Finite Group Databasebeta

← #29595923

Subgroups — #29595923

Order 15616.

758 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 127 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#21447508C212yesmaximal, complemented
2C2×#16839559C211yesmaximal, characteristic, complemented
3#21446357C212yesmaximal, complemented
4#21446223C211yesmaximal, characteristic, complemented
5C61×#3545C211yesmaximal, characteristic
6#16840240C2211yescharacteristic, complemented
7#16839779—22no—
8#16839767—21no—
9C2×#14519696C2212yescomplemented
10C2×#14519696C2211yescharacteristic, complemented
11#16839561—22no—
12#16839559—21no—
13#16839559—21no—
14#16839559—22no—
15#16839522—21no—
16#16839518—22no—
17C61×#563C411yescharacteristic, complemented
18C61×#562—21no—
19C61×H(3,𝔽4)C411yescharacteristic, complemented
20C61×#555C412yescomplemented
21C61×#551—21no—
22C61×#551—22no—
23C61×#546—22no—
24C61×#542—22no—
25C61×#538—22no—
26C61×#536C2212yes—
27C61×#534—21no—
28C61×#529—21no—
29C2×#124×C61C2211yescharacteristic
30C2×#11758194—21no—
31C2×#11758194—22no—
32Q8×Dic61—42no—
33#14519736—22no—
34#14519736—21no—
35#14519704—44no—
36#14519698—42no—
37#14519696—22no—
38#14519696—22no—
39#14519696—22no—
40C2×#41×C61D412yes—
41C22×Q8×C61C2311yescharacteristic
42C22×D4×C61—21no—
43C22×D4×C61D411yescharacteristic
44C4⋊Q8×C61C2×C412yes—
45C4⋊Q8×C61D412yes—
46#126×C61—22no—
47#126×C61—22no—
48#126×C61—22no—
49#125×C61—21no—
50#124×C61—22no—
51#124×C61—22no—
52#124×C61—22no—
53#124×C61—21no—
54#124×C61C2×C412yes—
55#124×C61C2×C411yescharacteristic
56#124×C61C2×C411yescharacteristic
57#124×C61D412yes—
58#124×C61D411yescharacteristic
59#124×C61D411yescharacteristic
60#123×C61—21no—
61#123×C61—22no—
62#123×C61—22no—
63#123×C61—22no—
64#123×C61—21no—
65#123×C61—22no—
66#123×C61—22no—
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—21no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71C22⋊Q8×C61—22no—
72#121×C61—22no—
73#121×C61—22no—
74#121×C61—21no—
75#121×C61—22no—
76#121×C61—22no—
77#120×C61—21no—
78#120×C61—22no—
79#120×C61—21no—
80#120×C61—22no—
81C4×Q8×C61—21no—
82C4×D4×C61—21no—
83C4×D4×C61—22no—
84C4×D4×C61—22no—
85C4×D4×C61—22no—
86#117×C61—22no—
87#117×C61—22no—
88#117×C61D412yes—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C42×C61D411yescharacteristic
93C24×C61—21no—
94C24×C61—21no—
95#41×C61—42no—
96#41×C61—42no—
97C2×Q8×C61—21no—
98C2×Q8×C61—41no—
99C2×Q8×C61—41no—
100C2×Q8×C61C22⋊C412yes—
101C2×Q8×C61C22×C411yescharacteristic
102C2×Q8×C61C2×D411yescharacteristic
103C2×Q8×C61C2×D412yes—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—21no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—81no—
113C2×D4×C61—81no—
114C2×D4×C61C22⋊C412yes—
115C2×D4×C61C22⋊C411yescharacteristic
116C2×D4×C61C22⋊C411yescharacteristic
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C22⋊C4×C61—22no—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—21no—
136C22⋊C4×C61—21no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C42×C61—22no—
149C42×C61—21no—
150C42×C61C22⋊C411yescharacteristic
151C42×C61C22⋊C411yescharacteristic
152C42×C61C22⋊C412yes—
153C42×C61C22⋊C412yes—
154C22⋊Dic61—41no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157C4×Dic61—24no—
158C4×Dic61—41no—
159#11758194—42no—
160#11758194—41no—
161#11758194—41no—
162#11758194—42no—
163C2×#92418—41no—
164C2×#92418—42no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—22no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—81no—
171C23×C61—82no—
172C23×C61—82no—
173C23×C61#12011yescharacteristic
174Q8×C61—41no—
175Q8×C61—41no—
176Q8×C61—42no—
177Q8×C61—42no—
178Q8×C61—81no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—81no—
185D4×C61—82no—
186D4×C61—82no—
187D4×C61—81no—
188D4×C61—82no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—82no—
199C2×C4×C61—82no—
200C2×C4×C61—81no—
201C2×C4×C61—82no—
202C2×C4×C61—81no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61#12012yes—
207C2×Dic61—42no—
208#92418—82no—
209#92418—81no—
210#12781—611nomaximal, Sylow
211C22×C61—41no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—41no—
218C22×C61—41no—
219C22×C61—161no—
220C22×C61—161no—
221C22×C61—162no—
222C22×C61#37911yescharacteristic
223C22×C61B(4,𝔽2)11yescharacteristic
224C22×C61#45811yescharacteristic
225C4×C61—22no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—22no—
229C4×C61—41no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—41no—
233C4×C61—42no—
234Dic61—82no—
235#3545D6111yescharacteristic
236#2133—612no—
237C2×#355—611no—
238#1912—612no—
239#1888—611no—
240C2×C61—21no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61—81no—
244C2×C61—82no—
245C2×C61—81no—
246C2×C61#179011yescharacteristic
247#563Dic6111yescharacteristic, complemented
248#562—21no—
249H(3,𝔽4)Dic6111yescharacteristic, complemented
250#555Dic6112yescomplemented
251#551—21no—
252#551—22no—
253#546—22no—
254#542—22no—
255#538—22no—
256#536C2×D6112yes—
257#534—21no—
258#529—21no—
259C2×#124C2×D6111yescharacteristic
260#519—611no—
261#421—1222no—
262#418—1221no—
263C2×#101—612no—
264C2×#101—611no—
265#356—1222no—
266#355—1221no—
267#355—1221no—
268#355—1222no—
269#339—1221no—
270H(3,ℤ/4)—1222no—
271C61#1278111yescharacteristic, complemented, Sylow
272C2×#41C22⋊D6112yes—
273C22×Q8C22×D6111yescharacteristic
274C22×D4—21no—
275C22×D4C22⋊D6111yescharacteristic
276C2×M16—1222no—
277C2×M16—1221no—
278C4⋊Q8C2×Dic6112yes—
279C4⋊Q8C22⋊D6112yes—
280#126—22no—
281#126—22no—
282#126—22no—
283#125—21no—
284#124—22no—
285#124—22no—
286#124—21no—
287#124—22no—
288#124C2×Dic6111yescharacteristic
289#124C2×Dic6111yescharacteristic
290#124C2×Dic6112yes—
291#124C22⋊D6111yescharacteristic
292#124C22⋊D6111yescharacteristic
293#124C22⋊D6112yes—
294#123—21no—
295#123—22no—
296#123—22no—
297#123—22no—
298#123—21no—
299#123—22no—
300#123—22no—
301C22⋊Q8—21no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306#121—21no—
307#121—22no—
308#121—22no—
309#121—22no—
310#121—22no—
311#120—21no—
312#120—22no—
313#120—21no—
314#120—22no—
315C4×Q8—21no—
316C4×Q8—2442no—
317C4×D4—21no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321#117—22no—
322#117—22no—
323#117—1222no—
324#117—1221no—
325#117C22⋊D6112yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42C22⋊D6111yescharacteristic
330#104—2444no—
331#102—2442no—
332#101—1222no—
333#101—1222no—
334#101—1222no—
335C24—21no—
336C24—21no—
337#41—42no—
338#41—42no—
339C2×Q8—21no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8C22⋊Dic6112yes—
343C2×Q8C22×Dic6111yescharacteristic
344C2×Q8C2×C22⋊D6111yescharacteristic
345C2×Q8C2×C22⋊D6112yes—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D4C22⋊Dic6112yes—
357C2×D4C22⋊Dic6111yescharacteristic
358C2×D4C22⋊Dic6111yescharacteristic
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4C2×C22⋊D6111yescharacteristic
365C22×C4C2×C22⋊D6112yes—
366M16—2442no—
367M16—2441no—
368M16—2441no—
369M16—2442no—
370C2×C8—2441no—
371C2×C8—2442no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—1222no—
382C4⋊C4—2442no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—2441no—
399C42—22no—
400C42—21no—
401C42—1224no—
402C42—2441no—
403C42C22⋊Dic6111yescharacteristic
404C42C22⋊Dic6111yescharacteristic
405C42C22⋊Dic6112yes—
406C42C22⋊Dic6112yes—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—81no—
413C23—82no—
414C23—82no—
415C23#1451974611yescharacteristic
416Q8—41no—
417Q8—41no—
418Q8—42no—
419Q8—42no—
420Q8—81no—
421D4—41no—
422D4—41no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—81no—
427D4—82no—
428D4—82no—
429D4—81no—
430D4—82no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—2442no—
446C2×C4C2×C22⋊Dic6112yes—
447C2×C4C2×C22⋊Dic6111yescharacteristic
448C2×C4#1451974611yescharacteristic
449C2×C4#1451974612yes—
450C8—4882no—
451C8—4881no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—41no—
459C22—41no—
460C22—161no—
461C22—161no—
462C22—162no—
463C22#1683961811yescharacteristic
464C22#1683994911yescharacteristic
465C22#1683995211yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—41no—
474C4—42no—
475C4—4882no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—81no—
480C2—81no—
481C2—82no—
482C2#2144576711yescharacteristic
subgroupG/H
centerC2#21445767
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
Fitting, prime radicalC61×#3545C2
socleC2×C61#1790
cosocleC22×Q8×C61C23
hypercenter#3545D61
hypocenterC61#12781
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ #3545

Complements

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

NH ≅ G/Ntype
1. #21447508C2semidirect
2. C2×#16839559C2semidirect
3. #21446357C2semidirect
4. #21446223C2semidirect
6. #16840240C22semidirect
9. C2×#14519696C22semidirect
10. C2×#14519696C22semidirect
17. C61×#563C4semidirect
19. C61×H(3,𝔽4)C4semidirect
20. C61×#555C4semidirect
247. #563Dic61semidirect
249. H(3,𝔽4)Dic61semidirect
250. #555Dic61semidirect
271. C61#12781semidirect

Maximal subgroups

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

subgroupindex
#214475082
C2×#168395592
#214463572
#214462232
C61×#35452
#1278161