Finite Group Databasebeta

← #11778289

Subgroups — #11778289

Order 1152.

579 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 78 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#10594974C211yesmaximal, characteristic
2#10592929C211yesmaximal, characteristic, complemented
3C3×#5228C211yesmaximal, characteristic, complemented
4C3×#5227C211yesmaximal, characteristic
5#10589152C211yesmaximal, characteristic, complemented
6#10589150C211yesmaximal, characteristic
7#10589148C211yesmaximal, characteristic, complemented
8#70982—31nomaximal
9#70974—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×#3750—21no—
12C2×#3750C2211yescharacteristic
13C2×C3×#216—21no—
14C2×C3×#215—21no—
15C3×#930C2211yescharacteristic
16C3×#924C2211yescharacteristic
17#67⋊Dic3—21no—
18#63962—21no—
19#63887—21no—
20#63884—21no—
21#63858—21no—
22#63846—21no—
23#63658—21no—
24#63658C2211yescharacteristic
25C3×#835—21no—
26C3×#835—21no—
27C3×#830C2211yescharacteristic
28C3×#830C2211yescharacteristic
29#5645—31no—
30#5602—31no—
31C3×#451—31no—
32C3×#448—31no—
33#5230—31no—
34#5230—31no—
35#5230—61no—
36#5228D311yescharacteristic, complemented
37#5227D311yescharacteristic
38#4538—31no—
39#4537—31no—
40#4470—31no—
41#4469—31no—
42#4469—31no—
43#4467—31no—
44C2×C32×Q8—21no—
45C2×C32×Q8—21no—
46C2×C32×Q8—42no—
47C22×C32×C4—21no—
48C22×C32×C4C2311yescharacteristic
49C22×C3⋊Dic3—21no—
50C22×C3⋊Dic3D411yescharacteristic
51C22×C3⋊Dic3D411yescharacteristic
52C2×C3×#67—21no—
53C2×C3×D12D411yescharacteristic
54C2×C3×Dic6D411yescharacteristic
55C2×D3×Dic3—21no—
56#3750—21no—
57#3750—21no—
58#3750—42no—
59C3×#217—21no—
60C3×#216—42no—
61C3×#215—42no—
62C3×(C2×D3)⋊C4—21no—
63C3×C4⋊Dic3D411yescharacteristic
64C3×C4⋊Dic3D411yescharacteristic
65C3×Dic3⋊C4—21no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68Dic3⋊Dic3—21no—
69Dic3⋊Dic3—21no—
70Dic3⋊Dic3—21no—
71(C2×D3)⋊Dic3—21no—
72(C2×D3)⋊Dic3—21no—
73(C2×D3)⋊Dic3—21no—
74Dic32—21no—
75#1603—91noSylow
76C22×C3×Q8—21no—
77C22×C3×Q8C2×D311yescharacteristic
78C22×C3×Q8C2×D311yescharacteristic
79C2×C3×Q16—61no—
80C2×C3×SD16—61no—
81C3×C22⋊Q8—31no—
82C3×#121—31no—
83#946—61no—
84C2×#216—21no—
85C2×#215—21no—
86#935—61no—
87#935—61no—
88D4×Dic3—61no—
89#930C2×D311yescharacteristic
90#924C2×D311yescharacteristic
91C2×Dic3⋊C4—31no—
92C2×Dic3⋊C4—31no—
93C2×Dic3⋊C4—61no—
94C2×Dic3⋊C4—61no—
95C2×Dic3⋊C4—61no—
96C2×Dic3⋊C4—62no—
97#898—61no—
98#894—61no—
99#890—61no—
100#889—61no—
101#886—61no—
102#67⋊C4—61no—
103#879—61no—
104#878—61no—
105C3×#103—61no—
106C3×#103—61no—
107C3×C22⋊C8—31no—
108C3×C22⋊C8—31no—
109#835—21no—
110#835—21no—
111C3⋊#95—31no—
112C3⋊#95—31no—
113C3⋊#95—61no—
114C3⋊#95—61no—
115C3⋊#95—61no—
116C3⋊#95—62no—
117#830C2×D311yescharacteristic
118#830C2×D311yescharacteristic
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—21no—
121C32×Q8—22no—
122C32×Q8—22no—
123C32×Q8—41no—
124C32×Q8—41no—
125C2×C32×C4—22no—
126C2×C32×C4—41no—
127C2×C32×C4—41no—
128C2×C32×C4—41no—
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—41no—
137C2×C3⋊Dic3—41no—
138C2×C3⋊Dic3—41no—
139C3×#67—41no—
140C3×#67—41no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C3×D12—41no—
145C3×Dic6—41no—
146D3×Dic3—81no—
147C3×C3⋊C8—41no—
148C3×C3⋊C8—41no—
149#539—91no—
150#451—31no—
151#448—31no—
152#395—91no—
153#328—91no—
154#327—91no—
155#324—91no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—31no—
161C2×C3×Q8—41no—
162C2×C3×Q8—41no—
163C2×C3×Q8—42no—
164C2×C3×Q8—42no—
165C2×C3×Q8—44no—
166C2×C3×D4—31no—
167C2×C3×D4—61no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C22×D311yescharacteristic
173C22×C3×C4C22×D311yescharacteristic
174C2×#67—21no—
175C22×Dic3—31no—
176C22×Dic3—31no—
177C22×Dic3—31no—
178C22×Dic3—31no—
179C22×Dic3—61no—
180C22×Dic3—61no—
181C22×Dic3—61no—
182C22×Dic3—62no—
183C22×Dic3—61no—
184C22×Dic3—61no—
185C2×D12#6711yescharacteristic
186C2×C4×D3—61no—
187C2×Dic6#6711yescharacteristic
188C3×Q16—122no—
189C3×SD16—122no—
190C2×C3×C8—61no—
191C2×C3×C8—61no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—31no—
194C3×C4⋊C4—61no—
195C3×C22⋊C4—61no—
196C3×C22⋊C4—61no—
197#217—21no—
198#217—61no—
199#217—61no—
200#217—61no—
201#216—42no—
202#215—42no—
203(C2×D3)⋊C4—21no—
204(C2×D3)⋊C4—61no—
205(C2×D3)⋊C4—61no—
206(C2×D3)⋊C4—61no—
207C4⋊Dic3—61no—
208C4⋊Dic3—61no—
209C4⋊Dic3—61no—
210C4⋊Dic3#6711yescharacteristic
211C4⋊Dic3#6711yescharacteristic
212Dic3⋊C4—21no—
213Dic3⋊C4—61no—
214Dic3⋊C4—61no—
215Dic3⋊C4—61no—
216Dic3⋊C4—61no—
217Dic3⋊C4—61no—
218Dic3⋊C4—61no—
219Dic3⋊C4—61no—
220Dic3⋊C4—121no—
221Dic3⋊C4—121no—
222Dic3⋊C4—122no—
223Dic3⋊C4—122no—
224Dic3⋊C4—124no—
225C4×Dic3—61no—
226C4×Dic3—61no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32#12011yescharacteristic
232C22×C32#13511yescharacteristic
233C22×C32#13711yescharacteristic
234C2×C3×D3—41no—
235C2×C3×D3—41no—
236C2×C3×D3—41no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—42no—
240C3⋊Dic3—42no—
241C3⋊Dic3—41no—
242C3⋊Dic3—41no—
243C3×Dic3—41no—
244C3×Dic3—41no—
245C3×Dic3—41no—
246C22×Q8D3211yescharacteristic
247C2×Q16—61no—
248C2×SD16—61no—
249#126—181no—
250#125—181no—
251#123—181no—
252#123—181no—
253C22⋊Q8—31no—
254C22⋊Q8—181no—
255#121—31no—
256C4×D4—181no—
257C2×C4⋊C4—91no—
258C2×C4⋊C4—181no—
259#103—61no—
260#103—61no—
261C22⋊C8—31no—
262C22⋊C8—31no—
263#95—91no—
264#95—181no—
265C23×C3—21no—
266C23×C3—61no—
267C23×C3D3×D411yescharacteristic
268C23×C3D3×D411yescharacteristic
269C22×D3—21no—
270C3×Q8—22no—
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—42no—
275C3×Q8—42no—
276C3×Q8—41no—
277C3×Q8—41no—
278C3×Q8—41no—
279C3×Q8—41no—
280C3×Q8—42no—
281C3×Q8—42no—
282C3×Q8—121no—
283C3×D4—121no—
284C3×D4—121no—
285C3×D4—121no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4D3×D411yescharacteristic
304C2×C3×C4D3×D411yescharacteristic
305C2×C3×C4C2×#6711yescharacteristic
306C2×C3×C4C2×#6711yescharacteristic
307#67—41no—
308#67—41no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—64no—
313C2×Dic3—62no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—61no—
324C2×Dic3—121no—
325C2×Dic3—121no—
326C2×Dic3—121no—
327C2×Dic3—122no—
328C2×Dic3—121no—
329C2×Dic3—121no—
330C2×Dic3—122no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333C2×Dic3—121no—
334C2×Dic3—121no—
335C2×Dic3—122no—
336C2×Dic3—241no—
337D12—41no—
338C4×D3—241no—
339Dic6—41no—
340C3×C8—121no—
341C3×C8—121no—
342C3⋊C8—41no—
343C3⋊C8—41no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32#45211yescharacteristic
347C2×C32#45611yescharacteristic
348C2×C32B(4,𝔽2)11yescharacteristic
349C3×D3—81no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—31no—
353C2×Q8—42no—
354C2×D4—31no—
355C2×D4—61no—
356C22×C4—21no—
357C22×C4—91no—
358C22×C4—91no—
359C22×C4—181no—
360C22×C4—181no—
361C22×C4C2×D3211yescharacteristic
362Q16—122no—
363SD16—122no—
364C2×C8—61no—
365C2×C8—61no—
366C4⋊C4—31no—
367C4⋊C4—31no—
368C4⋊C4—61no—
369C4⋊C4—181no—
370C4⋊C4—181no—
371C4⋊C4—181no—
372C4⋊C4—181no—
373C4⋊C4—181no—
374C4⋊C4—362no—
375C22⋊C4—61no—
376C22⋊C4—61no—
377C22⋊C4—181no—
378C22⋊C4—181no—
379C22⋊C4—181no—
380C42—181no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—121no—
391C22×C3—121no—
392C22×C3—121no—
393C22×C3#91511yescharacteristic
394C22×C3#91611yescharacteristic
395C22×C3#91811yescharacteristic
396C22×C3#91911yescharacteristic
397C22×C3#93711yescharacteristic
398C22×C3#93711yescharacteristic
399C2×D3—41no—
400C2×D3—41no—
401C2×D3—41no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—44no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—121no—
412C3×C4—121no—
413C3×C4—121no—
414Dic3—41no—
415Dic3—41no—
416Dic3—41no—
417Dic3—124no—
418Dic3—122no—
419Dic3—122no—
420Dic3—122no—
421Dic3—121no—
422Dic3—121no—
423Dic3—121no—
424Dic3—121no—
425Dic3—241no—
426C32#160311yescharacteristic, complemented, Sylow
427C23—61no—
428C23#381111yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—41no—
432Q8—41no—
433Q8—121no—
434D4—121no—
435D4—121no—
436D4—121no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—182no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—361no—
450C2×C4—361no—
451C2×C4—361no—
452C2×C4—721no—
453C2×C4D3×#6711yescharacteristic
454C2×C4D3×#6711yescharacteristic
455C8—121no—
456C8—121no—
457C2×C3—22no—
458C2×C3—22no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—241no—
467C2×C3#503211yescharacteristic
468C2×C3#503211yescharacteristic
469C2×C3#517311yescharacteristic
470C2×C3#518711yescharacteristic
471C2×C3#520511yescharacteristic
472C2×C3#520511yescharacteristic
473D3—81no—
474C22—21no—
475C22—21no—
476C22—121no—
477C22—121no—
478C22—121no—
479C22#6395011yescharacteristic
480C22#6395111yescharacteristic
481C22#6398311yescharacteristic
482C4—21no—
483C4—21no—
484C4—42no—
485C4—121no—
486C4—121no—
487C4—121no—
488C4—362no—
489C4—361no—
490C4—361no—
491C3—21no—
492C3#7097411yescharacteristic, complemented
493C3#7098211yescharacteristic, complemented
494C2—21no—
495C2—21no—
496C2—241no—
497C2#1059165611yescharacteristic
498C2#1059166011yescharacteristic
499C2#1059172511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#120
prime radicalC3×#930C22
hypercenterC22×Q8D32
hypocenterC32#1603
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. #10592929C2semidirect
3. C3×#5228C2semidirect
5. #10589152C2semidirect
7. #10589148C2semidirect
36. #5228D3semidirect
426. C32#1603semidirect
492. C3#70974semidirect
493. C3#70982semidirect

Maximal subgroups

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

subgroupindex
#105949742
#105929292
C3×#52282
C3×#52272
#105891522
#105891502
#105891482
#709823
#709743