Finite Group Databasebeta

← #10695511

Subgroups — #10695511

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#82678C211yesmaximal, characteristic, complemented
2#82627C211yesmaximal, characteristic
3#78624C211yesmaximal, characteristic, complemented
4C3×#1507C211yesmaximal, characteristic, complemented
5#69456C211yesmaximal, characteristic, complemented
6#69442C211yesmaximal, characteristic, complemented
7#67685C211yesmaximal, characteristic, complemented
8#9932—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5222—22no—
15#5221—21no—
16#5217—21no—
17#5212C2211yescharacteristic
18#5207C2211yescharacteristic
19#5131—21no—
20#5121—21no—
21#5039—22no—
22#5036—21no—
23#5034—21no—
24#4972—21no—
25#4968—21no—
26C2×#806—22no—
27#4953—21no—
28#4949—21no—
29#4737—22no—
30#4736—21no—
31#4735—21no—
32C3×#327—22no—
33#4523—22no—
34#3192—31no—
35#3094—31no—
36#2624—31no—
37#1964—31no—
38#1826—31no—
39#1507D311yescharacteristic, complemented
40#1305—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×D12D411yescharacteristic
46C2×C3×M16—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×#217—21no—
56C2×#217—22no—
57C2×#217—22no—
58C2×#217D412yes—
59C2×#213—22no—
60#930—21no—
61#930—21no—
62#930—21no—
63#930—21no—
64C2×(C2×D3)⋊C4—21no—
65#926—21no—
66#926—21no—
67#926—21no—
68#926—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4⋊Dic3D411yescharacteristic
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3—21no—
75C2×#208D412yes—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yes—
78C3×#95—21no—
79#832—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82#830—21no—
83#830—21no—
84#821—22no—
85#821—22no—
86#818—22no—
87#817—22no—
88#809—22no—
89#808—22no—
90#806—44no—
91C2×#121—31no—
92C2×#121C2×D311yescharacteristic
93C2×#117—31no—
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—62no—
115#327—22no—
116C24×C3—21no—
117C24×C3C2×D411yescharacteristic
118C23×D3—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—82no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C2×#67—42no—
135C2×#67—42no—
136C2×#67—41no—
137C2×#67—41no—
138C22×Dic3—21no—
139C22×Dic3—22no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12—42no—
145C3×M16—42no—
146C2×C3×C8—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—42no—
154#217—41no—
155#217—41no—
156#217—42no—
157#217—82no—
158#217—82no—
159#213—82no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—42no—
169Dic3⋊C4—82no—
170C4×Dic3—41no—
171C4×Dic3—41no—
172#208—42no—
173C2×C3⋊C8—22no—
174C22×D4—21no—
175C22×D4—31no—
176C22×D4—61no—
177C22×D4D1211yescharacteristic
178C23×C4C22×D311yescharacteristic
179C2×D8—62no—
180C2×M16—21no—
181C2×M16—32no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121—61no—
187#121—61no—
188#121—61no—
189#121—61no—
190#117—61no—
191#117—61no—
192#117—61no—
193#117—61no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—62no—
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—61no—
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—61no—
219C22⋊C8—61no—
220C22⋊C8#6712yes—
221#95—21no—
222#95—62no—
223#95—61no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3#12011yescharacteristic
234C23×C3#12111yescharacteristic
235C23×C3#12711yescharacteristic
236C22×D3—41no—
237C22×D3—42no—
238C22×D3—81no—
239C22×D3—81no—
240C3×D4—42no—
241C3×D4—81no—
242C3×D4—82no—
243C3×D4—82no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
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—81no—
261#67—81no—
262C2×Dic3—41no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273D12—42no—
274D12—42no—
275C3×C8—42no—
276C3⋊C8—42no—
277C24—21no—
278C24—61no—
279C24D3×D411yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—82no—
288C2×D4—122no—
289C2×D4—122no—
290C2×D4—122no—
291C2×D4—121no—
292C2×D4—121no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—61no—
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—61no—
314C4⋊C4—61no—
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—121no—
334C42—121no—
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—82no—
346C22×C3—82no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3#39211yescharacteristic
351C22×C3#45311yescharacteristic
352C22×C3#45411yescharacteristic
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#45812yes—
363Dic3—82no—
364Dic3—81no—
365Dic3—81no—
366Dic3—81no—
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—122no—
377C23—121no—
378C23—241no—
379C23—241no—
380C23#88211yescharacteristic
381C23#88411yescharacteristic
382C23#94011yescharacteristic
383D4—42no—
384D4—81no—
385D4—82no—
386D4—82no—
387D4—122no—
388D4—122no—
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—42no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—121no—
407C2×C4—122no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—121no—
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—161no—
425C2×C3#200611yescharacteristic
426C2×C3#200811yescharacteristic
427C2×C3#202211yescharacteristic
428D3—161no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—82no—
440C22—82no—
441C22—81no—
442C22—81no—
443C22—81no—
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—242no—
457C4—241no—
458C4—241no—
459C4—241no—
460C4#474212yes—
461C3#993211yescharacteristic, complemented, Sylow
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—161no—
467C2—481no—
468C2#7003111yescharacteristic
469C2#7033111yescharacteristic
470C2#7039211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#392
prime radical#82627C2
hypercenter#1507D3
hypocenterC3#9932
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
1. #82678C2semidirect
3. #78624C2semidirect
4. C3×#1507C2semidirect
5. #69456C2semidirect
6. #69442C2semidirect
7. #67685C2semidirect
39. #1507D3semidirect
461. C3#9932semidirect

Maximal subgroups

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

subgroupindex
#826782
#826272
#786242
C3×#15072
#694562
#694422
#676852
#99323