Finite Group Databasebeta

← #10695572

Subgroups — #10695572

Order 768.

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

normal subgroups only

This group has 600 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×#2484C211yesmaximal, characteristic, complemented
2#82217C211yesmaximal, characteristic
3#82182C211yesmaximal, characteristic
4#71590C211yesmaximal, characteristic, complemented
5#71552C211yesmaximal, characteristic, complemented
6#69531C211yesmaximal, characteristic
7#67628C211yesmaximal, characteristic, complemented
8#9941—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#400—21no—
13C3×#398—21no—
14C3×#387—22no—
15C3×#385—21no—
16C3×#381—21no—
17C3×#381C2211yescharacteristic
18#5213—21no—
19#5213—21no—
20C4×#217C2211yescharacteristic
21#5207C2211yescharacteristic
22#5207C2211yescharacteristic
23#5003—22no—
24#5001—21no—
25#4995—21no—
26#4991—21no—
27#4990—22no—
28#4986—21no—
29#4968—21no—
30#4967—21no—
31#4532—22no—
32#4528—22no—
33#4451—22no—
34#3094—31no—
35#2591—31no—
36#2484D311yescharacteristic, complemented
37#1826—31no—
38#1510—31no—
39#1463—31no—
40#1284—31no—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×D12D411yescharacteristic
46C3×#117—21no—
47C3×#117—21no—
48C3×#117—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42—21no—
62C2×#217—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66#930—21no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4×Dic3D412yes—
72C2×#208—21no—
73C2×#208—21no—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—22no—
77C3×#95—21no—
78C3×#95D412yes—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830D412yes—
86#809—22no—
87#809—22no—
88#808—22no—
89#807—22no—
90C2×#121—31no—
91C2×#117C2×D311yescharacteristic
92C22×C4⋊C4C2×D311yescharacteristic
93#482—62no—
94#481—61no—
95#480—61no—
96#428—61no—
97#418—61no—
98#407—31no—
99#407—31no—
100#400—21no—
101#400—61no—
102#398—21no—
103#394—61no—
104#388—62no—
105#387—22no—
106#386—61no—
107#386—61no—
108#385—21no—
109#381—21no—
110#381C2×D311yescharacteristic
111C4×C22⋊C4—31no—
112#339—62no—
113#327—62no—
114#324—62no—
115C24×C3C2×D411yescharacteristic
116C23×D3—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C2×#67—44no—
136C2×#67—41no—
137C2×#67—41no—
138C22×Dic3—21no—
139C22×Dic3—22no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—42no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—44no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—42no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C42—41no—
156C3×C42—41no—
157#217—42no—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—82no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4×Dic3—42no—
165#208—42no—
166#208—42no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C22×D4—31no—
170C22×D4—61no—
171C23×C4#6711yescharacteristic
172C23×C4#6711yescharacteristic
173C23×C4C22×D311yescharacteristic
174C2×M16—61no—
175C2×M16—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179#121—61no—
180#117—21no—
181#117—21no—
182#117—21no—
183#117—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4#6711yescharacteristic
193C2×C4⋊C4#6711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—62no—
200C2×C42—21no—
201C2×C42—32no—
202#107—62no—
203#106—62no—
204#102—62no—
205#102—62no—
206C22⋊C8—32no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—62no—
215#95—62no—
216#95—61no—
217#95#6712yes—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3#12011yescharacteristic
222C23×C3#12011yescharacteristic
223C23×C3#12111yescharacteristic
224C22×D3—42no—
225C22×D3—42no—
226C22×D3—41no—
227C22×D3—42no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—81no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256#67—82no—
257#67—82no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263D12—42no—
264D12—81no—
265C3⋊C8—42no—
266C3⋊C8—81no—
267C24—61no—
268C24C2×#6711yescharacteristic
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—122no—
272C2×D4—124no—
273C2×D4—121no—
274C2×D4—121no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4#23711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295M16—122no—
296M16—122no—
297C2×C8—62no—
298C2×C8—121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—44no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—124no—
314C22⋊C4—122no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—242no—
318C42—41no—
319C42—41no—
320C42—122no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—44no—
329C22×C3#39211yescharacteristic
330C22×C3#45412yes—
331C22×C3#45812yes—
332C2×D3—84no—
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—81no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—42no—
341C3×C4—82no—
342C3×C4—81no—
343C3×C4—81no—
344C3×C4—81no—
345Dic3—82no—
346Dic3—81no—
347C23—22no—
348C23—22no—
349C23—41no—
350C23—122no—
351C23—122no—
352C23—121no—
353C23—122no—
354C23#93711yescharacteristic
355C23#93911yescharacteristic
356C23#95311yescharacteristic
357D4—122no—
358D4—241no—
359D4—242no—
360D4—242no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—81no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—121no—
389C2×C4—122no—
390C2×C4#93711yescharacteristic
391C2×C4#93811yescharacteristic
392C2×C4#93911yescharacteristic
393C2×C4#94011yescharacteristic
394C8—122no—
395C8—241no—
396C2×C3—22no—
397C2×C3—22no—
398C2×C3—41no—
399C2×C3#200311yescharacteristic
400C2×C3#201011yescharacteristic
401C2×C3#202211yescharacteristic
402D3—161no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—44no—
411C22—244no—
412C22—241no—
413C22—241no—
414C22—241no—
415C22#505312yes—
416C22#522311yescharacteristic
417C22#524512yes—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—82no—
424C4—81no—
425C4—81no—
426C4—81no—
427C4—242no—
428C4—241no—
429C3#994111yescharacteristic, complemented, Sylow
430C2—22no—
431C2—22no—
432C2—41no—
433C2—481no—
434C2#7170611yescharacteristic
435C2#7171511yescharacteristic
436C2#7186711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2484C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#2484D3
hypocenterC3#9941
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
1. C3×#2484C2semidirect
4. #71590C2semidirect
5. #71552C2semidirect
7. #67628C2semidirect
36. #2484D3semidirect
429. C3#9941semidirect

Maximal subgroups

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

subgroupindex
C3×#24842
#822172
#821822
#715902
#715522
#695312
#676282
#99413