Finite Group Databasebeta

← #10695973

Subgroups — #10695973

Order 768.

692 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2540C211yesmaximal, characteristic, complemented
2#82947C211yesmaximal, characteristic, complemented
3#82310C211yesmaximal, characteristic, complemented
4#71718C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#69492C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#10004—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C3×C22⋊Q8—21no—
11C2×C3×#121—21no—
12C2×C3×C4×D4C2211yescharacteristic
13#5477C2211yescharacteristic
14C3×#399—21no—
15C3×#392—21no—
16C3×#390—22no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5227—21no—
19#5217—21no—
20#5217—21no—
21#5058—21no—
22#5057—22no—
23#5055—21no—
24#5015—21no—
25#5014—22no—
26#5008—21no—
27#4968—21no—
28C2×#805—22no—
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31C4×#208C2211yescharacteristic
32#4540—22no—
33D4⋊(C3⋊C8)—22no—
34#3213—31no—
35#3109—31no—
36#2540D311yescharacteristic, complemented
37#1980—31no—
38#1527—31no—
39#1490—31no—
40#1285—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4—22no—
43C22×C3×D4D411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C23×C3×C4—21no—
46C2×#235—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×C22⋊Q8—44no—
52C3×#121—44no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C42D412yes—
66C2×C3×C42C2311yescharacteristic
67C2×#215—22no—
68C2×#213—22no—
69C2×#213—22no—
70#930—21no—
71#930—21no—
72C2×#208—21no—
73#875—21no—
74#874D411yescharacteristic, complemented
75C4×D12—21no—
76C4×D12—21no—
77#869D411yescharacteristic, complemented
78C3×#95—21no—
79#835—22no—
80#832—22no—
81#832—22no—
82#830—21no—
83#830—21no—
84#809—22no—
85#808—22no—
86#805—44no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)D412yescomplemented
89#803—21no—
90#803—21no—
91#803D412yescomplemented
92C4×C3⋊C8—21no—
93#532—31no—
94C2×#124C2×D311yescharacteristic
95C2×C22⋊Q8—21no—
96C2×#121—21no—
97C2×C4×D4C2×D311yescharacteristic
98#496—61no—
99#489—61no—
100#481—61no—
101#476—61no—
102#463—62no—
103#459—61no—
104#448—61no—
105#447—61no—
106#447—61no—
107#442—62no—
108#432—31no—
109#432—31no—
110C2×#104—62no—
111C4×M16—31no—
112#399—21no—
113#392—21no—
114#390—22no—
115C4×C4⋊C4C2×D311yescharacteristic
116#329—62no—
117D4⋊C8—62no—
118C24×C3—21no—
119C24×C3—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—42no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—42no—
128C2×C3×D4—44no—
129C2×C3×D4—42no—
130C2×C3×D4—44no—
131C2×C3×D4—82no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C2×#67—41no—
143#235—42no—
144#235—42no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—21no—
148C2×C4×D3—41no—
149C2×Dic6—21no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—44no—
154C3×C22⋊C4—44no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C42—42no—
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4111yescharacteristic
167#215—82no—
168#213—44no—
169#213—82no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—41no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174#208—22no—
175#208—41no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—41no—
179C2×#41—61no—
180C22×Q8#6711yescharacteristic
181C22×D4—22no—
182C22×D4#6711yescharacteristic
183C22×D4#6711yescharacteristic
184C23×C4—21no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×D8—62no—
188C2×M16—61no—
189C4⋊Q8—31no—
190#127—31no—
191#124—21no—
192#124—21no—
193#124—21no—
194#124—21no—
195#124—61no—
196C22⋊Q8—44no—
197#121—44no—
198#121—61no—
199#121—61no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—61no—
205C4×D4—61no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4#6711yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42#6712yes—
215C2×C42C22×D311yescharacteristic
216#106—62no—
217C4⋊C8—32no—
218C4⋊C8—61no—
219#104—124no—
220#103—62no—
221#102—62no—
222#102—62no—
223#102—62no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226#97—32no—
227#97—61no—
228#97—61no—
229C4×C8—61no—
230#95—21no—
231C23×C3—22no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3#12011yescharacteristic
244C22×D3—41no—
245C3×Q8—42no—
246C3×Q8—81no—
247C3×D4—22no—
248C3×D4—22no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—42no—
252C3×D4—81no—
253C3×D4—88no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—22no—
257C2×C3×C4—24no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—82no—
267C2×C3×C4—82no—
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12011yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12711yescharacteristic
274#67—81no—
275#67—81no—
276C2×Dic3—41no—
277D12—42no—
278D12—42no—
279D12—42no—
280C4×D3—82no—
281Dic6—42no—
282C3⋊C8—42no—
283C3⋊C8—42no—
284C3⋊C8—42no—
285C24—21no—
286C24—21no—
287#41—122no—
288#41—122no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×Q8—61no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—42no—
298C2×D4—44no—
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—61no—
302C2×D4—61no—
303C2×D4—61no—
304C2×D4—82no—
305C2×D4—121no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—121no—
314C22×C4C2×#6711yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C22×C4C2×#6711yescharacteristic
317SD16—242no—
318D8—124no—
319D8—242no—
320M16—62no—
321M16—121no—
322C2×C8—62no—
323C2×C8—62no—
324C2×C8—121no—
325C4⋊C4—24no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—44no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C42—42no—
342C42D3×D411yescharacteristic
343C42D3×D411yescharacteristic
344C42D3×D411yescharacteristic
345C42#23711yescharacteristic
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3#39211yescharacteristic
361C22×C3#45311yescharacteristic
362C22×C3#45411yescharacteristic
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—44no—
372C3×C4—81no—
373C3×C4—81no—
374C3×C4#45312yes—
375Dic3—81no—
376C23—22no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—121no—
389C23#95311yescharacteristic
390Q8—42no—
391Q8—81no—
392Q8—122no—
393D4—22no—
394D4—22no—
395D4—41no—
396D4—41no—
397D4—42no—
398D4—81no—
399D4—88no—
400D4—122no—
401D4—122no—
402D4—122no—
403D4—241no—
404D4—241no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—121no—
420C2×C4—242no—
421C2×C4#93711yescharacteristic
422C2×C4#93711yescharacteristic
423C2×C4#93811yescharacteristic
424C2×C4#93911yescharacteristic
425C2×C4#93911yescharacteristic
426C2×C4#94011yescharacteristic
427C8—122no—
428C8—122no—
429C8—122no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3—161no—
434C2×C3#200211yescharacteristic
435C2×C3#200611yescharacteristic
436C2×C3#201011yescharacteristic
437D3—161no—
438C22—21no—
439C22—21no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—81no—
444C22—81no—
445C22—82no—
446C22—82no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22#522311yescharacteristic
456C22#524111yescharacteristic
457C22#524511yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—44no—
464C4—81no—
465C4—81no—
466C4—241no—
467C4#507712yes—
468C3#1000411yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—82no—
472C2—161no—
473C2—481no—
474C2#7171511yescharacteristic
475C2#7176911yescharacteristic
476C2#7187911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2540C2
socleC22×C3#392
hypercenter#2540D3
hypocenterC3#10004
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2540

Complements

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

NH ≅ G/Ntype
1. C3×#2540C2semidirect
2. #82947C2semidirect
3. #82310C2semidirect
4. #71718C2semidirect
5. #71599C2semidirect
6. #69492C2semidirect
7. #67621C2semidirect
36. #2540D3semidirect
74. #874D4semidirect
77. #869D4semidirect
88. C4⋊(C3⋊C8)D4semidirect
91. #803D4semidirect
468. C3#10004semidirect

Maximal subgroups

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

subgroupindex
C3×#25402
#829472
#823102
#717182
#715992
#694922
#676212
#100043