Finite Group Databasebeta

← #29595520

Subgroups — #29595520

Order 15616.

1018 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×C4⋊Q8×D61C211yesmaximal, characteristic
2C2×#16839561C211yesmaximal, characteristic, complemented
3C2×#16839560C212yesmaximal, complemented
4#21445741C212yesmaximal
5C2×C61×#356C211yesmaximal, characteristic, complemented
6C22×Q8×D61C411yescharacteristic
7C22×Q8×D61C2211yescharacteristic
8C2×#14519777C411yescharacteristic
9C2×#14519776—22no—
10C2×#14519775C411yescharacteristic
11C2×#14519775C2211yescharacteristic
12C4⋊Q8×D61—24no—
13C4⋊Q8×D61—22no—
14C4⋊Q8×D61—22no—
15C2×C4⋊C4×D61—22no—
16C2×C42×D61C411yescharacteristic
17C2×#14519696C2212yes—
18C2×#14519695—22no—
19#16839725—22no—
20#16839561—22no—
21#16839560—24no—
22C2×C4⋊Q8×C61C2211yescharacteristic
23C2×#101×C61C2212yes—
24C61×#356—22no—
25C22×Dic122—21no—
26C22×Dic122C2×C411yescharacteristic
27C22×Dic122C2×C411yescharacteristic
28C22×Dic122D412yes—
29C2×Q8×D61—22no—
30C2×Q8×D61—21no—
31C2×Q8×D61—22no—
32C2×Q8×D61—21no—
33C2×Q8×D61—22no—
34C2×Q8×D61—42no—
35C2×Q8×D61—42no—
36C2×Q8×D61—42no—
37C2×Q8×D61—42no—
38C22×C4×D61—21no—
39C22×C4×D61—21no—
40C22×C4×D61C2×C411yescharacteristic
41C22×C4×D61D412yes—
42C2×#11758194—22no—
43#14519777—22no—
44#14519777—22no—
45#14519776—28no—
46#14519776—28no—
47#14519775—22no—
48#14519775—21no—
49#14519775—21no—
50#14519775—22no—
51#14519775D414yes—
52C2×C4⋊Dic61—22no—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C4⋊C4×D61—28no—
56C4⋊C4×D61—28no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61C2×C411yescharacteristic
59C42×D61—22no—
60C42×D61—22no—
61#14519696—22no—
62#14519696—22no—
63#14519695—44no—
64#14519689—44no—
65C22×Q8×C61C2×C411yescharacteristic
66C22×Q8×C61C2311yescharacteristic
67C2×M16×C61—22no—
68C4⋊Q8×C61—22no—
69C4⋊Q8×C61D412yes—
70C4⋊Q8×C61D412yes—
71C2×C4⋊C4×C61—22no—
72C2×C42×C61C2×C411yescharacteristic
73#101×C61—22no—
74#101×C61—22no—
75C23×D61C22⋊C411yescharacteristic
76C2×Q8×C61—21no—
77C2×Q8×C61—41no—
78C2×Q8×C61—41no—
79C2×Q8×C61—41no—
80C2×Q8×C61—41no—
81C2×Q8×C61C22⋊C412yes—
82C2×Q8×C61C2×D411yescharacteristic
83C2×Q8×C61C2×D412yes—
84C2×Q8×C61C2×D411yescharacteristic
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D412yes—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61C22⋊C412yes—
92C22×Dic61C22⋊C411yescharacteristic
93Q8×D61—44no—
94Q8×D61—48no—
95Q8×D61—44no—
96Q8×D61—48no—
97Q8×D61—48no—
98C2×C4×D61—28no—
99C2×C4×D61—24no—
100C2×C4×D61—22no—
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×C4×D61—22no—
104C2×C4×D61—24no—
105C2×C4×D61—24no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—42no—
109C2×C4×D61—42no—
110C2×C4×D61—42no—
111C2×Dic122—24no—
112C2×Dic122—21no—
113C2×Dic122—21no—
114C2×Dic122—21no—
115C2×Dic122—22no—
116C2×Dic122—22no—
117C2×Dic122—24no—
118C2×Dic122—41no—
119C2×Dic122—41no—
120C2×Dic122—42no—
121C2×Dic122—42no—
122C2×Dic122—42no—
123C2×Dic122—41no—
124C2×Dic122—41no—
125C2×Dic122C22⋊C412yes—
126M16×C61—42no—
127M16×C61—42no—
128C2×C8×C61—42no—
129C4⋊C4×C61—24no—
130C4⋊C4×C61—24no—
131C42×C61—21no—
132C42×C61C22⋊C412yes—
133C4⋊Dic61—28no—
134#11758196—28no—
135#11758196—24no—
136#11758196—24no—
137C4×Dic61—24no—
138C4×Dic61—21no—
139C4×Dic61C22⋊C412yes—
140#11758194—42no—
141#11758194—42no—
142C2×#92418—42no—
143C23×C61C2×C22⋊C411yescharacteristic
144C22×D61—21no—
145C22×D61—21no—
146C22×D61—24no—
147C22×D61#9912yes—
148Q8×C61—41no—
149Q8×C61—42no—
150Q8×C61—41no—
151Q8×C61—42no—
152Q8×C61—42no—
153C2×C4×C61—24no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—22no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61#12012yes—
165C2×C4×C61#12012yes—
166C2×Dic61—24no—
167C2×Dic61—21no—
168C2×Dic61—22no—
169C2×Dic61—21no—
170C2×Dic61—22no—
171C2×Dic61—22no—
172C2×Dic61—22no—
173C2×Dic61—22no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61#9912yes—
177C4×D61—216no—
178C4×D61—24no—
179C4×D61—24no—
180C4×D61—44no—
181C4×D61—44no—
182C4×D61—48no—
183Dic122—44no—
184Dic122—44no—
185Dic122—48no—
186Dic122—42no—
187Dic122—42no—
188Dic122—42no—
189Dic122—42no—
190C8×C61—82no—
191#92418—82no—
192#12700—611nomaximal, Sylow
193C22×C61—22no—
194C22×C61#37911yescharacteristic
195C22×C61C2×#9911yescharacteristic
196C22×C61C2×#9911yescharacteristic
197C2×D61—24no—
198C2×D61—24no—
199C2×D61—22no—
200C2×D61—22no—
201C2×D61#35614yescomplemented
202C4×C61—24no—
203C4×C61—22no—
204C4×C61—22no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—42no—
208Dic61—24no—
209Dic61—22no—
210Dic61—42no—
211Dic61—42no—
212C22×C4⋊Q8—611no—
213C2×#356—612no—
214C2×#356—611no—
215C2×#356D6111yescharacteristic, complemented
216#1784—612no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61#178111yescharacteristic
220C2×C61C2×#35612yes—
221D61—24no—
222C23×Q8—611no—
223C23×Q8—611no—
224C2×C4⋊Q8—611no—
225C2×C4⋊Q8—611no—
226C2×C4⋊Q8—611no—
227C2×C4⋊Q8—1222no—
228C2×C4⋊Q8—1224no—
229C2×C4⋊Q8—1222no—
230C2×C4⋊Q8—1222no—
231C2×C4⋊Q8C2×D6111yescharacteristic
232C22×C4⋊C4—1222no—
233C22×C42—611no—
234C2×#101—612no—
235C2×#101—1222no—
236C2×#101C2×D6112yes—
237#407—1222no—
238#356—22no—
239#356—1224no—
240#356—1222no—
241C61#1270011yescharacteristic, complemented, Sylow
242C22×Q8—611no—
243C22×Q8—612no—
244C22×Q8—611no—
245C22×Q8—1221no—
246C22×Q8—1222no—
247C22×Q8—1222no—
248C22×Q8—1221no—
249C22×Q8—1222no—
250C22×Q8—1221no—
251C22×Q8—2442no—
252C22×Q8—2442no—
253C22×Q8—2442no—
254C22×Q8—2442no—
255C22×Q8C4×D6111yescharacteristic
256C22×Q8C22×D6111yescharacteristic
257C23×C4—611no—
258C23×C4—612no—
259C23×C4—1221no—
260C23×C4—1221no—
261C2×M16—22no—
262C2×M16—1222no—
263C4⋊Q8—22no—
264C4⋊Q8—614no—
265C4⋊Q8—1222no—
266C4⋊Q8—1222no—
267C4⋊Q8—1228no—
268C4⋊Q8—1228no—
269C4⋊Q8—1222no—
270C4⋊Q8—1221no—
271C4⋊Q8—1221no—
272C4⋊Q8—1222no—
273C4⋊Q8D24412yes—
274C4⋊Q8C22⋊D6112yes—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—1228no—
277C2×C4⋊C4—1228no—
278C2×C4⋊C4—1222no—
279C2×C4⋊C4—1222no—
280C2×C4⋊C4—1222no—
281C2×C42—611no—
282C2×C42—1222no—
283C2×C42—1221no—
284C2×C42—1222no—
285C2×C42C4×D6111yescharacteristic
286#101—22no—
287#101—22no—
288#101—1222no—
289#101—1222no—
290#101—2444no—
291C22⋊C8—2444no—
292C24—611no—
293C2×Q8—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—612no—
299C2×Q8—1224no—
300C2×Q8—1224no—
301C2×Q8—1221no—
302C2×Q8—1221no—
303C2×Q8—1221no—
304C2×Q8—1222no—
305C2×Q8—1222no—
306C2×Q8—2441no—
307C2×Q8—2442no—
308C2×Q8—2441no—
309C2×Q8—2441no—
310C2×Q8—2441no—
311C2×Q8—2442no—
312C2×Q8—2442no—
313C2×Q8—2444no—
314C2×Q8—2448no—
315C2×Q8—2444no—
316C2×Q8—2448no—
317C2×Q8—2448no—
318C2×Q8#1175819812yes—
319C2×Q8C2×D24411yescharacteristic
320C2×Q8D4×D6112yes—
321C2×Q8C2×C22⋊D6111yescharacteristic
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—611no—
325C22×C4—612no—
326C22×C4—1222no—
327C22×C4—1222no—
328C22×C4—1224no—
329C22×C4—1228no—
330C22×C4—1224no—
331C22×C4—1224no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—1221no—
335C22×C4—1221no—
336C22×C4—1222no—
337C22×C4—1222no—
338C22×C4—2442no—
339C22×C4—2442no—
340C22×C4—2442no—
341C22×C4C2×C4×D6111yescharacteristic
342C22×C4D4×D6112yes—
343M16—42no—
344M16—42no—
345M16—2442no—
346M16—2442no—
347C2×C8—42no—
348C2×C8—2442no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—1228no—
352C4⋊C4—1228no—
353C4⋊C4—1224no—
354C4⋊C4—1224no—
355C42—21no—
356C42—612no—
357C42—1224no—
358C42—1221no—
359C42#1175819812yes—
360C23—612no—
361C23—1224no—
362C23—1221no—
363C23—1221no—
364C23C22⋊C4×D6111yescharacteristic
365Q8—41no—
366Q8—42no—
367Q8—41no—
368Q8—42no—
369Q8—42no—
370Q8—2444no—
371Q8—2444no—
372Q8—2442no—
373Q8—2442no—
374Q8—2442no—
375Q8—2442no—
376Q8—2448no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—612no—
387C2×C4—12216no—
388C2×C4—1224no—
389C2×C4—1224no—
390C2×C4—1224no—
391C2×C4—1221no—
392C2×C4—1222no—
393C2×C4—1221no—
394C2×C4—1222no—
395C2×C4—1222no—
396C2×C4—1222no—
397C2×C4—1222no—
398C2×C4—2444no—
399C2×C4—2444no—
400C2×C4—2448no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4C22⋊C4×D6111yescharacteristic
404C2×C4C2×#1175819811yescharacteristic
405C2×C4#1451974512yes—
406C2×C4#1451974712yes—
407C8—82no—
408C8—4882no—
409C22—22no—
410C22—614no—
411C22—1224no—
412C22—1224no—
413C22—1222no—
414C22—1222no—
415C22#1683961711yescharacteristic
416C22#99×D6111yescharacteristic
417C22#99×D6111yescharacteristic
418C4—24no—
419C4—22no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—1224no—
425C4—1222no—
426C4—2442no—
427C4—2442no—
428C2—21no—
429C2—21no—
430C2—1224no—
431C2#2144573211yescharacteristic
432C2#356×D6112yes—
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC2×C61×#356C2
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC22×Q8×C61C23
hypercenterC2×#356D61
hypocenterC61#12700
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
2. C2×#16839561C2semidirect
3. C2×#16839560C2semidirect
5. C2×C61×#356C2semidirect
201. C2×D61#356semidirect
215. C2×#356D61semidirect
241. C61#12700semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168395612
C2×#168395602
#214457412
C2×C61×#3562
#1270061