22no—
49M16×D61—22no—
50M16×D61—22no—
51C2×C8×D61—22no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×#11758196Q812yes—
54C4⋊C4×D61—22no—
55C4⋊C4×D61—22no—
56C2×M16×C61C2×C412yes—
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—22no—
60C22×#6923—22no—
61C22×#6923—22no—
62C22×#6923—22no—
63C22×#6923—22no—
64C22×#6923—22no—
65C22×#6923—41no—
66C22×#6923—41no—
67C22×#6923—41no—
68C22×#6923—41no—
69C22×#6923—42no—
70C22×#6923—42no—
71C22×#6923—41no—
72C22×#6923—41no—
73C22×C4×C61C4⋊C412yes—
74C22×C4×C61C22×C411yescharacteristic
75C22×Dic61C4⋊C412yes—
76C22×Dic61C2×Q811yescharacteristic
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—42no—
80C2×C4×D61—42no—
81C2×C4×D61C2×D411yescharacteristic
82C2×C4×D61C2×D412yes—
83C2×C4×D61C2×Q811yescharacteristic
84C2×C4×D61#4112yes—
85C2×C4×D61#4112yes—
86C2×C4×D61#4111yescharacteristic
87C2×C4×D61#4112yes—
88C2×C4×D61#4111yescharacteristic
89C22⋊#6923—42no—
90C22⋊#6923—42no—
91C4⋊#6923—22no—
92C4⋊#6923—22no—
93C4⋊#6923—24no—
94C4⋊#6923—22no—
95C4⋊#6923—22no—
96C4⋊#6923—42no—
97C4⋊#6923—82no—
98C4⋊#6923—82no—
99C4×#6923—24no—
100C4×#6923—42no—
101C4×#6923—42no—
102C4×#6923—42no—
103M16×C61—22no—
104M16×C61—22no—
105C2×C8×C61—22no—
106C4⋊C4×C61—22no—
107C4⋊Dic61—22no—
108#11758196—24no—
109#11758194—22no—
110#11758194—22no—
111C2×#92418—22no—
112#11758189—42no—
113#11758189—42no—
114C8×D61—42no—
115C8×D61—42no—
116C23×C61C2×C4⋊C411yescharacteristic
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—21no—
121C22×D61#12011yescharacteristic
122C22×D61#12111yescharacteristic
123C22×D61C22⋊Q811yescharacteristic
124C22×D61#12311yescharacteristic
125C22×D61#12411yescharacteristic
126C22×D61#12411yescharacteristic
127C2×#6923—24no—
128C2×#6923—42no—
129C2×#6923—42no—
130C2×#6923—42no—
131C2×#6923—42no—
132C2×#6923—42no—
133C2×#6923—42no—
134C2×#6923—44no—
135C2×#6923—42no—
136C2×#6923—42no—
137C2×#6923—42no—
138C2×#6923—42no—
139C2×#6923—42no—
140C2×#6923—42no—
141C2×#6923—42no—
142C2×#6923—82no—
143C2×#6923—81no—
144C2×#6923—81no—
145C2×C4×C61—22no—
146C2×C4×C61—42no—
147C2×C4×C61#11712yes—
148C2×C4×C61C4×D412yes—
149C2×C4×C61C4×D411yescharacteristic
150C2×C4×C61C4×Q811yescharacteristic
151C2×Dic61—22no—
152C2×Dic61—42no—
153C2×Dic61C22⋊Q812yes—
154C2×Dic61#12512yes—
155C2×Dic61#12511yescharacteristic
156C2×Dic61C4⋊Q811yescharacteristic
157C4×D61—42no—
158C4×D61—42no—
159C4×D61—82no—
160C4×D61#12112yes—
161C4×D61C22⋊Q812yes—
162C4×D61#12312yes—
163C4×D61#12312yes—
164C4×D61#12412yes—
165C4×D61#12512yes—
166C4×D61#12612yes—
167C4×D61#12612yes—
168C8×C61—42no—
169#92418—42no—
170#11296—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61#38211yescharacteristic
174C22×C61#38511yescharacteristic
175C22×C61#38911yescharacteristic
176C2×D61—21no—
177C2×D61—21no—
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—21no—
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—21no—
186C2×D61#39311yescharacteristic
187C2×D61#39411yescharacteristic
188C2×D61#39711yescharacteristic
189C2×D61#40011yescharacteristic
190C2×D61#45311yescharacteristic
191C2×D61#45411yescharacteristic
192C2×D61#45511yescharacteristic
193C2×D61#45611yescharacteristic
194#6923—44no—
195#6923—82no—
196#6923—82no—
197#6923—82no—
198C4×C61—42no—
199C4×C61#38712yes—
200C4×C61#38812yes—
201Dic61—42no—
202Dic61#39812yes—
203Dic61#40012yes—
204#2316—611no—
205C2×#343—611no—
206#1736—612no—
207C2×#339—612no—
208C4⋊#95—611no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61#144211yescharacteristic
212C2×C61#185611yescharacteristic
213C2×C61#185711yescharacteristic
214D61—21no—
215D61—21no—
216D61#202711yescharacteristic
217D61#202811yescharacteristic
218D61#203911yescharacteristic
219D61#204011yescharacteristic
220C22×M16—612no—
221C2×#117—612no—
222C22×C4⋊C4—611no—
223C22×C4⋊C4—611no—
224C22×C42—611no—
225#389—1222no—
226#389—1222no—
227#380—1222no—
228C2×#95—1222no—
229#343—612no—
230#343—611no—
231#343#692311yescharacteristic, complemented
232#340—1222no—
233#340—1222no—
234#339—1224no—
235C61#1129611yescharacteristic, complemented, Sylow
236C23×C4—611no—
237C23×C4—612no—
238C23×C4—1221no—
239C23×C4—1221no—
240C2×M16—612no—
241C2×M16—1222no—
242C2×M16—1222no—
243C2×M16—1222no—
244C2×M16—1222no—
245C2×M16—1222no—
246C2×M16C2×#692312yes—
247C22×C8—1222no—
248#117—1222no—
249#117—1222no—
250#117—1222no—
251#117—1222no—
252C2×C4⋊C4—612no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—612no—
255C2×C4⋊C4—612no—
256C2×C4⋊C4—612no—
257C2×C4⋊C4—1222no—
258C2×C4⋊C4—1222no—
259C2×C4⋊C4—1222no—
260C2×C4⋊C4—1222no—
261C2×C4⋊C4—1222no—
262C2×C4⋊C4—1222no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4C2×#692311yescharacteristic
265C2×C22⋊C4—1222no—
266C2×C42—612no—
267C2×C42—1222no—
268C2×C42—1222no—
269C2×C42—1222no—
270C2×C42—1221no—
271#95—1224no—
272#95—2444no—
273C24—611no—
274C22×C4—611no—
275C22×C4—612no—
276C22×C4—612no—
277C22×C4—611no—
278C22×C4—612no—
279C22×C4—612no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—612no—
284C22×C4—1222no—
285C22×C4—1222no—
286C22×C4—1222no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4—1222no—
290C22×C4—1222no—
291C22×C4—1222no—
292C22×C4—2442no—
293C22×C4—2442no—
294C22×C4—2441no—
295C22×C4—2441no—
296C22×C4—2442no—
297C22×C4—2442no—
298C22×C4—2441no—
299C22×C4—2441no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4C4⋊#692312yes—
303C22×C4C22×#692311yescharacteristic
304M16—22no—
305M16—22no—
306M16—1222no—
307M16—1222no—
308M16—2442no—
309M16—2442no—
310C2×C8—22no—
311C2×C8—1222no—
312C2×C8—2442no—
313C2×C8—2442no—
314C4⋊C4—22no—
315C4⋊C4—1224no—
316C4⋊C4—1224no—
317C4⋊C4—1222no—
318C4⋊C4—1222no—
319C4⋊C4—1222no—
320C4⋊C4—1222no—
321C4⋊C4—1222no—
322C4⋊C4—4882no—
323C4⋊C4—4882no—
324C4⋊C4—2442no—
325C22⋊C4—2442no—
326C22⋊C4—2442no—
327C42—1224no—
328C42—2442no—
329C42—2442no—
330C42—2442no—
331C23—611no—
332C23—611no—
333C23—611no—
334C23—611no—
335C23—611no—
336C23—611no—
337C23—1221no—
338C23—1221no—
339C23—1221no—
340C23—1221no—
341C23C2×C4⋊#692311yescharacteristic
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—612no—
345C2×C4—612no—
346C2×C4—611no—
347C2×C4—611no—
348C2×C4—612no—
349C2×C4—612no—
350C2×C4—612no—
351C2×C4—612no—
352C2×C4—612no—
353C2×C4—612no—
354C2×C4—612no—
355C2×C4—612no—
356C2×C4—1224no—
357C2×C4—1222no—
358C2×C4—4882no—
359C2×C4—4881no—
360C2×C4—4881no—
361C2×C4—4882no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4—2444no—
372C2×C4—2442no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4#1451985612yes—
380C2×C4D4×#692312yes—
381C2×C4D4×#692311yescharacteristic
382C2×C4Q8×#692311yescharacteristic
383C8—42no—
384C8—2442no—
385C22—21no—
386C22—21no—
387C22—611no—
388C22—611no—
389C22—611no—
390C22—611no—
391C22—611no—
392C22—611no—
393C22—611no—
394C22—611no—
395C22—1221no—
396C22—1221no—
397C22—1221no—
398C22—1221no—
399C22—1221no—
400C22—1221no—
401C22—1221no—
402C22—1221no—
403C22—1221no—
404C22—1221no—
405C22#1684070611yescharacteristic
406C22#1684070911yescharacteristic
407C22#1684071311yescharacteristic
408C4—42no—
409C4—612no—
410C4—612no—
411C4—4882no—
412C4—4882no—
413C4—4882no—
414C4—2444no—
415C4—2442no—
416C4#1684071112yes—
417C4#1684071212yes—
418C2—21no—
419C2—21no—
420C2—611no—
421C2—611no—
422C2—611no—
423C2—611no—
424C2—1221no—
425C2—1221no—
426C2#2146100511yescharacteristic
427C2#2146177411yescharacteristic
428C2#2146177511yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#343C4
socleC22×C61#385
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#343#6923
hypocenterC61#11296
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
18. #16839531C4semidirect
19. #16839530C4semidirect
20. C61×#343C4semidirect
231. #343#6923semidirect
235. C61#11296semidirect

Maximal subgroups

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

subgroupindex
#214622992
#214615072
C2×#168406342
#214609962
#343×D612
#1129661