Finite Group Databasebeta

← #10695575

Subgroups — #10695575

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
1#78624C211yesmaximal, characteristic
2#74074C211yesmaximal, characteristic, complemented
3#73550C211yesmaximal, characteristic
4#69971C211yesmaximal, characteristic, complemented
5#69531C211yesmaximal, characteristic
6#69442C211yesmaximal, characteristic, complemented
7C3×#1284C211yesmaximal, characteristic, complemented
8#9941—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C22×Dic3⋊C4C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#381C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15#5207C2211yescharacteristic
16#5131—21no—
17#5121—21no—
18#4958—21no—
19#4958—21no—
20#4950—21no—
21#4943—21no—
22#4940—22no—
23#4737—22no—
24#4736—21no—
25#4735—21no—
26#4672—21no—
27#4661—21no—
28#4659—21no—
29#4659—21no—
30#4653—22no—
31#4523—22no—
32#4469—22no—
33#4451—22no—
34#3094—31no—
35#2591—31no—
36#2484—31no—
37#1826—31no—
38#1510—31no—
39#1463—31no—
40#1284D311yescharacteristic, complemented
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C23×Dic3D411yescharacteristic
44C23×Dic3D411yescharacteristic
45C22×D12D411yescharacteristic
46C2×C3×M16—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42D412yes—
50C2×#217—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58#926—21no—
59#926—21no—
60#926—21no—
61#926—21no—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4⋊Dic3D411yescharacteristic
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C2×C4×Dic3—21no—
73C2×#208—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76C3×#95—21no—
77C3×#95—21no—
78C3×#95D412yes—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84#830—21no—
85#830—21no—
86#821—22no—
87#821—22no—
88#818—22no—
89#817—22no—
90C2×#121—31no—
91C2×#117—31no—
92C22×C4⋊C4—31no—
93#482—62no—
94#481—61no—
95#480—61no—
96#428—61no—
97#418—61no—
98#407—31no—
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—61no—
110#381C2×D311yescharacteristic
111C4×C22⋊C4C2×D311yescharacteristic
112#339—62no—
113#327—62no—
114#324—62no—
115C24×C3C2×D411yescharacteristic
116C23×D3—21no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—42no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—44no—
127C2×#67—41no—
128C2×#67—41no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—42no—
139C22×Dic3—42no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—42no—
143C3×M16—42no—
144C2×C3×C8—22no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—42no—
147C3×C42—42no—
148#217—41no—
149#217—41no—
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—82no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—44no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165C4×Dic3—41no—
166C4×Dic3—41no—
167#208—42no—
168C2×C3⋊C8—41no—
169C22×D4—31no—
170C22×D4—61no—
171C23×C4—31no—
172C23×C4—31no—
173C23×C4C22×D311yescharacteristic
174C2×M16—21no—
175C2×M16—61no—
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—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
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—61no—
201C2×C42D1212yes—
202#107—62no—
203#106—62no—
204#102—62no—
205#102—62no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8D1212yes—
210#95—21no—
211#95—21no—
212#95—62no—
213#95—62no—
214#95—62no—
215#95—62no—
216#95—62no—
217#95D1212yes—
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—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244#67—82no—
245#67—82no—
246C2×Dic3—22no—
247C2×Dic3—22no—
248C2×Dic3—22no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—81no—
263D12—42no—
264D12—81no—
265C3×C8—42no—
266C3⋊C8—81no—
267C24—61no—
268C24D3×D411yescharacteristic
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—122no—
272C2×D4—124no—
273C2×D4—121no—
274C2×D4—121no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—122no—
288C22×C4—122no—
289C22×C4C2×D1211yescharacteristic
290C22×C4C2×D1211yescharacteristic
291C22×C4C2×D1211yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4#23911yescharacteristic
295M16—42no—
296M16—122no—
297C2×C8—22no—
298C2×C8—121no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—124no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4—121no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—124no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—242no—
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#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—82no—
341C3×C4—82no—
342Dic3—42no—
343Dic3—81no—
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#89511yescharacteristic
357D4—122no—
358D4—241no—
359D4—242no—
360D4—242no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—241no—
390C2×C4#87411yescharacteristic
391C2×C4#88211yescharacteristic
392C2×C4#89511yescharacteristic
393C2×C4#89511yescharacteristic
394C8—42no—
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#467111yescharacteristic
416C22#474212yes—
417C22#482712yes—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—82no—
423C4—82no—
424C4—122no—
425C4—241no—
426C4—241no—
427C4—241no—
428C4—241no—
429C3#994111yescharacteristic, complemented, Sylow
430C2—22no—
431C2—22no—
432C2—41no—
433C2—481no—
434C2#6870311yescharacteristic
435C2#6876711yescharacteristic
436C2#6896611yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1284D3
hypocenterC3#9941
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
2. #74074C2semidirect
4. #69971C2semidirect
6. #69442C2semidirect
7. C3×#1284C2semidirect
40. #1284D3semidirect
429. C3#9941semidirect

Maximal subgroups

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

subgroupindex
#786242
#740742
#735502
#699712
#695312
#694422
C3×#12842
#99413