Finite Group Databasebeta

← #30620427

Subgroups — #30620427

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758217C212yesmaximal, complemented
2#21462013C214yesmaximal, complemented
3#21462012C212yesmaximal, complemented
4#21461684C212yesmaximal
5#21461683C211yesmaximal, characteristic
6#21461569C212yesmaximal
7C4×#14519837C211yesmaximal, characteristic
8C2×#16840470C211yesmaximal, characteristic
9#16840813—24no—
10#16840813—22no—
11Q8×#92420—22no—
12#16840765C2212yes—
13#16840765C2214yes—
14C2×#14519837C2211yescharacteristic
15C2×#14519837C2212yes—
16C2×#14519837C2211yescharacteristic
17C4×#11758217C2212yes—
18C4×#11758217C2214yes—
19#16840642C2212yes—
20#16840642C2214yes—
21#16840642C2212yes—
22#16840642C2214yes—
23#16840470—22no—
24#16840470—22no—
25#16840470—24no—
26C2×#14519776C2211yescharacteristic
27C2×#14519767C2212yes—
28C2×#14519757C412yes—
29C2×#14519754C412yes—
30C2×#14519753C411yescharacteristic
31C2×Q8×Dic61C2211yescharacteristic
32C2×C4×C22⋊D61C2211yescharacteristic
33C2×#14519737C2212yes—
34C22×C4⋊Dic61C411yescharacteristic
35C2×C4⋊C4×D61C411yescharacteristic
36C2×C22⋊Q8×C61C411yescharacteristic
37C2×#11758217C2312yes—
38C2×#11758217C2314yes—
39C2×#11758217C2312yes—
40#14519844—24no—
41#14519844—22no—
42C4⋊#92420—24no—
43C4⋊#92420—22no—
44#14519837—22no—
45#14519837—24no—
46#14519837Q814yescomplemented
47#14519832—24no—
48#14519832—22no—
49C4×#92420—24no—
50C4×#92420—22no—
51C22×Dic122C2×C412yes—
52C22×C22⋊D61C2311yescharacteristic
53C23×Dic61C2×C412yes—
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C412yes—
56#14519776—22no—
57#14519776—22no—
58#14519767—24no—
59#14519767—24no—
60#14519757—24no—
61#14519757—24no—
62#14519754—24no—
63#14519754—24no—
64#14519753—22no—
65#14519753—22no—
66Q8×Dic61D412yes—
67Q8×Dic61D414yes—
68Q8×Dic61D412yes—
69C4×C22⋊D61—22no—
70C4×C22⋊D61—22no—
71#14519737—24no—
72#14519737—24no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×C4⋊Dic61C2×C412yes—
78C2×C4⋊Dic61C2×C411yescharacteristic
79C2×C4⋊Dic61C2×C412yes—
80C2×#11758196C2×C412yes—
81C2×#11758196C2×C412yes—
82C2×#11758196C2312yes—
83C2×#11758196C2311yescharacteristic
84C4⋊C4×D61—22no—
85C4⋊C4×D61—21no—
86C4⋊C4×D61—21no—
87C2×C22⋊Dic61C2×C411yescharacteristic
88C2×#11758198C2×C412yes—
89C2×#11758198C2×C411yescharacteristic
90C2×C4×Dic61C2312yes—
91C2×C4×Dic61C2311yescharacteristic
92C22×Q8×C61C2×C411yescharacteristic
93C23×C4×C61C2×C411yescharacteristic
94C22⋊Q8×C61—22no—
95C22⋊Q8×C61—22no—
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C4⋊C4×C61C2×C412yes—
98C2×C22⋊C4×C61C2×C412yes—
99C24×C61C22×C411yescharacteristic
100C23×D61C22×C411yescharacteristic
101C2×Q8×C61—42no—
102C2×Q8×C61C22⋊C412yes—
103C2×Q8×C61C22⋊C412yes—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—42no—
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C22×C412yes—
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C22×C412yes—
111C2×C22⋊D61—22no—
112C2×C22⋊D61—22no—
113C2×C22⋊D61C2×Q812yes—
114C2×C22⋊D61#4112yes—
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—82no—
118C22×Dic61C22×C411yescharacteristic
119C22×Dic61C22×C412yes—
120C22×Dic61C22×C412yes—
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C2411yescharacteristic
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—42no—
128C2×C4×D61—82no—
129C2×Dic122—24no—
130C2×Dic122—82no—
131#11758217—22no—
132#11758217—22no—
133#11758217—24no—
134#11758217—22no—
135#11758217—24no—
136#11758217C2×Q814yes—
137C2×#92420—22no—
138C2×#92420—24no—
139C2×#92420—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—24no—
143C22⋊C4×C61—24no—
144C22⋊Dic61—22no—
145#11758198—24no—
146#11758198—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61C22⋊C412yes—
152C4⋊Dic61C22⋊C411yescharacteristic
153C4⋊Dic61C22⋊C411yescharacteristic
154C4⋊Dic61C22⋊C414yes—
155C4⋊Dic61C22⋊C414yes—
156#11758196—24no—
157#11758196—22no—
158#11758196—22no—
159#11758196—24no—
160#11758196—22no—
161C4×Dic61C2×D414yes—
162C4×Dic61C2×D412yes—
163C4×Dic61C2×D414yes—
164C4×Dic61C2×D412yes—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61C4×Q812yes—
170C23×C61C23×C411yescharacteristic
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61C4×Q812yes—
176Q8×C61—24no—
177Q8×C61—42no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—21no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61C2×C22⋊C414yes—
187C2×C4×C61C2×C22⋊C411yescharacteristic
188C2×C4×C61C2×C22⋊C411yescharacteristic
189C22⋊D61—24no—
190C22⋊D61—24no—
191C2×Dic61—21no—
192C2×Dic61—22no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—82no—
198C2×Dic61—82no—
199C2×Dic61C2×C22⋊C412yes—
200C2×Dic61C2×C22⋊C412yes—
201C2×Dic61C2×C22⋊C412yes—
202C2×Dic61C22×D412yes—
203C2×Dic61C22×Q811yescharacteristic
204C2×Dic61C2×#4111yescharacteristic
205C2×Dic612−1+412yes—
206C4×D61—42no—
207C4×D61—42no—
208C4×D61—82no—
209C4×D61—82no—
210Dic122—84no—
211#92420—24no—
212#92420—44no—
213#92420—42no—
214#22515—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—22no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61C22×C22⋊C411yescharacteristic
223C22×C61C2×C4×Q811yescharacteristic
224C22×C61#51911yescharacteristic
225C2×D61—22no—
226C2×D61—42no—
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—42no—
230C4×C61—21no—
231C4×C61—24no—
232C4×C61—41no—
233C4×C61—42no—
234C4×C61#41312yes—
235Dic61—22no—
236Dic61—22no—
237Dic61—22no—
238Dic61—41no—
239Dic61—42no—
240Dic61#54812yes—
241Dic61D4×Q812yes—
242C2×#554—611no—
243Q8×M16—612no—
244#1998—614no—
245#1997—612no—
246#1800—611no—
247#1800—612no—
248C4×#100—611no—
249C4×#100—612no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—42no—
253C2×C61Q8×C22⋊C411yescharacteristic
254C2×C61C2×#41311yescharacteristic
255C2×C61#288811yescharacteristic
256D61—42no—
257#554—1222no—
258#554—1224no—
259#554—1222no—
260C2×C4⋊Q8—611no—
261C2×#125—612no—
262C2×C22⋊Q8—611no—
263C2×C22⋊Q8—612no—
264C2×C22⋊Q8—612no—
265C2×C22⋊Q8#692311yescharacteristic
266C2×C4×Q8—611no—
267C2×C4×D4—611no—
268C2×C4×D4—612no—
269C22×C4⋊C4—611no—
270C22×C4⋊C4—611no—
271#446—1224no—
272#446—1222no—
273C8×Q8—1222no—
274#423—612no—
275#423—614no—
276C2×#100—611no—
277C2×#100—612no—
278C2×#100—611no—
279C4×M16—612no—
280C4×M16—614no—
281#343—614no—
282#343—612no—
283#343—614no—
284#343—612no—
285C61#2251511yescharacteristic, complemented, Sylow
286C22×Q8—612no—
287C22×Q8C2×#692311yescharacteristic
288C22×D4—611no—
289C23×C4—612no—
290C23×C4—612no—
291C23×C4—611no—
292C23×C4C2×#692311yescharacteristic
293C2×M16—612no—
294C2×M16—612no—
295C2×M16—614no—
296C4⋊Q8—1222no—
297C4⋊Q8—1222no—
298#125—1224no—
299#125—1224no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—1224no—
303C22⋊Q8—1224no—
304C22⋊Q8—1224no—
305C22⋊Q8—1224no—
306C22⋊Q8—1222no—
307C22⋊Q8—1222no—
308C4×Q8—612no—
309C4×Q8—614no—
310C4×Q8—612no—
311C4×D4—1224no—
312C4×D4—1224no—
313C4×D4—1222no—
314C4×D4—1222no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—612no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—612no—
323C2×C4⋊C4—1222no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4C2×#692311yescharacteristic
330C2×C4⋊C4C2×#692312yes—
331C2×C22⋊C4—611no—
332C2×C22⋊C4—611no—
333C2×C22⋊C4—612no—
334C2×C22⋊C4C2×#692312yes—
335C2×C42—612no—
336C2×C42—611no—
337C4⋊C8—1224no—
338C4⋊C8—1222no—
339C4⋊C8—1224no—
340C4⋊C8—1222no—
341#100—614no—
342#100—1222no—
343#100—1224no—
344#97—1224no—
345#97—1222no—
346C4×C8—1224no—
347C4×C8—1222no—
348C24—611no—
349C24C22×#692311yescharacteristic
350C2×Q8—42no—
351C2×Q8—1224no—
352C2×Q8—4882no—
353C2×Q8C22⋊#692312yes—
354C2×Q8C22⋊#692312yes—
355C2×D4—612no—
356C2×D4—612no—
357C2×D4—1222no—
358C2×D4—1222no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—42no—
362C22×C4—611no—
363C22×C4—611no—
364C22×C4—612no—
365C22×C4—611no—
366C22×C4—612no—
367C22×C4—1222no—
368C22×C4—1222no—
369C22×C4—1222no—
370C22×C4—1222no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—4882no—
374C22×C4—4882no—
375C22×C4—2442no—
376C22×C4C22×#692311yescharacteristic
377C22×C4C22×#692311yescharacteristic
378C22×C4C22×#692312yes—
379C22×C4C22×#692312yes—
380M16—614no—
381M16—1222no—
382M16—1222no—
383M16—1224no—
384M16—1222no—
385M16—1224no—
386C2×C8—1222no—
387C2×C8—1224no—
388C2×C8—1222no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—24no—
392C4⋊C4—612no—
393C4⋊C4—611no—
394C4⋊C4—611no—
395C4⋊C4—614no—
396C4⋊C4—614no—
397C4⋊C4—1222no—
398C4⋊C4—1222no—
399C4⋊C4—1222no—
400C4⋊C4—1222no—
401C4⋊C4—1222no—
402C4⋊C4—1224no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1224no—
406C22⋊C4—24no—
407C22⋊C4—1222no—
408C22⋊C4—1222no—
409C22⋊C4—1224no—
410C42—614no—
411C42—612no—
412C42—614no—
413C42—612no—
414C23—21no—
415C23—21no—
416C23—41no—
417C23—41no—
418C23—612no—
419C23—1221no—
420C23—1221no—
421C23—2441no—
422C23—2441no—
423C23Q8×#692312yes—
424C23C23×#692311yescharacteristic
425Q8—24no—
426Q8—42no—
427Q8—4884no—
428D4—1224no—
429D4—1224no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—612no—
439C2×C4—611no—
440C2×C4—611no—
441C2×C4—612no—
442C2×C4—612no—
443C2×C4—612no—
444C2×C4—612no—
445C2×C4—1221no—
446C2×C4—1222no—
447C2×C4—4882no—
448C2×C4—4882no—
449C2×C4—4882no—
450C2×C4—4882no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2441no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2441no—
457C2×C4C2×C22⋊#692314yes—
458C2×C4C2×C22⋊#692311yescharacteristic
459C2×C4C2×C22⋊#692311yescharacteristic
460C8—1224no—
461C8—2444no—
462C8—2442no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—42no—
467C22—42no—
468C22—41no—
469C22—41no—
470C22—1222no—
471C22—2442no—
472C22—2442no—
473C22—2441no—
474C22—2441no—
475C22C22×C22⋊#692311yescharacteristic
476C22C2×Q8×#692311yescharacteristic
477C22#1684083011yescharacteristic
478C4—21no—
479C4—24no—
480C4—41no—
481C4—42no—
482C4—612no—
483C4—612no—
484C4—1222no—
485C4—1222no—
486C4—1222no—
487C4—2441no—
488C4—2442no—
489C4#1684074912yes—
490C2—21no—
491C2—21no—
492C2—42no—
493C2—2442no—
494C2Q8×C22⋊#692311yescharacteristic
495C2C2×#1684074911yescharacteristic
496C2#2146243011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C23
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#22515
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#11758217C2semidirect
2. #21462013C2semidirect
3. #21462012C2semidirect
46. #14519837Q8semidirect
285. C61#22515semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582172
#214620132
#214620122
#214616842
#214616832
#214615692
C4×#145198372
C2×#168404702
#2251561