Finite Group Databasebeta

← #10694720

Subgroups — #10694720

Order 768.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#5048C211yesmaximal, characteristic
2C3×#1520C211yesmaximal, characteristic, complemented
3#71701C211yesmaximal, characteristic, complemented
4#69948C211yesmaximal, characteristic, complemented
5#69944C211yesmaximal, characteristic
6#69528C211yesmaximal, characteristic
7#69512C211yesmaximal, characteristic, complemented
8#9786—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×#215C411yescharacteristic, complemented
11C2×#930C2211yescharacteristic
12C23×C3⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×#835C411yescharacteristic, complemented
16#5212C2211yescharacteristic
17C4×#217C2211yescharacteristic
18C2×#815—21no—
19#5048—21no—
20#5048—21no—
21#5048—21no—
22#5048—21no—
23C2×#809C411yescharacteristic, complemented
24C2×#808C411yescharacteristic, complemented
25C2×#807—21no—
26C2×#806—22no—
27C3×#328—22no—
28#4550—21no—
29#4529—21no—
30#4526—22no—
31#4470—22no—
32C2×#465—31no—
33#1996—31no—
34#1964—31no—
35#1808—31no—
36#1752—31no—
37#1520D311yescharacteristic, complemented
38#1519—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×D12C2×C411yescharacteristic
43C22×C3×C8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54C2×#217D412yes—
55C2×#215—21no—
56C2×#215—21no—
57C2×#215—42no—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3C2×C411yescharacteristic
65C2×C4×Dic3—21no—
66C22×C3⋊C8—42no—
67C22×C3⋊C8C2×C411yescharacteristic
68C22×C3⋊C8C2×C411yescharacteristic
69C22×C3⋊C8D411yescharacteristic
70C22×C3⋊C8D411yescharacteristic
71C22×C3⋊C8D411yescharacteristic
72C22×C3⋊C8D411yescharacteristic
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75C3×#95—21no—
76#835—21no—
77#835—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#815—42no—
81#809—21no—
82#809—21no—
83#808—21no—
84#808—21no—
85#807—42no—
86#806—44no—
87C22×SD16—31no—
88C23×C8—31no—
89C2×C22⋊Q8C2×D311yescharacteristic
90C2×#121—31no—
91#465—61no—
92#465—61no—
93#465—61no—
94#465—61no—
95C2×#107—61no—
96C2×#106—31no—
97C2×#103—31no—
98C2×#102—31no—
99C2×#99—62no—
100C2×C22⋊C8C2×D311yescharacteristic
101C2×#97—61no—
102#380—31no—
103#380C2×D311yescharacteristic
104C4×C22⋊C4—31no—
105#342—62no—
106#340—61no—
107#336—61no—
108#328—22no—
109#327—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—42no—
130C2×#67—42no—
131C22×Dic3—21no—
132C22×Dic3—22no—
133C22×Dic3—41no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—41no—
137C2×D12—41no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—42no—
150#217—22no—
151#217—22no—
152#217—42no—
153#215—42no—
154#215—42no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—41no—
160C4⋊Dic3—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8C22⋊C411yescharacteristic
170C2×C3⋊C8C22⋊C411yescharacteristic
171C2×C3⋊C8C22⋊C411yescharacteristic
172C2×C3⋊C8C22⋊C411yescharacteristic
173C22×Q8C4×D311yescharacteristic
174C22×D4—31no—
175C22×D4—61no—
176C23×C4C22×D311yescharacteristic
177C2×SD16—61no—
178C2×SD16—61no—
179C2×SD16—122no—
180C22×C8—21no—
181C22×C8—31no—
182C22×C8—31no—
183C22×C8—31no—
184C22×C8—31no—
185C22×C8—31no—
186C22×C8—31no—
187C22×C8—122no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192#121—61no—
193#121—61no—
194#121—61no—
195#121—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4C4×D311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—32no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C42—61no—
207#107—122no—
208#106—61no—
209#106—61no—
210#103—61no—
211#103—61no—
212#102—61no—
213#102—61no—
214#99—124no—
215C22⋊C8—21no—
216C22⋊C8#6712yes—
217#97—122no—
218#95—21no—
219#95—61no—
220#95—61no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12711yescharacteristic
226C23×C3AGL(1,ℤ/8)12yes—
227C23×C3#13712yes—
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—81no—
231C22×D3—81no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—81no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4C2×C22⋊C411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12411yescharacteristic
255#67—82no—
256#67—81no—
257#67—81no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265C2×Dic3—81no—
266D12—41no—
267D12—41no—
268D12—81no—
269C3×C8—42no—
270C3⋊C8—21no—
271C3⋊C8—21no—
272C3⋊C8—21no—
273C3⋊C8#9912yescomplemented
274C24—61no—
275C24D3×D411yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—122no—
286C2×D4—122no—
287C2×D4—121no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—61no—
294C22×C4—62no—
295C22×C4—121no—
296C22×C4C2×C4×D311yescharacteristic
297C22×C4#23511yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—122no—
303SD16—122no—
304C2×C8—22no—
305C2×C8—31no—
306C2×C8—31no—
307C2×C8—31no—
308C2×C8—31no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—62no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—121no—
326C4⋊C4—121no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—62no—
331C22⋊C4—62no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—122no—
335C42—121no—
336C42—121no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3#39011yescharacteristic
346C22×C3#44012yes—
347C22×C3#46812yes—
348C22×C3#49611yescharacteristic
349C22×C3#49711yescharacteristic
350C2×D3—81no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—162no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—81no—
358C3×C4—81no—
359C3×C4—161no—
360C3×C4C2×#9911yescharacteristic
361C3×C4#41011yescharacteristic
362Dic3—82no—
363Dic3—81no—
364Dic3—161no—
365C23—41no—
366C23—41no—
367C23—121no—
368C23—122no—
369C23—241no—
370C23—241no—
371C23#67⋊C411yescharacteristic
372C23#88411yescharacteristic
373C23#91412yes—
374C23#91812yes—
375C23#94011yescharacteristic
376Q8—41no—
377Q8—41no—
378Q8—81no—
379D4—121no—
380D4—121no—
381D4—241no—
382D4—241no—
383D4—241no—
384D4—242no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—241no—
407C2×C4—241no—
408C2×C4—241no—
409C2×C4D3×C22⋊C411yescharacteristic
410C2×C4#88511yescharacteristic
411C2×C4C4×#6711yescharacteristic
412C2×C4#93911yescharacteristic
413C8—32no—
414C8—42no—
415C8—61no—
416C8—61no—
417C8—61no—
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3#196411yescharacteristic
423C2×C3#197311yescharacteristic
424C2×C3#197611yescharacteristic
425D3—161no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—42no—
433C22—42no—
434C22—241no—
435C22—241no—
436C22—241no—
437C22—482no—
438C22#481212yes—
439C22#481712yes—
440C22#495711yescharacteristic
441C22#517611yescharacteristic
442C22#519011yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—81no—
447C4—81no—
448C4—161no—
449C4—242no—
450C4—241no—
451C4—481no—
452C4#474011yescharacteristic
453C4D3×#9911yescharacteristic
454C3#978611yescharacteristic, complemented, Sylow
455C2—21no—
456C2—22no—
457C2—22no—
458C2—21no—
459C2—481no—
460C2#7003011yescharacteristic
461C2#7026011yescharacteristic
462C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#390
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1520D3
hypocenterC3#9786
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
2. C3×#1520C2semidirect
3. #71701C2semidirect
4. #69948C2semidirect
7. #69512C2semidirect
10. C22×#215C4semidirect
15. C2×#835C4semidirect
23. C2×#809C4semidirect
24. C2×#808C4semidirect
37. #1520D3semidirect
273. C3⋊C8#99semidirect
454. C3#9786semidirect

Maximal subgroups

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

subgroupindex
C2×#50482
C3×#15202
#717012
#699482
#699442
#695282
#695122
#97863