Finite Group Databasebeta

← #11778223

Subgroups — #11778223

Order 1152.

789 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 88 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#10595964C211yesmaximal, characteristic
2C3×#4947C211yesmaximal, characteristic, complemented
3#10591174C212yesmaximal, complemented
4C3×#4525C211yesmaximal, characteristic, complemented
5#10589151C212yesmaximal, complemented
6#69699—31nomaximal
7#68108—31nomaximal
8C22×#638—21no—
9C22×C3×Dic6C2211yescharacteristic, complemented
10C2×C32×C22⋊C4C2211yescharacteristic
11#64139—21no—
12#64139C2212yes—
13#64095—22no—
14#64091—22no—
15C2×C3×#217—21no—
16C2×#3717—22no—
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic, complemented
19C3×#830C2212yescomplemented
20#63566—22no—
21#5493—31no—
22#5493—31no—
23#5493—61no—
24C3×#393—31no—
25#5227—32no—
26#4947D311yescharacteristic, complemented
27#4738—32no—
28C3×#323—31no—
29#4525D311yescharacteristic, complemented
30#4470—32no—
31#4451—32no—
32C24×C32D411yescharacteristic
33C23×C3⋊D3—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4C2311yescharacteristic
36C2×#638—22no—
37C2×#638—22no—
38C2×#638—42no—
39C2×#638—42no—
40C22×C3⋊Dic3—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)D412yes—
43C22×C3×Dic3—21no—
44C22×C3×Dic3—21no—
45C2×C3×Dic6—22no—
46C2×C3×Dic6—42no—
47C32×C22⋊C4—22no—
48#3752—22no—
49#3752—22no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)D412yes—
52C3×#217—42no—
53C2×C3×C3⋊C8—22no—
54#3717—44no—
55#1601—91noSylow
56C22×C3×Q8—31no—
57C22×#67—62no—
58C22×#67—61no—
59C22×#67—61no—
60C22×Dic6C2×D311yescharacteristic, complemented
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—61no—
63C2×C3×C22⋊C4C2×D311yescharacteristic
64C2×C3×C22⋊C4C2×D311yescharacteristic
65C2×#217—21no—
66C2×#215—62no—
67#930—32no—
68#930—32no—
69#930—62no—
70#930—62no—
71#930—61no—
72#930—61no—
73C2×#204—62no—
74#886—62no—
75#886—62no—
76#886—64no—
77#882—62no—
78#882—62no—
79#882—64no—
80C3×C22⋊C8—32no—
81C3×#95—31no—
82C3×#95—61no—
83#835—62no—
84C3⋊#95—21no—
85C3⋊#95C2×D311yescharacteristic, complemented
86#830C2×D312yescomplemented
87#816—62no—
88C23×C32—21no—
89C23×C32—21no—
90C23×C32—41no—
91C23×C32—41no—
92C23×C32SD1612yescomplemented
93C23×C32C2×D411yescharacteristic
94C22×C3⋊D3—22no—
95C22×C3⋊D3—42no—
96C22×C3⋊D3—81no—
97C22×C3⋊D3—81no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4C2×D412yes—
102#638—44no—
103#638—42no—
104#638—42no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—42no—
107C4⋊(C3⋊D3)—42no—
108C4⋊(C3⋊D3)—81no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—41no—
111C2×C3×Dic3—41no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—41no—
115C3×Dic6—24no—
116C3×Dic6—42no—
117C3×C3⋊C8—42no—
118#534—91no—
119#448—92no—
120#393—31no—
121#327—92no—
122#323—31no—
123C24×C3—21no—
124C24×C3D1211yescharacteristic
125C24×C3#6711yescharacteristic
126C23×D3—62no—
127C23×D3—61no—
128C23×D3—61no—
129C2×C3×Q8—62no—
130C2×C3×Q8—122no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—61no—
134C22×C3×C4—61no—
135C22×C3×C4D1211yescharacteristic
136C22×C3×C4#6711yescharacteristic
137C22×C3×C4C22×D311yescharacteristic
138C22×C3×C4C22×D311yescharacteristic
139C2×#67—64no—
140C2×#67—62no—
141C2×#67—62no—
142C2×#67—64no—
143C2×#67—62no—
144C2×#67—62no—
145C2×#67—124no—
146C2×#67—122no—
147C2×#67—122no—
148C2×#67—124no—
149C2×#67—122no—
150C2×#67—122no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—62no—
154C22×Dic3—61no—
155C22×Dic3—61no—
156C2×D12—32no—
157C2×D12—32no—
158C2×D12—62no—
159C2×D12—62no—
160C2×D12—61no—
161C2×D12—61no—
162C2×Dic6—22no—
163C2×Dic6—42no—
164C2×C3×C8—62no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—22no—
168C3×C22⋊C4—122no—
169#217—42no—
170#215—124no—
171(C2×D3)⋊C4—64no—
172(C2×D3)⋊C4—62no—
173(C2×D3)⋊C4—62no—
174(C2×D3)⋊C4—64no—
175(C2×D3)⋊C4—62no—
176(C2×D3)⋊C4—62no—
177C4⋊Dic3—32no—
178C4⋊Dic3—32no—
179C4⋊Dic3—62no—
180C4⋊Dic3—62no—
181C4⋊Dic3—61no—
182C4⋊Dic3—61no—
183C2×C3⋊C8—22no—
184#204—124no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—42no—
191C22×C32—81no—
192C22×C32#12011yescharacteristic
193C22×C32C2×SD1611yescharacteristic
194C22×C32AGL(1,ℤ/8)11yescharacteristic
195C2×C3⋊D3—42no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—42no—
198C2×C3⋊D3—84no—
199C32×C4—22no—
200C32×C4—41no—
201C3⋊Dic3—42no—
202C3×Dic3—42no—
203C3×Dic3—81no—
204C22×Q8—31no—
205C22×D4—181no—
206C2×SD16—182no—
207#123—182no—
208#121—92no—
209#121—181no—
210#120—182no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4D3211yescharacteristic
213#103—182no—
214C22⋊C8—32no—
215#95—31no—
216#95—61no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—22no—
224C23×C3—21no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—81no—
230C23×C3—81no—
231C23×C3#20412yescomplemented
232C23×C3#21512yescomplemented
233C23×C3C2×D1211yescharacteristic
234C23×C3C2×#6711yescharacteristic
235C22×D3—64no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—124no—
239C22×D3—122no—
240C22×D3—122no—
241C22×D3—242no—
242C22×D3—242no—
243C22×D3—241no—
244C22×D3—241no—
245C22×D3—241no—
246C22×D3—241no—
247C3×Q8—64no—
248C3×Q8—122no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—22no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—62no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4—121no—
261C2×C3×C4—121no—
262C2×C3×C4—121no—
263C2×C3×C4—121no—
264C2×C3×C4—121no—
265C2×C3×C4D3×D412yes—
266C2×C3×C4D3×D412yes—
267#67—128no—
268#67—124no—
269#67—124no—
270#67—124no—
271#67—124no—
272#67—122no—
273#67—122no—
274#67—122no—
275#67—122no—
276C2×Dic3—22no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—64no—
283C2×Dic3—62no—
284C2×Dic3—62no—
285C2×Dic3—124no—
286C2×Dic3—122no—
287C2×Dic3—122no—
288D12—122no—
289D12—122no—
290D12—242no—
291D12—242no—
292D12—241no—
293D12—241no—
294Dic6—24no—
295Dic6—42no—
296C3×C8—122no—
297C3⋊C8—42no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—42no—
301C2×C32#45011yescharacteristic
302C2×C32#45511yescharacteristic
303C2×C32B(4,𝔽2)11yescharacteristic
304C3⋊D3—82no—
305C24—181no—
306C24#62611yescharacteristic
307C2×Q8—62no—
308C2×Q8—122no—
309C2×D4—92no—
310C2×D4—181no—
311C2×D4—182no—
312C2×D4—182no—
313C2×D4—362no—
314C2×D4—362no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—181no—
318C22×C4#62611yescharacteristic
319C22×C4C2×D3211yescharacteristic
320SD16—364no—
321C2×C8—62no—
322C4⋊C4—92no—
323C4⋊C4—181no—
324C22⋊C4—22no—
325C22⋊C4—122no—
326C22⋊C4—182no—
327C22⋊C4—182no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—24no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—82no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—82no—
352C22×C3#88211yescharacteristic
353C22×C3C2×#20411yescharacteristic
354C22×C3#90811yescharacteristic
355C22×C3#93711yescharacteristic
356C22×C3C2×#21511yescharacteristic
357C22×C3#94911yescharacteristic
358C2×D3—124no—
359C2×D3—124no—
360C2×D3—124no—
361C2×D3—122no—
362C2×D3—122no—
363C2×D3—122no—
364C2×D3—122no—
365C2×D3—122no—
366C2×D3—122no—
367C2×D3—248no—
368C2×D3—244no—
369C2×D3—244no—
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—122no—
377C3×C4—241no—
378Dic3—42no—
379Dic3—81no—
380Dic3—124no—
381Dic3—122no—
382Dic3—122no—
383C32#160111yescharacteristic, complemented, Sylow
384C23—21no—
385C23—21no—
386C23—41no—
387C23—41no—
388C23—182no—
389C23—362no—
390C23—721no—
391C23—721no—
392C23#371712yescomplemented
393C23C2×#62611yescharacteristic
394Q8—64no—
395Q8—122no—
396D4—362no—
397D4—364no—
398D4—362no—
399D4—362no—
400D4—721no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—62no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—182no—
411C2×C4—362no—
412C2×C4#381112yes—
413C8—122no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—44no—
424C2×C3—42no—
425C2×C3—42no—
426C2×C3#473111yescharacteristic
427C2×C3#474111yescharacteristic
428C2×C3#474611yescharacteristic
429C2×C3#503211yescharacteristic
430C2×C3#504611yescharacteristic
431C2×C3#506111yescharacteristic
432D3—244no—
433D3—242no—
434D3—242no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—81no—
442C22—362no—
443C22—362no—
444C22—362no—
445C22—724no—
446C22#6382511yescharacteristic
447C22C2×#371711yescharacteristic
448C22#6398111yescharacteristic
449C4—22no—
450C4—41no—
451C4—122no—
452C4—241no—
453C4—362no—
454C3—21no—
455C3#6810811yescharacteristic, complemented
456C3#6969911yescharacteristic, complemented
457C2—21no—
458C2—21no—
459C2—42no—
460C2—722no—
461C2#1059118311yescharacteristic
462C2#1059151411yescharacteristic
463C2#1059153211yescharacteristic
subgroupG/H
centerC22#63981
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C22⋊C4C22
socleC22×C32#120
prime radical#10595964C2
hypercenterC2×C22⋊C4D32
hypocenterC32#1601
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4947C2semidirect
3. #10591174C2semidirect
4. C3×#4525C2semidirect
5. #10589151C2semidirect
9. C22×C3×Dic6C22semidirect
18. C3×C3⋊#95C22semidirect
19. C3×#830C22semidirect
26. #4947D3semidirect
29. #4525D3semidirect
60. C22×Dic6C2×D3semidirect
85. C3⋊#95C2×D3semidirect
86. #830C2×D3semidirect
92. C23×C32SD16semidirect
231. C23×C3#204semidirect
232. C23×C3#215semidirect
383. C32#1601semidirect
392. C23#3717semidirect
455. C3#68108semidirect
456. C3#69699semidirect

Maximal subgroups

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

subgroupindex
#105959642
C3×#49472
#105911742
C3×#45252
#105891512
#696993
#681083