Finite Group Databasebeta

← #10696529

Subgroups — #10696529

Order 768.

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

normal subgroups only

This group has 770 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×#2657C211yesmaximal, characteristic, complemented
2#83163C211yesmaximal, characteristic, complemented
3#82947C211yesmaximal, characteristic, complemented
4#71760C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10088—31nomaximal, Sylow
9C2×C3×#127—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×#124—21no—
12C2×C3×#124C2211yescharacteristic
13C2×#932—22no—
14#5477C2211yescharacteristic
15C3×#392—22no—
16C3×#392—22no—
17C3×#376C2211yescharacteristic
18#5227—21no—
19#5217—21no—
20#5076—21no—
21#5074—22no—
22#5073—21no—
23#5058—21no—
24#5057—22no—
25#5055—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4544—22no—
30#4540—22no—
31#4451—22no—
32#3213—31no—
33#3151—31no—
34#2657D311yescharacteristic, complemented
35#1555—31no—
36#1527—31no—
37#1505—31no—
38#1300—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—24no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C22×C3×D4D412yes—
45C2×#235—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—44no—
51C3×#124—21no—
52C3×#124—21no—
53C3×#124—21no—
54C3×#124—21no—
55C3×#124—44no—
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#215—22no—
62C2×#214—24no—
63#932—48no—
64C2×#213—24no—
65C2×#213—22no—
66#930—21no—
67#930—21no—
68C2×#208—21no—
69#875—21no—
70#874D411yescharacteristic, complemented
71C4×D12—21no—
72C4×D12—21no—
73#869D411yescharacteristic, complemented
74C3×#95—21no—
75C3×#95D412yes—
76#835—22no—
77#832—22no—
78#830—21no—
79#830—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yescomplemented
82#803—21no—
83#803—21no—
84#803D412yescomplemented
85C4×C3⋊C8—21no—
86C2×AGL(1,ℤ/8)—62no—
87#532—31no—
88C2×#127—21no—
89C2×#127C2×D311yescharacteristic
90C2×#124—21no—
91C2×#124C2×D311yescharacteristic
92#496—62no—
93#496—61no—
94#493—61no—
95#492—61no—
96#489—61no—
97#463—62no—
98#448—61no—
99#447—61no—
100#432—31no—
101#432—31no—
102C4×M16—31no—
103#392—22no—
104#392—22no—
105#376C2×D311yescharacteristic
106#331—62no—
107#329—62no—
108#327—62no—
109C24×C3—22no—
110C24×C3—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—42no—
114C2×C3×D4—21no—
115C2×C3×D4—22no—
116C2×C3×D4—21no—
117C2×C3×D4—22no—
118C2×C3×D4—44no—
119C2×C3×D4—42no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—48no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—82no—
128C2×C3×D4—84no—
129C2×C3×D4—84no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C2×#67—41no—
136#235—44no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×C4×D3—41no—
141C2×Dic6—21no—
142C3×C22⋊C4—48no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42—24no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#215—82no—
153#214—48no—
154#213—48no—
155#213—82no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160#208—24no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C2×#41—61no—
165C22×Q8#6711yescharacteristic
166C22×D4—24no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4#6711yescharacteristic
170C22×D4#6712yes—
171AGL(1,ℤ/8)—128no—
172C2×SD16—64no—
173C2×SD16—62no—
174C2×D8—64no—
175C2×D8—62no—
176C2×M16—61no—
177C4⋊Q8—31no—
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#127—31no—
183#127—44no—
184#124—21no—
185#124—21no—
186#124—21no—
187#124—21no—
188#124—44no—
189#124—61no—
190#121—61no—
191#121—61no—
192C4×D4—61no—
193C4×D4—61no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C42—21no—
198C2×C42C22×D311yescharacteristic
199C4⋊C8—32no—
200C4⋊C8—61no—
201#103—62no—
202#102—62no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205#97—32no—
206#97—61no—
207#97—61no—
208C4×C8—61no—
209#95—21no—
210#95#6712yes—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—44no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—44no—
219C23×C3—42no—
220C23×C3#12011yescharacteristic
221C22×D3—41no—
222C3×Q8—42no—
223C3×Q8—81no—
224C3×D4—44no—
225C3×D4—42no—
226C3×D4—82no—
227C3×D4—81no—
228C3×D4—82no—
229C3×D4—82no—
230C3×D4—82no—
231C3×D4—816no—
232C3×D4—82no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12711yescharacteristic
247#67—82no—
248C2×Dic3—41no—
249D12—42no—
250D12—42no—
251D12—81no—
252C4×D3—82no—
253Dic6—42no—
254C3⋊C8—24no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C24—22no—
258C24—21no—
259#41—124no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—42no—
263C2×Q8—61no—
264C2×D4—21no—
265C2×D4—22no—
266C2×D4—21no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—48no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—82no—
281C2×D4—84no—
282C2×D4—84no—
283C2×D4—121no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—121no—
287C22×C4C2×#6711yescharacteristic
288C22×C4C2×#6711yescharacteristic
289C22×C4C2×#6711yescharacteristic
290SD16—128no—
291SD16—242no—
292D8—128no—
293D8—242no—
294M16—64no—
295C2×C8—62no—
296C2×C8—62no—
297C2×C8—121no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C22⋊C4—48no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C42—24no—
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42D3×D411yescharacteristic
311C42#23711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—84no—
315C22×C3—84no—
316C22×C3—84no—
317C22×C3—81no—
318C22×C3—82no—
319C22×C3—82no—
320C22×C3—81no—
321C22×C3—82no—
322C22×C3—81no—
323C22×C3#39211yescharacteristic
324C22×C3#45312yes—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—81no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—44no—
334C3×C4—81no—
335C3×C4#45312yes—
336Dic3—81no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—44no—
342C23—41no—
343C23—42no—
344C23—44no—
345C23—42no—
346C23—121no—
347C23#95311yescharacteristic
348Q8—42no—
349Q8—81no—
350Q8—122no—
351D4—44no—
352D4—42no—
353D4—81no—
354D4—82no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—816no—
359D4—82no—
360D4—122no—
361D4—122no—
362D4—241no—
363D4—242no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—121no—
373C2×C4—242no—
374C2×C4#93711yescharacteristic
375C2×C4#93711yescharacteristic
376C2×C4#93811yescharacteristic
377C2×C4#93911yescharacteristic
378C2×C4#93911yescharacteristic
379C2×C4#94011yescharacteristic
380C8—64no—
381C8—122no—
382C8—241no—
383C2×C3—22no—
384C2×C3—161no—
385C2×C3—162no—
386C2×C3#200211yescharacteristic
387C2×C3#200211yescharacteristic
388C2×C3#200811yescharacteristic
389D3—161no—
390C22—21no—
391C22—21no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—81no—
396C22—82no—
397C22—82no—
398C22—81no—
399C22—82no—
400C22—81no—
401C22—241no—
402C22—241no—
403C22—241no—
404C22#522311yescharacteristic
405C22#524112yes—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—44no—
412C4—81no—
413C4—241no—
414C4#507712yes—
415C3#1008811yescharacteristic, complemented, Sylow
416C2—22no—
417C2—161no—
418C2—162no—
419C2—481no—
420C2#7176911yescharacteristic
421C2#7185611yescharacteristic
422C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2657C2
socleC22×C3#392
hypercenter#2657D3
hypocenterC3#10088
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2657

Complements

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

NH ≅ G/Ntype
1. C3×#2657C2semidirect
2. #83163C2semidirect
3. #82947C2semidirect
4. #71760C2semidirect
5. #71718C2semidirect
6. #69432C2semidirect
7. #67615C2semidirect
34. #2657D3semidirect
70. #874D4semidirect
73. #869D4semidirect
81. C4⋊(C3⋊C8)D4semidirect
84. #803D4semidirect
415. C3#10088semidirect

Maximal subgroups

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

subgroupindex
C3×#26572
#831632
#829472
#717602
#717182
#694322
#676152
#100883