Finite Group Databasebeta

← #355×#6923 #30606553

Subgroups — #355×#6923 #30606553

Order 15616.

532 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 96 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#124×#6923C211yesmaximal, characteristic
2D61⋊#355C211yesmaximal, characteristic
3#101×#6923C212yesmaximal, complemented
4#21460915C212yesmaximal, complemented
5#355×D61C211yesmaximal, characteristic
6C2×Q8×#6923C2211yescharacteristic
7C2×D4×#6923C411yescharacteristic
8D61⋊#101C2212yes—
9M16×#6923—22no—
10(C2×#6923)⋊Q8C2211yescharacteristic
11#16840714C411yescharacteristic
12#16840712—22no—
13#16840706C411yescharacteristic
14C22⋊C4×#6923—22no—
15C42×#6923C411yescharacteristic
16#16840671C412yescomplemented
17#16840670C412yescomplemented
18#16840641—22no—
19#124×D61C2211yescharacteristic
20#101×D61C2212yes—
21#16839559C411yescharacteristic, complemented
22#16839558C412yescomplemented
23C61×#355C411yescharacteristic, complemented
24C23×#6923—21no—
25D61⋊M16—22no—
26Q8×#6923—41no—
27Q8×#6923—41no—
28D4×#6923—41no—
29D4×#6923—41no—
30C2×C4⋊#6923C2×C411yescharacteristic
31C2×C4⋊#6923C2×C411yescharacteristic
32C2×C4⋊#6923D412yes—
33C2×C22⋊#6923—21no—
34C2×C22⋊#6923—22no—
35C2×C4×#6923—21no—
36C2×C4×#6923—21no—
37C2×C4×#6923C2×C411yescharacteristic
38C2×C4×#6923C2×C411yescharacteristic
39C2×C4×#6923D412yes—
40#14519838—22no—
41#14519838C2×C414yescomplemented
42#14519831—42no—
43C8×#6923—42no—
44(C2×#6923)⋊C4—22no—
45(C2×#6923)⋊C4—22no—
46C2×Q8×D61C2311yescharacteristic
47C2×D4×D61C2×C411yescharacteristic
48M16×D61—22no—
49#14519766C2×C411yescharacteristic
50#14519765C2×C411yescharacteristic
51#14519764C2×C411yescharacteristic
52#14519763—22no—
53C22⋊C4×D61—22no—
54C42×D61C2×C411yescharacteristic
55#14519696C2×C412yescomplemented
56#14519695—22no—
57#124×C61C2×C411yescharacteristic
58#101×C61C2×C412yescomplemented
59C23×D61—21no—
60C22×#6923—21no—
61C22×#6923—21no—
62C22×#6923—44no—
63C22×#6923—41no—
64C22×#6923—41no—
65C22×#6923C22⋊C411yescharacteristic
66C22×#6923C22⋊C412yes—
67C22×#6923C22⋊C411yescharacteristic
68C2×Q8×C61C22×C411yescharacteristic
69C2×D4×C61C4211yescharacteristic
70C2×C22⋊D61—21no—
71Q8×D61—41no—
72Q8×D61—41no—
73D4×D61—41no—
74D4×D61—41no—
75C2×D244C4211yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61C22×C411yescharacteristic
78C2×C4×D61C2×D412yes—
79C2×Dic122C22×C411yescharacteristic
80C2×Dic122#4112yes—
81C22⋊#6923—44no—
82C22⋊#6923—42no—
83#11758217—24no—
84C4⋊#6923—41no—
85C4⋊#6923—42no—
86C4⋊#6923—41no—
87C4×#6923—22no—
88C4×#6923—22no—
89C4×#6923—21no—
90C4×#6923—21no—
91C4×#6923—41no—
92C4×#6923—42no—
93C4×#6923—42no—
94#11758213—42no—
95D61⋊C8—42no—
96M16×C61—22no—
97C22⋊C4×C61—22no—
98C42×C61C4211yescharacteristic
99C22⋊Dic61—22no—
100#11758198—22no—
101#11758198—22no—
102C4×Dic61—21no—
103C4×Dic61C4211yescharacteristic
104#11758194—22no—
105#11758189—42no—
106C8×D61—42no—
107C23×C61—21no—
108C22×D61—21no—
109C22×D61—42no—
110C22×D61—42no—
111C22×D61—41no—
112C22×D61—41no—
113C22×D61C2×C22⋊C411yescharacteristic
114C2×#6923—22no—
115C2×#6923—22no—
116C2×#6923—21no—
117C2×#6923—22no—
118C2×#6923—41no—
119C2×#6923—41no—
120C2×#6923—41no—
121C2×#6923—42no—
122C2×#6923—42no—
123C2×#6923—41no—
124C2×#6923—82no—
125C2×#6923—82no—
126C2×#6923#9911yescharacteristic
127C2×#6923#9911yescharacteristic
128Q8×C61—41no—
129D4×C61—41no—
130C2×C4×C61—21no—
131C2×C4×C61C2×C4211yescharacteristic
132C2×C4×C61C4×D412yes—
133C22⋊D61—42no—
134C2×Dic61—21no—
135C2×Dic61#11711yescharacteristic
136C2×Dic61C4×D412yes—
137D244—41no—
138C4×D61—22no—
139C4×D61—21no—
140C4×D61—22no—
141C4×D61—21no—
142C4×D61—41no—
143C4×D61—41no—
144Dic122—41no—
145Dic122—42no—
146#92420—44no—
147C8×C61—42no—
148#92418—42no—
149C4×#355—611nomaximal, Sylow
150C22×C61—42no—
151C22×C61—41no—
152C22×C61C4×C22⋊C411yescharacteristic
153C2×D61—21no—
154C2×D61—21no—
155C2×D61—42no—
156C2×D61—41no—
157C2×D61—82no—
158C2×D61—81no—
159C2×D61—81no—
160C2×D61C2×#9911yescharacteristic
161C2×D61#41011yescharacteristic
162#6923—21no—
163#6923—42no—
164#6923—41no—
165#6923—42no—
166#6923—81no—
167#6923#35512yesdirect product factor
168C4×C61—22no—
169C4×C61—21no—
170C4×C61—41no—
171Dic61—22no—
172Dic61—21no—
173Dic61—41no—
174C4×#124—611no—
175C2×#355—611no—
176C2×#355—611no—
177C4×#101—612no—
178#1394—612no—
179C2×C61—21no—
180C2×C61—81no—
181C2×C61C4×#9911yescharacteristic
182D61—21no—
183D61—81no—
184D61C2×#35511yescharacteristic
185D61#213211yescharacteristic
186C2×#124—611no—
187C2×C4×Q8—611no—
188C2×C4×D4—611no—
189C2×#101—612no—
190C2×#101—612no—
191C4×M16—1222no—
192#391—611no—
193#390—611no—
194#388—1222no—
195#382—611no—
196C4×C22⋊C4—1222no—
197C43—611no—
198#355—612no—
199#355—612no—
200#355—611no—
201#355—612no—
202#355#692311yescharacteristic, direct product factor
203#343—1222no—
204C61C4×#35511yescharacteristic, complemented, Sylow
205C22×Q8—611no—
206C22×D4—611no—
207C23×C4—1221no—
208C2×M16—1222no—
209C2×M16—1222no—
210#124—611no—
211#124—611no—
212#124—611no—
213#124—1222no—
214#124C2×#692311yescharacteristic
215C4×Q8—2441no—
216C4×Q8—2441no—
217C4×D4—2441no—
218C4×D4—2441no—
219C2×C4⋊C4—611no—
220C2×C4⋊C4—612no—
221C2×C4⋊C4—611no—
222C2×C22⋊C4—1222no—
223C2×C22⋊C4—1221no—
224C2×C22⋊C4—1222no—
225C2×C42—611no—
226C2×C42—611no—
227C2×C42—612no—
228C2×C42—611no—
229C2×C42—1221no—
230C2×C42—1221no—
231#101—614no—
232#101—612no—
233#101—1222no—
234#101—1222no—
235#101C2×#692312yescomplemented
236#97—2442no—
237C4×C8—2442no—
238#95—1222no—
239#95—1222no—
240C24—1221no—
241C2×Q8—611no—
242C2×Q8—612no—
243C2×Q8—2441no—
244C2×Q8—2441no—
245C2×Q8C22×#692311yescharacteristic
246C2×D4—611no—
247C2×D4—1221no—
248C2×D4—2441no—
249C2×D4—2441no—
250C2×D4C4×#692311yescharacteristic
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—611no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—1221no—
259C22×C4—2441no—
260C22×C4—2441no—
261C22×C4—2444no—
262M16—22no—
263M16—1224no—
264M16—1222no—
265M16—2442no—
266M16—2442no—
267C2×C8—2442no—
268C2×C8—2442no—
269C4⋊C4—2441no—
270C4⋊C4—2442no—
271C4⋊C4—2441no—
272C22⋊C4—22no—
273C22⋊C4—1222no—
274C22⋊C4—1222no—
275C22⋊C4—1222no—
276C22⋊C4—2442no—
277C22⋊C4—2444no—
278C42—611no—
279C42—1221no—
280C42—1222no—
281C42—1222no—
282C42—1221no—
283C42—1221no—
284C42—2441no—
285C42—2442no—
286C42—2442no—
287C42C4×#692311yescharacteristic
288C23—21no—
289C23—611no—
290C23—1221no—
291C23—2441no—
292C23—2441no—
293C23—2442no—
294C23—2442no—
295Q8—41no—
296Q8—2442no—
297Q8—2441no—
298D4—41no—
299D4—2441no—
300D4—2442no—
301C2×C4—21no—
302C2×C4—611no—
303C2×C4—611no—
304C2×C4—611no—
305C2×C4—612no—
306C2×C4—1221no—
307C2×C4—1222no—
308C2×C4—1221no—
309C2×C4—1222no—
310C2×C4—1222no—
311C2×C4—1221no—
312C2×C4—1222no—
313C2×C4—1222no—
314C2×C4—1221no—
315C2×C4—4882no—
316C2×C4—4882no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2442no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2442no—
325C2×C4C2×C4×#692311yescharacteristic
326C2×C4D4×#692312yes—
327C8—42no—
328C8—2442no—
329C8—2444no—
330C22—42no—
331C22—41no—
332C22—611no—
333C22—611no—
334C22—1221no—
335C22—1221no—
336C22—4881no—
337C22—4881no—
338C22—4882no—
339C22—2442no—
340C22—2441no—
341C22C22⋊C4×#692311yescharacteristic
342C4—22no—
343C4—21no—
344C4—41no—
345C4—612no—
346C4—1221no—
347C4—1222no—
348C4—1221no—
349C4—4881no—
350C4—2441no—
351C4—2441no—
352C4—2442no—
353C4—2442no—
354C2—21no—
355C2—81no—
356C2—611no—
357C2—611no—
358C2—1221no—
359C2—4881no—
360C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×Q8C22×#6923
FittingC61×#355C4
socleC2×C61C4×#99
prime radicalC2×D4×D61C2×C4
cosocleC2×Q8×D61C23
hypercenter#355#6923
hypocenterC61C4×#355
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
3. #101×#6923C2semidirect
4. #21460915C2semidirect
16. #16840671C4semidirect
17. #16840670C4semidirect
21. #16839559C4semidirect
22. #16839558C4semidirect
23. C61×#355C4semidirect
41. #14519838C2×C4semidirect
55. #14519696C2×C4semidirect
58. #101×C61C2×C4semidirect
167. #6923#355direct (N × H)
202. #355#6923direct (N × H)
204. C61C4×#355semidirect
235. #101C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#124×#69232
D61⋊#3552
#101×#69232
#214609152
#355×D612
C4×#35561