Finite Group Databasebeta

← #11775407

Subgroups — #11775407

Order 1152.

737 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 94 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#10596098C211yesmaximal, characteristic
2#10591355C211yesmaximal, characteristic, complemented
3#10591159C211yesmaximal, characteristic, complemented
4C3×#4538C212yesmaximal, complemented
5#10589151C212yesmaximal, complemented
6#68337—32nomaximal
7C2×#3832C411yescharacteristic, complemented
8C22×C4×C3⋊D3C2211yescharacteristic
9C32×C22⋊Q8C2211yescharacteristic
10#64155C411yescharacteristic, complemented
11#64139C2211yescharacteristic
12#64104C411yescharacteristic, complemented
13#64103—21no—
14#64102—21no—
15#64101C411yescharacteristic, complemented
16#64091—21no—
17#64090—21no—
18C2×#3724—21no—
19C2×#3722—21no—
20C2×#3712—21no—
21C3×C3⋊#95C2212yes—
22C3×#830C2212yes—
23#63557—21no—
24#5627—32no—
25#5627—61no—
26#4726—32no—
27#4667—32no—
28C3×#328—32no—
29#4538D312yescomplemented
30#4470—32no—
31#4451—32no—
32C23×C3⋊D3D411yescharacteristic
33C2×C32×Q8C2×C411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C2×#638—21no—
36C22×C3⋊Dic3D411yescharacteristic
37#3832—42no—
38C2×C4⋊(C3⋊D3)—21no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
41C2×C4×C3⋊D3—21no—
42C2×C4×C3⋊D3—42no—
43C2×C4×C3⋊D3C2×C411yescharacteristic
44C2×C4×C3⋊D3C2×C411yescharacteristic
45C2×C4×C3⋊D3D411yescharacteristic
46C2×C4×C3⋊D3D411yescharacteristic
47C22×C3×Dic3—22no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4C2×C411yescharacteristic
50C32×C22⋊C4—21no—
51#3752—21no—
52#3752—21no—
53#3752—21no—
54#3752—21no—
55C4⋊(C3⋊Dic3)C2×C411yescharacteristic
56#3750—21no—
57C4×C3⋊Dic3—21no—
58C2×C3×C3⋊C8—22no—
59#3724—42no—
60#3722—42no—
61#3712—42no—
62#1513—91noSylow
63C2×#239—32no—
64C2×#239—61no—
65C22×C4×D3—32no—
66C22×C4×D3—61no—
67C3×C22⋊Q8—21no—
68C3×C22⋊Q8C2×D312yes—
69#946—32no—
70#946—61no—
71#930—32no—
72#930—61no—
73C2×(C2×D3)⋊C4—62no—
74C2×Dic3⋊C4—62no—
75C2×ΓL(1,𝔽25)—62no—
76#895—32no—
77#895—61no—
78#894—62no—
79#894—62no—
80#893—62no—
81#893—62no—
82#892—32no—
83#892—61no—
84#882—62no—
85#882—62no—
86#67⋊C4—62no—
87#67⋊C4—62no—
88C3×C22⋊C8—32no—
89C3×#95—32no—
90C3⋊#95C2×D312yes—
91#830C2×D312yes—
92(C2×D3)⋊C8—62no—
93C23×C32C2×D411yescharacteristic
94C22×C3⋊D3—21no—
95C22×C3⋊D3—21no—
96C22×C3⋊D3—21no—
97C22×C3⋊D3—41no—
98C22×C3⋊D3—41no—
99C22×C3⋊D3C22⋊C411yescharacteristic
100C22×C3⋊D3C22⋊C411yescharacteristic
101C32×Q8—41no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—41no—
105C2×C32×C4C22×C411yescharacteristic
106C2×C32×C4C2×D411yescharacteristic
107#638—41no—
108#638—41no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3C22⋊C411yescharacteristic
112C2×C3⋊Dic3C22⋊C411yescharacteristic
113C4⋊(C3⋊D3)—41no—
114C4⋊(C3⋊D3)—42no—
115C4⋊(C3⋊D3)—42no—
116C4×C3⋊D3—22no—
117C4×C3⋊D3—22no—
118C4×C3⋊D3—41no—
119C4×C3⋊D3—41no—
120C4×C3⋊D3—42no—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—42no—
124C3×C3⋊C8—42no—
125#525—91no—
126#407—91no—
127#380—91no—
128#328—32no—
129#327—92no—
130C23×D3—32no—
131C23×D3—61no—
132C2×C3×Q8—21no—
133C2×C3×Q8C4×D312yes—
134C22×C3×C4—21no—
135C22×C3×C4—62no—
136C22×C3×C4C22×D312yes—
137C2×#67—62no—
138C2×#67—62no—
139C22×Dic3—22no—
140C22×Dic3—32no—
141C22×Dic3—61no—
142#239—124no—
143#239—124no—
144C2×D12—32no—
145C2×D12—61no—
146C2×D12—62no—
147C2×D12—62no—
148C2×D12—62no—
149C2×D12—62no—
150C2×C4×D3—32no—
151C2×C4×D3—32no—
152C2×C4×D3—32no—
153C2×C4×D3—32no—
154C2×C4×D3—61no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×C4×D3—61no—
158C2×C4×D3—62no—
159C2×C4×D3—62no—
160C2×C4×D3—124no—
161C2×C4×D3—124no—
162C2×C3×C8—62no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4C4×D312yes—
167C3×C22⋊C4—22no—
168C3×C22⋊C4—22no—
169(C2×D3)⋊C4—62no—
170(C2×D3)⋊C4—62no—
171(C2×D3)⋊C4—62no—
172(C2×D3)⋊C4—62no—
173(C2×D3)⋊C4—62no—
174(C2×D3)⋊C4—62no—
175(C2×D3)⋊C4—62no—
176(C2×D3)⋊C4—62no—
177(C2×D3)⋊C4—124no—
178C4⋊Dic3—32no—
179C4⋊Dic3—61no—
180Dic3⋊C4—62no—
181Dic3⋊C4—62no—
182Dic3⋊C4—124no—
183C4×Dic3—62no—
184C4×Dic3—62no—
185C2×C3⋊C8—22no—
186ΓL(1,𝔽25)—124no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32C2×C22⋊C411yescharacteristic
190C22×C32AGL(1,ℤ/8)11yescharacteristic
191C22×C32#13711yescharacteristic
192C2×C3⋊D3—21no—
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3⋊D3—21no—
196C2×C3⋊D3—21no—
197C2×C3⋊D3—21no—
198C2×C3⋊D3—42no—
199C2×C3⋊D3—42no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—41no—
206C32×C4—41no—
207C3⋊Dic3—21no—
208C3⋊Dic3—41no—
209C3⋊Dic3#9912yes—
210C3×Dic3—82no—
211C2×#41—91no—
212C23×C4—91no—
213C2×M16—181no—
214#123—181no—
215C22⋊Q8—91no—
216C22⋊Q8D3211yescharacteristic
217#121—91no—
218#121—91no—
219#120—181no—
220C4×D4—181no—
221C4×D4—181no—
222#117—91no—
223C2×C4⋊C4—181no—
224C2×C22⋊C4—181no—
225C22⋊C8—32no—
226C22⋊C8—181no—
227#95—32no—
228C23×C3—21no—
229C23×C3D3×D412yes—
230C22×D3—32no—
231C22×D3—32no—
232C22×D3—61no—
233C22×D3—61no—
234C22×D3—62no—
235C22×D3—62no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—121no—
239C22×D3—121no—
240C22×D3—122no—
241C22×D3—122no—
242C22×D3—122no—
243C22×D3—122no—
244C3×Q8—42no—
245C3×Q8—42no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—122no—
255C2×C3×C4—122no—
256C2×C3×C4—122no—
257C2×C3×C4C2×C4×D312yes—
258C2×C3×C4D3×D412yes—
259#67—122no—
260#67—122no—
261#67—122no—
262#67—122no—
263C2×Dic3—32no—
264C2×Dic3—32no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—64no—
271C2×Dic3—62no—
272C2×Dic3—62no—
273C2×Dic3—122no—
274D12—122no—
275D12—122no—
276D12—124no—
277D12—124no—
278D12—124no—
279D12—124no—
280C4×D3—64no—
281C4×D3—64no—
282C4×D3—64no—
283C4×D3—122no—
284C4×D3—124no—
285C4×D3—122no—
286C4×D3—122no—
287C4×D3—124no—
288C4×D3—122no—
289C4×D3—122no—
290C3×C8—122no—
291C3⋊C8—42no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32C2×#9911yescharacteristic
295C2×C32#41711yescharacteristic
296C2×C32#42111yescharacteristic
297C3⋊D3—41no—
298C3⋊D3—41no—
299C3⋊D3—81no—
300C24—91no—
301#41—362no—
302C2×Q8#62411yescharacteristic
303C2×D4—91no—
304C2×D4—181no—
305C2×D4—181no—
306C2×D4—181no—
307C22×C4—62no—
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—91no—
311C22×C4—91no—
312C22×C4—91no—
313C22×C4—181no—
314C22×C4—362no—
315C22×C4C2×D3211yescharacteristic
316M16—362no—
317C2×C8—62no—
318C4⋊C4—21no—
319C4⋊C4—91no—
320C4⋊C4—181no—
321C4⋊C4—362no—
322C4⋊C4#62411yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—181no—
325C22⋊C4—181no—
326C22⋊C4—181no—
327C22⋊C4—181no—
328C22⋊C4—362no—
329C42—181no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3D3×C22⋊C412yes—
338C22×C3#91412yes—
339C22×C3#91812yes—
340C2×D3—62no—
341C2×D3—62no—
342C2×D3—62no—
343C2×D3—62no—
344C2×D3—62no—
345C2×D3—62no—
346C2×D3—62no—
347C2×D3—62no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—124no—
353C2×D3—124no—
354C2×D3—124no—
355C2×D3—124no—
356C2×D3—122no—
357C2×D3—122no—
358C2×D3—122no—
359C2×D3—122no—
360C2×D3—122no—
361C2×D3—122no—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—41no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—242no—
371Dic3—34no—
372Dic3—62no—
373Dic3—62no—
374Dic3—82no—
375Dic3—121no—
376Dic3—122no—
377Dic3—122no—
378C32#151311yescharacteristic, complemented, Sylow
379C23—91no—
380C23—91no—
381C23—181no—
382C23—181no—
383C23—181no—
384C23—361no—
385C23—361no—
386C23#381111yescharacteristic
387Q8—41no—
388D4—361no—
389D4—362no—
390D4—362no—
391D4—361no—
392D4—361no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4—182no—
404C2×C4—181no—
405C2×C4—361no—
406C2×C4—361no—
407C2×C4—362no—
408C2×C4C2×#62411yescharacteristic
409C2×C4#381111yescharacteristic
410C8—122no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3D3×#9912yes—
419C2×C3#480312yes—
420C2×C3#482112yes—
421D3—122no—
422D3—122no—
423D3—122no—
424D3—122no—
425D3—242no—
426D3—242no—
427C22—21no—
428C22—21no—
429C22—181no—
430C22—181no—
431C22—181no—
432C22—181no—
433C22—181no—
434C22—181no—
435C22—362no—
436C22—362no—
437C22—361no—
438C22—361no—
439C22—361no—
440C22#6394111yescharacteristic
441C22#6394511yescharacteristic
442C22#6397411yescharacteristic
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—92no—
448C4—181no—
449C4—242no—
450C4—361no—
451C3—21no—
452C3#6833712yescomplemented
453C2—21no—
454C2—21no—
455C2—361no—
456C2—361no—
457C2—721no—
458C2#1059152711yescharacteristic
459C2#1059155411yescharacteristic
460C2#1059167211yescharacteristic
subgroupG/H
centerC22#63974
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32C2×C22⋊C4
prime radical#10596098C2
cosocleC22×C32×C4C23
hypercenterC22⋊Q8D32
hypocenterC32#1513
LCSG ▷ C2×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. #10591355C2semidirect
3. #10591159C2semidirect
4. C3×#4538C2semidirect
5. #10589151C2semidirect
7. C2×#3832C4semidirect
10. #64155C4semidirect
12. #64104C4semidirect
15. #64101C4semidirect
29. #4538D3semidirect
378. C32#1513semidirect
452. C3#68337semidirect

Maximal subgroups

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

subgroupindex
#105960982
#105913552
#105911592
C3×#45382
#105891512
#683373