Finite Group Databasebeta

← #10695841

Subgroups — #10695841

Order 768.

544 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2816C211yesmaximal, characteristic
2#82727C211yesmaximal, characteristic, complemented
3#82616C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic, complemented
5#71674C211yesmaximal, characteristic
6#69534C211yesmaximal, characteristic, complemented
7#67629C211yesmaximal, characteristic, complemented
8#9984—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#399—21no—
13C3×#398—21no—
14C3×#394—21no—
15C3×#394—22no—
16C3×#393—21no—
17C3×#381C2211yescharacteristic
18#5225—21no—
19#5215—21no—
20#5211C2211yescharacteristic
21#5207C2211yescharacteristic
22#5207C2211yescharacteristic
23#5045—22no—
24#5044—21no—
25#5043—21no—
26#5035—22no—
27#5034—21no—
28#5033—21no—
29#4964—21no—
30#4964—21no—
31#4538—22no—
32#4537—22no—
33#4527—22no—
34#3095—31no—
35#3094—31no—
36#2816D311yescharacteristic
37#1820—31no—
38#1511—31no—
39#1510—31no—
40#1306—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4D412yes—
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×C4×Dic3—21no—
71C2×#208—21no—
72C2×#208—21no—
73C3×#95—22no—
74C3×#95—21no—
75C3×#95—21no—
76#835—22no—
77#835—22no—
78#832—22no—
79#832—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#830D412yes—
86#808—22no—
87#808—22no—
88#807—22no—
89#807—22no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#121C2×D311yescharacteristic
92C2×#117—31no—
93#485—62no—
94#484—61no—
95#483—61no—
96#482—62no—
97#481—61no—
98#480—61no—
99#428—61no—
100#428—61no—
101#419—61no—
102#418—61no—
103#407—31no—
104#407—31no—
105#399—21no—
106#398—21no—
107#394—21no—
108#394—22no—
109#393—21no—
110#381—31no—
111#381C2×D311yescharacteristic
112#342—62no—
113#328—62no—
114#327—62no—
115C24×C3—21no—
116C24×C3C2×D411yescharacteristic
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—42no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—82no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—42no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—82no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—42no—
152C3×C22⋊C4—82no—
153C3×C22⋊C4—82no—
154#217—41no—
155#217—41no—
156#217—42no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163#208—42no—
164#208—42no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C22×Q8#6711yescharacteristic
168C22×D4—21no—
169C22×D4#6711yescharacteristic
170C23×C4C22×D311yescharacteristic
171C2×M16—61no—
172C2×M16—61no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177#121—21no—
178#121—21no—
179#121—21no—
180#121—21no—
181#117—61no—
182#117—61no—
183#117—61no—
184#117—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C4⋊C4#6711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4#6712yes—
198C2×C42—61no—
199#107—62no—
200#107—62no—
201#106—62no—
202#106—62no—
203#103—62no—
204#103—62no—
205#102—62no—
206#102—62no—
207C22⋊C8—32no—
208C22⋊C8—32no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—61no—
215#95—61no—
216C23×C3—22no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—42no—
223C23×C3#12011yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12711yescharacteristic
226C3×Q8—42no—
227C3×Q8—81no—
228C3×D4—42no—
229C3×D4—81no—
230C3×D4—82no—
231C3×D4—82no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258C2×Dic3—81no—
259C3⋊C8—42no—
260C3⋊C8—42no—
261C24—21no—
262C24C2×#6711yescharacteristic
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—82no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—121no—
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4#23711yescharacteristic
284C22×C4C2×#6711yescharacteristic
285C22×C4C2×#6711yescharacteristic
286M16—122no—
287M16—122no—
288C2×C8—62no—
289C2×C8—62no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—82no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—121no—
309C22⋊C4—121no—
310C22⋊C4—122no—
311C42—121no—
312C42—121no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—82no—
322C22×C3—84no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3#39211yescharacteristic
327C22×C3#45812yes—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333C3×C4—81no—
334C3×C4—81no—
335C3×C4—81no—
336Dic3—81no—
337Dic3—81no—
338Dic3—161no—
339C23—22no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—42no—
344C23—41no—
345C23—42no—
346C23#93811yescharacteristic
347C23#94011yescharacteristic
348C23#95311yescharacteristic
349Q8—42no—
350Q8—81no—
351D4—42no—
352D4—81no—
353D4—82no—
354D4—82no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4—241no—
377C2×C4—241no—
378C2×C4#93711yescharacteristic
379C2×C4#93711yescharacteristic
380C2×C4#93911yescharacteristic
381C2×C4#93911yescharacteristic
382C8—122no—
383C8—122no—
384C2×C3—22no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3—161no—
388C2×C3#200811yescharacteristic
389C2×C3#200911yescharacteristic
390C2×C3#202211yescharacteristic
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—82no—
400C22—84no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22#522311yescharacteristic
405C22#524312yes—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—82no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—241no—
415C4—241no—
416C4—481no—
417C3#998411yescharacteristic, complemented, Sylow
418C2—22no—
419C2—41no—
420C2—41no—
421C2—161no—
422C2#7184911yescharacteristic
423C2#7185811yescharacteristic
424C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2816C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#2816D3
hypocenterC3#9984
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2816

Complements

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

NH ≅ G/Ntype
2. #82727C2semidirect
4. #71689C2semidirect
6. #69534C2semidirect
7. #67629C2semidirect
417. C3#9984semidirect

Maximal subgroups

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

subgroupindex
C3×#28162
#827272
#826162
#716892
#716742
#695342
#676292
#99843