Finite Group Databasebeta

← #10695574

Subgroups — #10695574

Order 768.

600 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#82182C211yesmaximal, characteristic
2#82176C211yesmaximal, characteristic, complemented
3#78624C211yesmaximal, characteristic
4C3×#1463C211yesmaximal, characteristic, complemented
5#69978C211yesmaximal, characteristic
6#69975C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9941—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14#5213—21no—
15#5213—21no—
16#5211—21no—
17#5211C2211yescharacteristic
18#5210—21no—
19C4×#217C2211yescharacteristic
20#5131—21no—
21#5121—21no—
22#4991—21no—
23#4990—22no—
24#4986—21no—
25#4986—21no—
26#4981—21no—
27#4978—22no—
28#4737—22no—
29#4736—21no—
30#4735—21no—
31C3×#324—22no—
32#4555—22no—
33#4470—22no—
34#3094—31no—
35#2591—31no—
36#2484—31no—
37#1826—31no—
38#1510—31no—
39#1463D311yescharacteristic, complemented
40#1284—31no—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×D12D411yescharacteristic
46C2×C3×M16—21no—
47C2×C3×M16—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—22no—
64#926—21no—
65#926—21no—
66#926—21no—
67#926—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×C4⋊Dic3D411yescharacteristic
71C2×Dic3⋊C4—22no—
72C2×C4×Dic3—21no—
73C2×C4×Dic3D412yes—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yes—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95D412yes—
86#821—22no—
87#821—22no—
88#818—22no—
89#817—22no—
90C2×#121—31no—
91C2×#117—31no—
92C22×C4⋊C4C2×D311yescharacteristic
93#482—62no—
94#481—61no—
95#480—61no—
96#428—61no—
97#418—61no—
98#407C2×D311yescharacteristic
99#407C2×D311yescharacteristic
100#400—61no—
101#400—61no—
102#398—61no—
103#394—61no—
104#388—62no—
105#387—62no—
106#386—61no—
107#386—61no—
108#385—61no—
109#381—31no—
110#381—61no—
111C4×C22⋊C4—31no—
112#339—62no—
113#327—62no—
114#324—22no—
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—21no—
122C22×C3×C4—21no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C2×#67—44no—
133C2×#67—41no—
134C2×#67—41no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—22no—
139C22×Dic3—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—42no—
143C3×M16—42no—
144C3×M16—42no—
145C2×C3×C8—22no—
146C2×C3×C8—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151#217—42no—
152#217—42no—
153#217—41no—
154#217—41no—
155(C2×D3)⋊C4—44no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—82no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165Dic3⋊C4—44no—
166C4×Dic3—41no—
167C4×Dic3—41no—
168C4×Dic3—42no—
169C22×D4—31no—
170C22×D4—61no—
171C23×C4D1211yescharacteristic
172C23×C4D1211yescharacteristic
173C23×C4C22×D311yescharacteristic
174C2×M16—21no—
175C2×M16—21no—
176#121—61no—
177#121—61no—
178#121—61no—
179#121—61no—
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—31no—
193C2×C4⋊C4—62no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—62no—
200C2×C42—32no—
201C2×C42—61no—
202#107—62no—
203#106—62no—
204#102—62no—
205#102—62no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8#6712yes—
210#95—32no—
211#95—62no—
212#95—62no—
213#95—62no—
214#95—62no—
215#95—62no—
216#95—61no—
217#95—61no—
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—41no—
243C2×C3×C4—42no—
244C2×C3×C4—81no—
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12711yescharacteristic
249#67—82no—
250#67—82no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263D12—42no—
264D12—81no—
265C3×C8—42no—
266C3×C8—81no—
267C24—61no—
268C24C2×D1211yescharacteristic
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—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—62no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—62no—
288C22×C4—61no—
289C22×C4#23511yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4C2×#6711yescharacteristic
295M16—42no—
296M16—42no—
297C2×C8—22no—
298C2×C8—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—124no—
310C22⋊C4—124no—
311C22⋊C4—122no—
312C22⋊C4—122no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—242no—
318C42—122no—
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#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—81no—
342Dic3—82no—
343Dic3—82no—
344Dic3—81no—
345Dic3—81no—
346Dic3—81no—
347C23—22no—
348C23—22no—
349C23—41no—
350C23—122no—
351C23—122no—
352C23—121no—
353C23—122no—
354C23#88211yescharacteristic
355C23#88211yescharacteristic
356C23#93011yescharacteristic
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—41no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4#88411yescharacteristic
391C2×C4#88411yescharacteristic
392C2×C4#93711yescharacteristic
393C2×C4#94011yescharacteristic
394C8—42no—
395C8—81no—
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#474212yes—
416C22#475312yes—
417C22#496011yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—81no—
424C4—242no—
425C4—242no—
426C4—241no—
427C4—241no—
428C4—241no—
429C3#994111yescharacteristic, complemented, Sylow
430C2—22no—
431C2—22no—
432C2—41no—
433C2—481no—
434C2#6999311yescharacteristic
435C2#7003111yescharacteristic
436C2#7011611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1463C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1463D3
hypocenterC3#9941
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #82176C2semidirect
4. C3×#1463C2semidirect
6. #69975C2semidirect
7. #67682C2semidirect
39. #1463D3semidirect
429. C3#9941semidirect

Maximal subgroups

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

subgroupindex
#821822
#821762
#786242
C3×#14632
#699782
#699752
#676822
#99413