Finite Group Databasebeta

← #10695570

Subgroups — #10695570

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×#2591C211yesmaximal, characteristic
2#82616C211yesmaximal, characteristic
3#82176C211yesmaximal, characteristic, complemented
4#71674C211yesmaximal, characteristic
5#71552C211yesmaximal, characteristic, complemented
6#69442C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9941—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#400—21no—
13C3×#394—21no—
14C3×#388—22no—
15C3×#386—21no—
16C3×#386—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5215—21no—
19#5211—21no—
20#5211C2211yescharacteristic
21#5210—21no—
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5035—22no—
25#5034—21no—
26#5033—21no—
27#4986—21no—
28#4981—21no—
29#4978—22no—
30#4964—21no—
31#4537—22no—
32#4528—22no—
33#4523—22no—
34#3094—31no—
35#2591D311yescharacteristic
36#2484—31no—
37#1826—31no—
38#1510—31no—
39#1463—31no—
40#1284—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C2×C3×C4⋊C4—21no—
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⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C42D412yes—
62C2×#217—21no—
63C2×#217—21no—
64#926—21no—
65#926—21no—
66#926—21no—
67#926—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×Dic3⋊C4—22no—
71C2×C4×Dic3—21no—
72C2×#208—21no—
73C2×#208—21no—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—21no—
77#832—22no—
78#832—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86#830—21no—
87#830D412yes—
88#808—22no—
89#807—22no—
90C2×#121C2×D311yescharacteristic
91C2×#117—31no—
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—61no—
103#394—21no—
104#388—22no—
105#387—62no—
106#386—21no—
107#386—21no—
108#385—61no—
109#381—31no—
110#381—61no—
111C4×C22⋊C4C2×D311yescharacteristic
112#339—62no—
113#327—62no—
114#324—62no—
115C24×C3—21no—
116C24×C3C2×D411yescharacteristic
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C2×C3×D4—44no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×Dic3—22no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C22⋊C4—44no—
150C3×C22⋊C4—42no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—82no—
154C3×C42—42no—
155#217—42no—
156#217—41no—
157#217—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162Dic3⋊C4—44no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165#208—42no—
166#208—42no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C22×D4—21no—
170C22×D4#6711yescharacteristic
171C23×C4#6711yescharacteristic
172C23×C4#6711yescharacteristic
173C23×C4C22×D311yescharacteristic
174C2×M16—61no—
175C2×M16—61no—
176#121—21no—
177#121—21no—
178#121—21no—
179#121—21no—
180#117—61no—
181#117—61no—
182#117—61no—
183#117—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4#6711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C42—61no—
201C2×C42#6712yes—
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—21no—
213#95—32no—
214#95—62no—
215#95—62no—
216#95—62no—
217#95—61no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3#12011yescharacteristic
226C23×C3#12011yescharacteristic
227C23×C3#12111yescharacteristic
228C3×D4—42no—
229C3×D4—81no—
230C3×D4—82no—
231C3×D4—82no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
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—41no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—81no—
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12711yescharacteristic
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C3⋊C8—42no—
266C3⋊C8—81no—
267C24—21no—
268C24C2×#6711yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—61no—
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—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—124no—
310C22⋊C4—44no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—82no—
315C22⋊C4—122no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C42—42no—
319C42—121no—
320C42—121no—
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—84no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3#39211yescharacteristic
334C22×C3#45412yes—
335C22×C3#45812yes—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—42no—
341C3×C4—82no—
342C3×C4—81no—
343C3×C4—81no—
344Dic3—82no—
345Dic3—81no—
346Dic3—81no—
347C23—22no—
348C23—22no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—42no—
354C23#93711yescharacteristic
355C23#93911yescharacteristic
356C23#95311yescharacteristic
357D4—42no—
358D4—81no—
359D4—82no—
360D4—82no—
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—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—122no—
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—161no—
400C2×C3#200311yescharacteristic
401C2×C3#201011yescharacteristic
402C2×C3#202211yescharacteristic
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—44no—
411C22—84no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22#506212yes—
416C22#522311yescharacteristic
417C22#524312yes—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—82no—
424C4—81no—
425C4—81no—
426C4—242no—
427C4—241no—
428C4—241no—
429C3#994111yescharacteristic, complemented, Sylow
430C2—22no—
431C2—22no—
432C2—41no—
433C2—161no—
434C2#7171511yescharacteristic
435C2#7173011yescharacteristic
436C2#7184911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2591C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#2591D3
hypocenterC3#9941
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
3. #82176C2semidirect
5. #71552C2semidirect
6. #69442C2semidirect
7. #67616C2semidirect
429. C3#9941semidirect

Maximal subgroups

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

subgroupindex
C3×#25912
#826162
#821762
#716742
#715522
#694422
#676162
#99413