Finite Group Databasebeta

← #10695510

Subgroups — #10695510

Order 768.

610 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 83 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#82288C211yesmaximal, characteristic, complemented
2#82217C211yesmaximal, characteristic, complemented
3#78641C211yesmaximal, characteristic
4C3×#1826C211yesmaximal, characteristic, complemented
5#69456C211yesmaximal, characteristic, complemented
6#69444C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9932—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5212C2211yescharacteristic
15#5207C2211yescharacteristic
16#5135—22no—
17#5003—22no—
18#5001—21no—
19#4995—21no—
20#4993—21no—
21#4992—21no—
22#4968—21no—
23#4967—21no—
24#4959—22no—
25C2×#806—22no—
26#4939—21no—
27#4928—21no—
28#4735—21no—
29#4735—21no—
30#4732—21no—
31#4732—21no—
32C3×#339—22no—
33#4451—22no—
34#3192—31no—
35#3094—31no—
36#2624—31no—
37#1964—31no—
38#1826D311yescharacteristic, complemented
39#1507—31no—
40#1305—31no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×#67—21no—
44C22×D12D411yescharacteristic
45C22×D12D411yescharacteristic
46C2×C3×M16D412yes—
47C3×#117—21no—
48C3×#117—21no—
49C3×#117—21no—
50C3×#117—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57C2×#217D412yes—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4⋊Dic3D411yescharacteristic
73C2×Dic3⋊C4—22no—
74C2×#208—21no—
75C2×D24—22no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×#95—21no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81#830—21no—
82#830D412yes—
83#821—22no—
84#821—22no—
85#817—22no—
86#809—22no—
87#809—22no—
88#808—22no—
89#807—22no—
90#806—44no—
91C2×#121—31no—
92C2×#121—31no—
93C2×#117C2×D311yescharacteristic
94#482—62no—
95#481—61no—
96#481—61no—
97#481—61no—
98#480—61no—
99#469—62no—
100#447—61no—
101#447—61no—
102#428—61no—
103#418—61no—
104C2×#99—62no—
105#407—31no—
106#407C2×D311yescharacteristic
107#399—61no—
108#397—61no—
109#394—62no—
110#390—61no—
111#389—61no—
112#380—31no—
113C4×C22⋊C4C2×D311yescharacteristic
114#339—22no—
115#327—62no—
116C24×C3C2×D411yescharacteristic
117C23×D3—21no—
118C23×D3—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—42no—
130C2×#67—42no—
131C2×#67—41no—
132C2×#67—41no—
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—82no—
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12—42no—
144C2×D12—42no—
145C3×M16—42no—
146C2×C3×C8—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C42—42no—
156C3×C42—41no—
157C3×C42—41no—
158#217—42no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—82no—
164(C2×D3)⋊C4—82no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—42no—
170Dic3⋊C4—82no—
171#208—42no—
172C2×C3⋊C8—22no—
173D24—82no—
174C22×D4—31no—
175C22×D4—31no—
176C22×D4—61no—
177C22×D4—61no—
178C23×C4C22×D311yescharacteristic
179C2×D8—62no—
180C2×M16—61no—
181C2×M16#6712yes—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186#121—61no—
187#121—61no—
188#121—61no—
189#121—61no—
190#117—21no—
191#117—21no—
192#117—21no—
193#117—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4D1211yescharacteristic
199C2×C4⋊C4D1211yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—32no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—62no—
207C2×C42—21no—
208C2×C42—21no—
209#107—62no—
210#106—62no—
211#106—62no—
212#102—62no—
213#102—62no—
214#102—62no—
215#102—62no—
216#99—124no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—32no—
220C22⋊C8—61no—
221#95—21no—
222#95—62no—
223#95—61no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3#12011yescharacteristic
229C23×C3#12111yescharacteristic
230C23×C3#12711yescharacteristic
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—42no—
238C22×D3—81no—
239C22×D3—81no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259#67—82no—
260#67—82no—
261#67—82no—
262#67—81no—
263#67—81no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271D12—42no—
272D12—42no—
273D12—42no—
274D12—81no—
275C3×C8—42no—
276C3⋊C8—42no—
277C24—61no—
278C24—61no—
279C24D3×D411yescharacteristic
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—122no—
285C2×D4—122no—
286C2×D4—122no—
287C2×D4—122no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—242no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—62no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4C2×D1211yescharacteristic
301C22×C4#23511yescharacteristic
302C22×C4D3×D411yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4C2×#6711yescharacteristic
306D8—242no—
307M16—42no—
308M16—122no—
309C2×C8—22no—
310C2×C8—62no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—122no—
320C4⋊C4—242no—
321C22⋊C4—22no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C22⋊C4—122no—
330C22⋊C4—242no—
331C22⋊C4—242no—
332C42—42no—
333C42—41no—
334C42—41no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3#39211yescharacteristic
346C22×C3#45311yescharacteristic
347C22×C3#45411yescharacteristic
348C2×D3—82no—
349C2×D3—82no—
350C2×D3—81no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—81no—
356C2×D3—162no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—82no—
361C3×C4—81no—
362C3×C4—81no—
363C3×C4#45812yes—
364Dic3—82no—
365Dic3—81no—
366Dic3—81no—
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—122no—
372C23—122no—
373C23—122no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23—121no—
378C23—241no—
379C23—241no—
380C23#88211yescharacteristic
381C23#88411yescharacteristic
382C23#94011yescharacteristic
383D4—122no—
384D4—122no—
385D4—122no—
386D4—241no—
387D4—242no—
388D4—242no—
389D4—242no—
390D4—241no—
391D4—241no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4#88211yescharacteristic
415C2×C4#88411yescharacteristic
416C2×C4#93011yescharacteristic
417C2×C4#93711yescharacteristic
418C8—42no—
419C8—122no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3#200611yescharacteristic
425C2×C3#200811yescharacteristic
426C2×C3#202211yescharacteristic
427D3—161no—
428D3—161no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—242no—
440C22—242no—
441C22—241no—
442C22—241no—
443C22—241no—
444C22—241no—
445C22—241no—
446C22—241no—
447C22—482no—
448C22#482211yescharacteristic
449C22#482711yescharacteristic
450C22#496011yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—82no—
455C4—81no—
456C4—81no—
457C4—242no—
458C4—241no—
459C4—241no—
460C4#505312yes—
461C3#993211yescharacteristic, complemented, Sylow
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—481no—
467C2—481no—
468C2#6953911yescharacteristic
469C2#7039211yescharacteristic
470C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#392
prime radical#78641C2
hypercenter#1826D3
hypocenterC3#9932
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
1. #82288C2semidirect
2. #82217C2semidirect
4. C3×#1826C2semidirect
5. #69456C2semidirect
6. #69444C2semidirect
7. #67684C2semidirect
38. #1826D3semidirect
461. C3#9932semidirect

Maximal subgroups

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

subgroupindex
#822882
#822172
#786412
C3×#18262
#694562
#694442
#676842
#99323