Finite Group Databasebeta

← #10694717

Subgroups — #10694717

Order 768.

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

normal subgroups only

This group has 594 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×#5038C211yesmaximal, characteristic
2C3×#1964C211yesmaximal, characteristic, complemented
3#71675C211yesmaximal, characteristic, complemented
4#69511C211yesmaximal, characteristic, complemented
5#69509C211yesmaximal, characteristic, complemented
6#69456C211yesmaximal, characteristic, complemented
7#69447C211yesmaximal, characteristic, complemented
8#9785—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×#213C411yescharacteristic, complemented
11C2×#930C2211yescharacteristic
12C23×C3⋊C8C2211yescharacteristic
13C2×C3×#99—22no—
14C3×#380C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16C2×#832C411yescharacteristic, complemented
17#5212C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#5038—21no—
20#5038—21no—
21#5038—21no—
22#5038—21no—
23C2×#809C411yescharacteristic, complemented
24C2×#808—21no—
25C2×#807C411yescharacteristic, complemented
26C2×#806—22no—
27C2×#803—21no—
28#4537—22no—
29#4529—21no—
30#4524—21no—
31#4451—22no—
32C2×#466—31no—
33#1964—31no—
34#1964D311yescharacteristic, complemented
35#1807—31no—
36#1752—31no—
37#1519—31no—
38#1519—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12C2×C411yescharacteristic
44C3×#121—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D412yes—
53C2×C3×C42—21no—
54C2×#217—21no—
55C2×#213—21no—
56C2×#213—21no—
57C2×#213—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
65C22×C3⋊C8—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×#99—44no—
74C3×#95—21no—
75C3×#95—21no—
76#832—21no—
77#832—21no—
78C3⋊#95—21no—
79#830—21no—
80#830D412yes—
81#809—21no—
82#809—21no—
83#808—42no—
84#807—21no—
85#807—21no—
86#806—44no—
87#803—42no—
88C22×D8—31no—
89C23×C8—31no—
90C2×#121—31no—
91C2×#121C2×D311yescharacteristic
92#466—61no—
93#466—61no—
94#466—61no—
95#466—61no—
96C2×#107—31no—
97C2×#106—61no—
98C2×#102—31no—
99C2×#102—31no—
100C2×#99—22no—
101C2×#99—62no—
102C2×C22⋊C8—31no—
103C2×#97—61no—
104#380—31no—
105#380C2×D311yescharacteristic
106C4×C22⋊C4C2×D311yescharacteristic
107#340—61no—
108#336—61no—
109#327—62no—
110#327—62no—
111C24×C3—21no—
112C24×C3C2×D411yescharacteristic
113C23×D3—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—42no—
136C2×#67—42no—
137C22×Dic3—21no—
138C22×Dic3—41no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—41no—
142C2×D12—41no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—22no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—42no—
152C3×C42—41no—
153C3×C42—41no—
154#217—42no—
155#213—42no—
156#213—42no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—41no—
162C4⋊Dic3—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8C22⋊C411yescharacteristic
173C2×C3⋊C8C22⋊C411yescharacteristic
174C2×C3⋊C8C22⋊C411yescharacteristic
175C2×C3⋊C8C22⋊C411yescharacteristic
176C22×D4—21no—
177C22×D4—31no—
178C22×D4—61no—
179C22×D4C4×D311yescharacteristic
180C23×C4C22×D311yescharacteristic
181C2×D8—61no—
182C2×D8—61no—
183C2×D8—122no—
184C22×C8—31no—
185C22×C8—31no—
186C22×C8—31no—
187C22×C8—31no—
188C22×C8—31no—
189C22×C8—31no—
190C22×C8—61no—
191C22×C8—122no—
192#121—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4C4×D311yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4#6712yes—
209C2×C42—21no—
210#107—61no—
211#107—61no—
212#106—122no—
213#102—61no—
214#102—61no—
215#102—61no—
216#102—61no—
217#99—44no—
218#99—124no—
219C22⋊C8—32no—
220C22⋊C8—61no—
221#97—122no—
222#95—21no—
223#95—21no—
224#95—61no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—42no—
229C23×C3—81no—
230C23×C3—81no—
231C23×C3C4×D411yescharacteristic
232C23×C3#12111yescharacteristic
233C23×C3#12711yescharacteristic
234C23×C3AGL(1,ℤ/8)12yes—
235C23×C3AGL(1,ℤ/8)12yes—
236C22×D3—41no—
237C22×D3—42no—
238C22×D3—81no—
239C22×D3—81no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—81no—
243C3×D4—82no—
244C3×D4—81no—
245C3×D4—81no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—81no—
261C2×C3×C4—81no—
262C2×C3×C4—81no—
263C2×C3×C4C2×C22⋊C411yescharacteristic
264C2×C3×C4C4×D411yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12411yescharacteristic
267#67—82no—
268#67—81no—
269#67—81no—
270C2×Dic3—41no—
271C2×Dic3—42no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275D12—41no—
276D12—41no—
277D12—81no—
278C3⋊C8—21no—
279C3⋊C8—21no—
280C3⋊C8—21no—
281C3⋊C8—42no—
282C3⋊C8#9912yescomplemented
283C24—21no—
284C24—61no—
285C24C2×#6711yescharacteristic
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—121no—
297C2×D4—121no—
298C2×D4—122no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—122no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—61no—
308C22×C4—121no—
309C22×C4C2×C4×D311yescharacteristic
310C22×C4#23511yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4#23711yescharacteristic
315D8—122no—
316D8—122no—
317C2×C8—31no—
318C2×C8—31no—
319C2×C8—31no—
320C2×C8—31no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8—121no—
326C2×C8—121no—
327C2×C8—121no—
328C2×C8—121no—
329C2×C8—121no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—61no—
335C4⋊C4—61no—
336C4⋊C4—121no—
337C4⋊C4—121no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—42no—
343C22⋊C4—121no—
344C22⋊C4—121no—
345C22⋊C4—122no—
346C42—41no—
347C42—41no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—162no—
360C22×C3#39011yescharacteristic
361C22×C3#44212yes—
362C22×C3#46912yes—
363C22×C3#49611yescharacteristic
364C22×C3#49611yescharacteristic
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C2×D3—162no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—82no—
373C3×C4—81no—
374C3×C4—161no—
375C3×C4C2×#9911yescharacteristic
376C3×C4#41011yescharacteristic
377Dic3—81no—
378Dic3—161no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—81no—
384C23—81no—
385C23—121no—
386C23—122no—
387C23—241no—
388C23—241no—
389C23C4×#6711yescharacteristic
390C23#93212yes—
391C23#93911yescharacteristic
392C23#94011yescharacteristic
393C23#94912yes—
394D4—41no—
395D4—41no—
396D4—81no—
397D4—82no—
398D4—81no—
399D4—81no—
400D4—121no—
401D4—121no—
402D4—241no—
403D4—242no—
404D4—241no—
405D4—241no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—121no—
424C2×C4—122no—
425C2×C4—241no—
426C2×C4—241no—
427C2×C4—241no—
428C2×C4D3×C22⋊C411yescharacteristic
429C2×C4#67⋊C411yescharacteristic
430C2×C4#88411yescharacteristic
431C2×C4#88511yescharacteristic
432C8—32no—
433C8—61no—
434C8—61no—
435C8—61no—
436C8—122no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—21no—
440C2×C3—22no—
441C2×C3—161no—
442C2×C3#196411yescharacteristic
443C2×C3#197211yescharacteristic
444C2×C3#197611yescharacteristic
445D3—161no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—42no—
453C22—42no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—162no—
458C22—241no—
459C22—241no—
460C22—241no—
461C22—482no—
462C22#495711yescharacteristic
463C22#501412yes—
464C22#503912yes—
465C22#505811yescharacteristic
466C22#507411yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—82no—
471C4—81no—
472C4—161no—
473C4—241no—
474C4—481no—
475C4#474011yescharacteristic
476C4D3×#9911yescharacteristic
477C3#978511yescharacteristic, complemented, Sylow
478C2—22no—
479C2—21no—
480C2—21no—
481C2—22no—
482C2—161no—
483C2—481no—
484C2#6964811yescharacteristic
485C2#7003011yescharacteristic
486C2#7004211yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1964C2
socleC22×C3#390
prime radicalC2×#5038C2
cosocleC23×C3×C4C23
hypercenter#1964D3
hypocenterC3#9785
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. C3×#1964C2semidirect
3. #71675C2semidirect
4. #69511C2semidirect
5. #69509C2semidirect
6. #69456C2semidirect
7. #69447C2semidirect
10. C22×#213C4semidirect
16. C2×#832C4semidirect
23. C2×#809C4semidirect
25. C2×#807C4semidirect
34. #1964D3semidirect
282. C3⋊C8#99semidirect
477. C3#9785semidirect

Maximal subgroups

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

subgroupindex
C2×#50382
C3×#19642
#716752
#695112
#695092
#694562
#694472
#97853