Finite Group Databasebeta

← #11778250

Subgroups — #11778250

Order 1152.

586 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 82 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#10594971C211yesmaximal, characteristic
2#10592927C211yesmaximal, characteristic, complemented
3C3×#5231C211yesmaximal, characteristic
4C3×#5228C211yesmaximal, characteristic, complemented
5#10589154C211yesmaximal, characteristic, complemented
6#10589153C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic
8#70980—31nomaximal
9#69701—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×#3756—21no—
12#64133C2211yescharacteristic
13C2×C3×#216—21no—
14C2×C3×(C2×D3)⋊C4—21no—
15C2×C3×(C2×D3)⋊C4C2211yescharacteristic
16C3×#924C2211yescharacteristic
17D3×#217—21no—
18#63964—21no—
19#63899—21no—
20#63896—21no—
21#63893—21no—
22#63887—21no—
23#63662—21no—
24#63657C2211yescharacteristic
25C3×#835—21no—
26C3×C3⋊#95—21no—
27C3×C3⋊#95C2211yescharacteristic
28C3×#830C2211yescharacteristic
29#5634—31no—
30#5483—31no—
31C3×#451—31no—
32C3×#393—31no—
33#5231D311yescharacteristic
34#5228—31no—
35#5228—31no—
36#5228—61no—
37#5228D311yescharacteristic, complemented
38#4538—31no—
39#4525—31no—
40#4468—31no—
41#4467—31no—
42#4452—31no—
43#4452—31no—
44C23×C3×D3D411yescharacteristic
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—42no—
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C2×#634D411yescharacteristic
51C22×C3×Dic3—21no—
52C22×C3×Dic3D411yescharacteristic
53C2×C3×Dic6D411yescharacteristic
54C2×#627—21no—
55C2×D3×Dic3—21no—
56C2×D3×Dic3—21no—
57#3757—21no—
58#3756—42no—
59C4⋊(C3⋊Dic3)D411yescharacteristic
60#3750—21no—
61C2×C3⋊(C3⋊C8)—21no—
62C3×#217—21no—
63C3×#216—42no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—42no—
67C3×C4⋊Dic3D411yescharacteristic
68C3×Dic3⋊C4—21no—
69C2×C3×C3⋊C8—21no—
70#3724—21no—
71Dic3⋊Dic3—21no—
72Dic3⋊Dic3—21no—
73(C2×D3)⋊Dic3—21no—
74(C2×D3)⋊Dic3—21no—
75#1602—91noSylow
76C22×C3×Q8—21no—
77C22×C3×Q8C2×D311yescharacteristic
78C22×C3×Q8C2×D311yescharacteristic
79C2×C3×Q16—61no—
80C3×C22⋊Q8—31no—
81C2×C3×C22⋊C4—31no—
82C2×C3×C22⋊C4—61no—
83C2×#217—61no—
84#946—61no—
85C2×#216—21no—
86C2×#216—61no—
87C2×#216—61no—
88C2×#216—62no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4C2×D311yescharacteristic
91#924—31no—
92#924—31no—
93#924—61no—
94#924—61no—
95#924C2×D311yescharacteristic
96#897—61no—
97#896—61no—
98#896—61no—
99D3×C22⋊C4—61no—
100#878—61no—
101#878—61no—
102#878—61no—
103#878—61no—
104#878—61no—
105C3×#103—61no—
106C3×C22⋊C8—31no—
107C3×#95—31no—
108C3×#95—61no—
109#835—21no—
110#835—61no—
111#835—61no—
112#835—62no—
113C3⋊#95—21no—
114C3⋊#95C2×D311yescharacteristic
115#830—31no—
116#830—31no—
117#830—61no—
118#830C2×D311yescharacteristic
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—21no—
121C22×C3×D3—21no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C22×C3×D3Q1612yescomplemented
125C32×Q8—22no—
126C32×Q8—22no—
127C32×Q8—41no—
128C32×Q8—41no—
129C2×C32×C4—22no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137#634—41no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C3×Dic6—41no—
146#627—81no—
147D3×Dic3—81no—
148D3×Dic3—81no—
149C3⋊(C3⋊C8)—41no—
150C3×C3⋊C8—41no—
151#543—91no—
152#451—31no—
153#451—91no—
154#393—31no—
155#328—91no—
156#328—91no—
157#323—91no—
158C24×C3—31no—
159C23×D3#6711yescharacteristic
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—31no—
166C2×C3×Q8—41no—
167C2×C3×Q8—42no—
168C2×C3×Q8—42no—
169C2×C3×Q8—44no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—31no—
174C22×C3×C4—41no—
175C22×C3×C4—61no—
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C22×D311yescharacteristic
178C22×Dic3—21no—
179C22×Dic3—61no—
180C22×Dic3—61no—
181C22×Dic3#6711yescharacteristic
182C2×C4×D3—61no—
183C2×C4×D3—61no—
184C2×Dic6—31no—
185C2×Dic6—31no—
186C2×Dic6—61no—
187C2×Dic6—61no—
188C2×Dic6—61no—
189C2×Dic6#6711yescharacteristic
190C3×Q16—122no—
191C2×C3×C8—61no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—61no—
194C3×C22⋊C4—61no—
195C3×C22⋊C4—61no—
196C3×C22⋊C4—61no—
197C3×C22⋊C4—122no—
198#217—21no—
199#217—61no—
200#217—61no—
201#217—61no—
202#217—61no—
203#217—121no—
204#216—42no—
205#216—122no—
206#216—122no—
207#216—124no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—21no—
210(C2×D3)⋊C4—42no—
211(C2×D3)⋊C4—61no—
212(C2×D3)⋊C4—61no—
213C4⋊Dic3—31no—
214C4⋊Dic3—31no—
215C4⋊Dic3—61no—
216C4⋊Dic3—61no—
217C4⋊Dic3—61no—
218C4⋊Dic3#6711yescharacteristic
219Dic3⋊C4—21no—
220Dic3⋊C4—61no—
221Dic3⋊C4—61no—
222Dic3⋊C4—61no—
223Dic3⋊C4—61no—
224Dic3⋊C4—61no—
225Dic3⋊C4—61no—
226Dic3⋊C4—121no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—62no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8—61no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32#12011yescharacteristic
234C22×C32C2×Q1611yescharacteristic
235C22×C32#13711yescharacteristic
236C2×C3×D3—22no—
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C2×C3×D3—42no—
240C2×C3×D3—81no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—42no—
244C3⋊Dic3—41no—
245C3⋊Dic3—41no—
246C3×Dic3—41no—
247C3×Dic3—41no—
248C3×Dic3—41no—
249C3×Dic3—81no—
250C22×Q8D3211yescharacteristic
251C2×Q16—61no—
252C2×Q16—181no—
253C22⋊Q8—31no—
254C22⋊Q8—91no—
255C22⋊Q8—181no—
256C22⋊Q8—181no—
257C22⋊Q8—181no—
258C22⋊Q8—181no—
259C22⋊Q8—181no—
260C2×C22⋊C4—31no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—181no—
263#103—61no—
264#103—181no—
265C22⋊C8—31no—
266C22⋊C8—91no—
267#95—31no—
268#95—61no—
269C23×C3—21no—
270C23×C3—32no—
271C23×C3—61no—
272C23×C3—61no—
273C23×C3—121no—
274C23×C3—121no—
275C23×C3D3×D411yescharacteristic
276C23×C3C2×#6711yescharacteristic
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3#21612yescomplemented
282C3×Q8—22no—
283C3×Q8—24no—
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—22no—
287C3×Q8—42no—
288C3×Q8—42no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×Q8—41no—
292C3×Q8—41no—
293C3×Q8—42no—
294C3×Q8—121no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—121no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4—121no—
316C2×C3×C4D3×D411yescharacteristic
317C2×C3×C4D3×D411yescharacteristic
318C2×C3×C4D3×D411yescharacteristic
319C2×C3×C4C2×#6711yescharacteristic
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333C2×Dic3—241no—
334C2×Dic3—241no—
335C4×D3—241no—
336C4×D3—241no—
337Dic6—41no—
338Dic6—121no—
339Dic6—121no—
340Dic6—122no—
341Dic6—241no—
342Dic6—241no—
343C3×C8—121no—
344C3⋊C8—41no—
345C3⋊C8—122no—
346C3⋊C8—121no—
347C3⋊C8—121no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32#45111yescharacteristic
351C2×C32#45511yescharacteristic
352C2×C32B(4,𝔽2)11yescharacteristic
353C3×D3—42no—
354C24—31no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—31no—
358C2×Q8—42no—
359C2×Q8—91no—
360C2×Q8—181no—
361C22×C4—21no—
362C22×C4—31no—
363C22×C4—61no—
364C22×C4—181no—
365C22×C4—181no—
366C22×C4C2×D3211yescharacteristic
367Q16—122no—
368Q16—362no—
369C2×C8—61no—
370C2×C8—181no—
371C4⋊C4—31no—
372C4⋊C4—61no—
373C4⋊C4—91no—
374C4⋊C4—181no—
375C4⋊C4—181no—
376C4⋊C4—181no—
377C4⋊C4—181no—
378C22⋊C4—61no—
379C22⋊C4—61no—
380C22⋊C4—61no—
381C22⋊C4—122no—
382C22⋊C4—181no—
383C22⋊C4—181no—
384C22⋊C4—181no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—62no—
395C22×C3—122no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3—241no—
399C22×C3D3×Q1611yescharacteristic
400C22×C3#91811yescharacteristic
401C22×C3#93711yescharacteristic
402C22×C3#93711yescharacteristic
403C22×C3C2×#21611yescharacteristic
404C22×C3#95111yescharacteristic
405C2×D3—22no—
406C2×D3—41no—
407C2×D3—41no—
408C2×D3—42no—
409C2×D3—81no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—41no—
416C3×C4—44no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—121no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4—241no—
423Dic3—41no—
424Dic3—41no—
425Dic3—41no—
426Dic3—81no—
427Dic3—121no—
428Dic3—121no—
429Dic3—121no—
430Dic3—121no—
431Dic3—241no—
432Dic3—241no—
433C32#160211yescharacteristic, complemented, Sylow
434C23—32no—
435C23—61no—
436C23—61no—
437C23—121no—
438C23—121no—
439C23D3×#6711yescharacteristic
440Q8—22no—
441Q8—22no—
442Q8—41no—
443Q8—41no—
444Q8—121no—
445Q8—361no—
446Q8—721no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—721no—
461C2×C4—721no—
462C2×C4D3×#6711yescharacteristic
463C2×C4#381111yescharacteristic
464C8—121no—
465C8—361no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—122no—
476C2×C3#503211yescharacteristic
477C2×C3#503211yescharacteristic
478C2×C3#505011yescharacteristic
479C2×C3#506111yescharacteristic
480C2×C3#518611yescharacteristic
481C2×C3#520211yescharacteristic
482D3—42no—
483C22—21no—
484C22—21no—
485C22—62no—
486C22—122no—
487C22—121no—
488C22—121no—
489C22—241no—
490C22D3×#21611yescharacteristic
491C22#6394311yescharacteristic
492C22#6398311yescharacteristic
493C4—21no—
494C4—21no—
495C4—42no—
496C4—121no—
497C4—121no—
498C4—121no—
499C4—241no—
500C4—361no—
501C4—361no—
502C3—21no—
503C3#6970111yescharacteristic, complemented
504C3#7098011yescharacteristic, complemented
505C2—21no—
506C2—21no—
507C2—122no—
508C2#1059165311yescharacteristic
509C2#1059165911yescharacteristic
510C2#1059172511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#120
prime radicalC23×C3×D3D4
hypercenterC22×Q8D32
hypocenterC32#1602
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #10592927C2semidirect
4. C3×#5228C2semidirect
5. #10589154C2semidirect
6. #10589153C2semidirect
37. #5228D3semidirect
124. C22×C3×D3Q16semidirect
281. C22×D3#216semidirect
433. C32#1602semidirect
503. C3#69701semidirect
504. C3#70980semidirect

Maximal subgroups

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

subgroupindex
#105949712
#105929272
C3×#52312
C3×#52282
#105891542
#105891532
#105891452
#709803
#697013