Finite Group Databasebeta

← #10695938

Subgroups — #10695938

Order 768.

554 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C3×#2581C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic
3#82441C211yesmaximal, characteristic
4#71622C211yesmaximal, characteristic, complemented
5#71621C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic, complemented
8#9999—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#942—21no—
12#5477C2211yescharacteristic
13C3×#400—21no—
14C3×#395—21no—
15C3×#391—21no—
16C3×#389—21no—
17C3×#387—22no—
18C3×#382C2211yescharacteristic
19#5227—21no—
20#5031—21no—
21#5028—22no—
22#5027—21no—
23#5027—21no—
24C4×#215—22no—
25#5020—21no—
26#4968—21no—
27#4966—21no—
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4535—22no—
32#4535—22no—
33#4451—22no—
34#3103—31no—
35#3037—31no—
36#2581D311yescharacteristic, complemented
37#1517—31no—
38#1487—31no—
39#1479—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C2×#235—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#216—22no—
62#942—42no—
63#942—42no—
64C2×#215—22no—
65C2×#215—22no—
66#930—21no—
67#930—21no—
68C2×#208—21no—
69#875—21no—
70#874D411yescharacteristic
71C4×D12—21no—
72C4×D12—21no—
73#869D411yescharacteristic
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—22no—
77C3×#95—21no—
78C3×#95D412yes—
79#835—22no—
80#830—21no—
81#830—21no—
82#810—22no—
83#809—22no—
84#809—22no—
85#808—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yes—
90C4×C3⋊C8—21no—
91C4×C3⋊C8D412yes—
92C2×#137—61no—
93#532—31no—
94C2×C4×Q8C2×D311yescharacteristic
95C2×C4×Q8C2×D311yescharacteristic
96#481—61no—
97#475—61no—
98#464—62no—
99#462—61no—
100#460—61no—
101#460—61no—
102#448—61no—
103C4×SD16—62no—
104#431—31no—
105#431—31no—
106#423—31no—
107#417—61no—
108#400—21no—
109#395—21no—
110#391—21no—
111#389—21no—
112#387—22no—
113#382C2×D311yescharacteristic
114#327—62no—
115Q8⋊C8—62no—
116Q8⋊C8—62no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—42no—
125C2×C3×Q8SD1612yescomplemented
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C2×#67—41no—
135#235—41no—
136#235—41no—
137#235—41no—
138#235—41no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×C4×D3—41no—
143C2×Dic6—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—44no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42—41no—
158C3×C42—41no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163#216—44no—
164#215—44no—
165#215—44no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170#208—41no—
171#208—41no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C2×#41—61no—
176C22×Q8#6711yescharacteristic
177C22×Q8#6711yescharacteristic
178#137—122no—
179#137—122no—
180C2×Q16—62no—
181C2×SD16—62no—
182C2×SD16—62no—
183C2×M16—61no—
184C4⋊Q8—31no—
185#127—31no—
186#124—61no—
187#121—61no—
188#121—61no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×D4—61no—
198C4×D4—61no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4#6711yescharacteristic
204C2×C4⋊C4#6711yescharacteristic
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C22×D311yescharacteristic
208#106—62no—
209C4⋊C8—32no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212C4⋊C8—61no—
213#103—62no—
214#103—62no—
215#102—62no—
216#102—62no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C4×C8—32no—
220C4×C8—61no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95#6712yes—
226C23×C3#12011yescharacteristic
227C22×D3—41no—
228C3×Q8—22no—
229C3×Q8—22no—
230C3×Q8—22no—
231C3×Q8—22no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—41no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12711yescharacteristic
257C2×C3×C4C2×SD1611yescharacteristic
258C2×C3×C4#13511yescharacteristic
259C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
260C2×C3×C4#13711yescharacteristic
261#67—81no—
262#67—81no—
263C2×Dic3—41no—
264D12—42no—
265D12—41no—
266D12—41no—
267D12—81no—
268C4×D3—81no—
269C4×D3—81no—
270Dic6—41no—
271Dic6—41no—
272C3⋊C8—42no—
273C3⋊C8—42no—
274C3⋊C8—81no—
275#41—121no—
276#41—121no—
277#41—121no—
278#41—121no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—42no—
287C2×Q8—61no—
288C2×Q8#21512yescomplemented
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—121no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—121no—
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—124no—
303SD16—124no—
304SD16—124no—
305M16—121no—
306M16—121no—
307C2×C8—62no—
308C2×C8—62no—
309C2×C8—121no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—44no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42D3×D411yescharacteristic
330C42D3×D411yescharacteristic
331C42D3×D411yescharacteristic
332C42#23711yescharacteristic
333C22×C3—21no—
334C22×C3—21no—
335C22×C3#39211yescharacteristic
336C22×C3#44811yescharacteristic
337C22×C3#44911yescharacteristic
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—81no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—42no—
346C3×C4—42no—
347C3×C4—82no—
348C3×C4—81no—
349C3×C4—81no—
350C3×C4#45011yescharacteristic
351C3×C4#45211yescharacteristic
352C3×C4#46511yescharacteristic
353C3×C4#46811yescharacteristic
354Dic3—81no—
355C23—121no—
356C23#95311yescharacteristic
357Q8—22no—
358Q8—22no—
359Q8—22no—
360Q8—22no—
361Q8—41no—
362Q8—41no—
363Q8—41no—
364Q8—41no—
365Q8—121no—
366Q8—121no—
367D4—122no—
368D4—121no—
369D4—121no—
370D4—241no—
371D4—241no—
372D4—241no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4#93711yescharacteristic
392C2×C4#93711yescharacteristic
393C2×C4#93811yescharacteristic
394C2×C4#93911yescharacteristic
395C2×C4#93911yescharacteristic
396C2×C4#94011yescharacteristic
397C2×C4C2×#21511yescharacteristic
398C2×C4#94211yescharacteristic
399C2×C4#94911yescharacteristic
400C2×C4#95011yescharacteristic
401C8—122no—
402C8—122no—
403C8—241no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3#200011yescharacteristic
407C2×C3#200911yescharacteristic
408C2×C3#201011yescharacteristic
409D3—161no—
410C22—21no—
411C22—21no—
412C22—241no—
413C22—241no—
414C22—241no—
415C22#522311yescharacteristic
416C22#522711yescharacteristic
417C22#523811yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—42no—
424C4—82no—
425C4—81no—
426C4—81no—
427C4—241no—
428C4#504611yescharacteristic
429C4#504711yescharacteristic
430C4#504811yescharacteristic
431C4#504911yescharacteristic
432C3#999911yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—481no—
436C2#7169911yescharacteristic
437C2#7171511yescharacteristic
438C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2581C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2581D3
hypocenterC3#9999
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
1. C3×#2581C2semidirect
4. #71622C2semidirect
5. #71621C2semidirect
7. #67631C2semidirect
36. #2581D3semidirect
125. C2×C3×Q8SD16semidirect
288. C2×Q8#215semidirect
432. C3#9999semidirect

Maximal subgroups

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

subgroupindex
C3×#25812
#824422
#824412
#716222
#716212
#695452
#676312
#99993