Finite Group Databasebeta

← #29658211

Subgroups — #29658211

Order 15616.

870 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457156C212yesmaximal, complemented
2#21456199C211yesmaximal, characteristic, complemented
3#21445047C212yesmaximal, complemented
4#21445040C212yesmaximal, complemented
5#21445033C212yesmaximal, complemented
6#21445026C212yesmaximal, complemented
7C2×#16839511C211yesmaximal, characteristic, complemented
8#21444232C212yesmaximal, complemented
9C61×#3544C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840130—22no—
12#16840112C2212yes—
13#16840112C2212yes—
14#16840066—22no—
15#16840047C2212yes—
16#16840042C2212yes—
17#16840001—22no—
18#16839989—22no—
19#16839981—22no—
20#16839972C2212yes—
21#16839967C2212yes—
22C2×C4⋊#92418C2212yes—
23#16839771—22no—
24C4×#11758194C2211yescharacteristic
25#16839511C2211yescharacteristic
26#16839511C2211yescharacteristic
27#16839511C2212yes—
28Q8⋊#92418C2212yes—
29Q8⋊#92418C2212yes—
30D4⋊#92418C2212yes—
31D4⋊#92418C2212yes—
32C61×#563—22no—
33Q82×C61C2211yescharacteristic
34C61×#555C2211yescharacteristic
35C61×#554C2212yes—
36C61×#552—22no—
37C61×#547—22no—
38C61×#544—22no—
39C61×#542—22no—
40C61×#536—22no—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#529C2212yes—
43C2×#11758202—22no—
44C2×#11758200—22no—
45C2×#11758194—21no—
46C22×#92418—22no—
47#14519777C2312yes—
48#14519769—21no—
49#14519756—22no—
50C4×Dic122—22no—
51#14519736—21no—
52#14519710—22no—
53#14519708—22no—
54C4⋊#92418—22no—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57#14519701—22no—
58#14519701—24no—
59#14519701—24no—
60#14519700—22no—
61#14519700—22no—
62#14519698—22no—
63#14519688—21no—
64C4×#92418C2312yes—
65C2×#41×C61—22no—
66C22×Q8×C61D412yes—
67C4⋊Q8×C61—22no—
68C4⋊Q8×C61—22no—
69C4⋊Q8×C61—22no—
70C4⋊Q8×C61—22no—
71C4⋊Q8×C61D412yes—
72C4⋊Q8×C61C2311yescharacteristic
73C4⋊Q8×C61C2311yescharacteristic
74#126×C61—24no—
75#126×C61—22no—
76#125×C61—24no—
77#125×C61—22no—
78#125×C61D412yes—
79#124×C61—22no—
80#124×C61—22no—
81#124×C61D412yes—
82#123×C61—24no—
83#123×C61—24no—
84#123×C61—22no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61D412yes—
90#121×C61—22no—
91#121×C61D412yes—
92C4×Q8×C61—22no—
93C4×Q8×C61C2312yes—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61C2312yes—
97#117×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61—22no—
102C2×C42×C61C2311yescharacteristic
103#41×C61—84no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—44no—
107C2×Q8×C61C2×D412yes—
108C2×Q8×C61C2×D412yes—
109C2×Q8×C61C2×D412yes—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61C2×D412yes—
113C22×C4×C61—22no—
114C22×C4×C61—24no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D411yescharacteristic
118C2×Dic122—22no—
119C4⋊C4×C61—24no—
120C4⋊C4×C61—24no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D412yes—
130C4⋊C4×C61C2×D412yes—
131C22⋊C4×C61—24no—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—22no—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C42×C61—22no—
139C42×C61—22no—
140C42×C61—22no—
141C42×C61C2×D412yes—
142C42×C61C2×D411yescharacteristic
143C42×C61C2411yescharacteristic
144C22⋊Dic61—41no—
145#11758202—82no—
146#11758200—82no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149#11758196—42no—
150C4×Dic61—41no—
151#11758194—42no—
152C2×#92418—22no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—82no—
156C23×C61—22no—
157C23×C61#12011yescharacteristic
158Q8×C61—22no—
159Q8×C61—22no—
160Q8×C61—42no—
161Q8×C61—42no—
162Q8×C61—82no—
163Q8×C61—82no—
164D4×C61—22no—
165D4×C61—22no—
166D4×C61—44no—
167D4×C61—82no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—44no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—84no—
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12012yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D412yes—
184C2×Dic61—42no—
185Dic122—42no—
186Dic122—42no—
187#92418—42no—
188#92418—42no—
189#92418—42no—
190#18721—611nomaximal, Sylow
191C22×C61—22no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—42no—
195C22×C61#45512yes—
196C22×C61C2×#12011yescharacteristic
197C4×C61—22no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—42no—
201C4×C61—42no—
202C4×C61—42no—
203C4×C61—42no—
204C4×C61—42no—
205C4×C61#57712yes—
206C4×C61#57812yes—
207Dic61—82no—
208#3544D6111yescharacteristic
209#3248—612no—
210#3139—611no—
211#1646—612no—
212#1645—612no—
213#1644—612no—
214#1643—612no—
215C2×#333—611no—
216#1483—612no—
217C2×C61—22no—
218C2×C61—82no—
219C2×C61#301312yes—
220C2×C61C2×#45511yescharacteristic
221#563—22no—
222Q82C2×D6111yescharacteristic
223#555C2×D6111yescharacteristic
224#554C2×D6112yes—
225#552—22no—
226#547—22no—
227#544—22no—
228#542—22no—
229#536—22no—
230#533—611no—
231C2×C4⋊Q8C2×D6111yescharacteristic
232#529C2×D6112yes—
233#491—1222no—
234#487—612no—
235#487—612no—
236#479—1222no—
237#475—612no—
238#474—612no—
239#467—1222no—
240#465—1222no—
241#464—1222no—
242#462—612no—
243#461—612no—
244C2×C4⋊C8—612no—
245#419—1222no—
246C4×M16—611no—
247#333—611no—
248#333—611no—
249#333—612no—
250Q8⋊C8—612no—
251Q8⋊C8—612no—
252D4⋊C8—612no—
253D4⋊C8—612no—
254C61#1872111yescharacteristic, complemented, Sylow
255C2×#41—22no—
256C22×Q8C22⋊D6112yes—
257C2×Q16—1222no—
258C2×SD16—1222no—
259C2×M16—1221no—
260C22×C8—1222no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—612no—
266C4⋊Q8C22⋊D6112yes—
267C4⋊Q8C22×D6111yescharacteristic
268C4⋊Q8C22×D6111yescharacteristic
269#126—24no—
270#126—22no—
271#125—24no—
272#125—22no—
273#125—1221no—
274#125C22⋊D6112yes—
275#124—22no—
276#124—22no—
277#124C22⋊D6112yes—
278#123—24no—
279#123—24no—
280#123—22no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—22no—
285C22⋊Q8—1222no—
286C22⋊Q8C22⋊D6112yes—
287#121—22no—
288#121C22⋊D6112yes—
289C4×Q8—22no—
290C4×Q8—1222no—
291C4×Q8C22×D6112yes—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4C22×D6112yes—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—1221no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C42C22×D6111yescharacteristic
302#107—1222no—
303#106—1222no—
304C4⋊C8—612no—
305C4⋊C8—612no—
306C4⋊C8—1222no—
307#103—1224no—
308#103—1224no—
309#103—1222no—
310#103—1222no—
311#103—1222no—
312#102—1222no—
313#97—1221no—
314C4×C8—612no—
315#41—84no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—44no—
319C2×Q8—1222no—
320C2×Q8C2×C22⋊D6112yes—
321C2×Q8C2×C22⋊D6112yes—
322C2×Q8C2×C22⋊D6112yes—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4C2×C22⋊D6112yes—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4C2×C22⋊D6112yes—
330C22×C4C2×C22⋊D6111yescharacteristic
331Q16—4882no—
332SD16—4882no—
333M16—2442no—
334C2×C8—1222no—
335C2×C8—1222no—
336C2×C8—1222no—
337C2×C8—4882no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—1222no—
349C4⋊C4—1222no—
350C4⋊C4—2442no—
351C4⋊C4C2×C22⋊D6112yes—
352C4⋊C4C2×C22⋊D6112yes—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—2441no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—2441no—
365C42C2×C22⋊D6111yescharacteristic
366C42C2×C22⋊D6112yes—
367C42C23×D6111yescharacteristic
368C23—22no—
369C23#1451974611yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—42no—
373Q8—42no—
374Q8—82no—
375Q8—82no—
376Q8—2442no—
377Q8—2442no—
378D4—22no—
379D4—22no—
380D4—44no—
381D4—82no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—84no—
394C2×C4—2442no—
395C2×C4#1451974611yescharacteristic
396C2×C4#1451974612yes—
397C2×C4C22×C22⋊D6111yescharacteristic
398C2×C4C22×C22⋊D6112yes—
399C8—2442no—
400C8—2442no—
401C8—2442no—
402C22—22no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22#1683994012yes—
407C22C2×#1451974611yescharacteristic
408C4—22no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—4882no—
417C4#1684057012yes—
418C4#1684057712yes—
419C2—22no—
420C2—82no—
421C2#2145513412yes—
422C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3544C2
socleC22×C61C2×#120
hypercenter#3544D61
hypocenterC61#18721
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3544

Complements

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

NH ≅ G/Ntype
1. #21457156C2semidirect
2. #21456199C2semidirect
3. #21445047C2semidirect
4. #21445040C2semidirect
5. #21445033C2semidirect
6. #21445026C2semidirect
7. C2×#16839511C2semidirect
8. #21444232C2semidirect
254. C61#18721semidirect

Maximal subgroups

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

subgroupindex
#214571562
#214561992
#214450472
#214450402
#214450332
#214450262
C2×#168395112
#214442322
C61×#35442
#1872161