Finite Group Databasebeta

← #10696533

Subgroups — #10696533

Order 768.

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

normal subgroups only

This group has 628 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×#3151C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic, complemented
3#83180C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic, complemented
5#71760C211yesmaximal, characteristic, complemented
6#69863C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10089—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—22no—
10C3×#532C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12C2×#870C2211yescharacteristic
13C3×#496—22no—
14C3×#493—21no—
15C3×#492—21no—
16C3×C4×M16C2211yescharacteristic
17#5222—22no—
18#5222—22no—
19#5221—21no—
20#5221—21no—
21#5081—21no—
22#5071—21no—
23#5034—21no—
24#5033—21no—
25#5009—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4544—22no—
30#4537—22no—
31#4465—22no—
32#3234—31no—
33#3151D311yescharacteristic
34#2641—31no—
35#1555—31no—
36#1528—31no—
37#1505—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4D412yes—
43C3×AGL(1,ℤ/8)—48no—
44C2×C3×SD16—24no—
45C2×C3×D8—24no—
46C2×C3×M16—21no—
47C3×C4⋊Q8D411yescharacteristic
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127—21no—
52C3×#127D411yescharacteristic
53C3×#124—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—24no—
60C2×#217—24no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66#870—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70C3×#97D412yes—
71C3×C4×C8—21no—
72#832—22no—
73#832—22no—
74C3⋊#95—21no—
75C3⋊#95D412yes—
76#830—21no—
77#830—21no—
78#808—22no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82#803—21no—
83#803—21no—
84C2×AGL(1,ℤ/8)—22no—
85#532C2×D311yescharacteristic
86C2×#127C2×D311yescharacteristic
87C2×#125—31no—
88#501—61no—
89#496—22no—
90#493—21no—
91#492—21no—
92#490—61no—
93#481—61no—
94#480—61no—
95#478—62no—
96#432—31no—
97#432—31no—
98C4×M16C2×D311yescharacteristic
99#394—62no—
100#394—62no—
101#390—61no—
102#390—61no—
103#376—31no—
104#331—62no—
105#330—62no—
106#327—62no—
107C24×C3—22no—
108C3×#41—44no—
109C2×C3×Q8—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—22no—
113C2×C3×D4—21no—
114C2×C3×D4—22no—
115C2×C3×D4—44no—
116C2×C3×D4—44no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—82no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×Dic3—22no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C3×SD16—48no—
131C3×D8—48no—
132C3×M16—24no—
133C2×C3×C8—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#217—84no—
143#217—84no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—42no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×#41—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4#6712yes—
159AGL(1,ℤ/8)—48no—
160C2×SD16—24no—
161C2×D8—24no—
162C2×M16—21no—
163C4⋊Q8#6711yescharacteristic
164#127—21no—
165#127—21no—
166#127—21no—
167#127—21no—
168#127#6711yescharacteristic
169#125—61no—
170#125—61no—
171#125—61no—
172#125—61no—
173#124—21no—
174#121—21no—
175#121—21no—
176C4×D4—21no—
177C4×D4—21no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C22⋊C4—64no—
183C2×C22⋊C4—64no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186#107—62no—
187#106—62no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190#102—62no—
191#102—62no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194#97—61no—
195#97—61no—
196#97#6712yes—
197C4×C8—21no—
198#95—32no—
199#95—61no—
200C23×C3—44no—
201C23×C3—44no—
202C23×C3—41no—
203C23×C3—42no—
204C23×C3—44no—
205C23×C3#12011yescharacteristic
206C3×Q8—42no—
207C3×D4—42no—
208C3×D4—44no—
209C3×D4—42no—
210C3×D4—81no—
211C3×D4—82no—
212C3×D4—82no—
213C3×D4—82no—
214C3×D4—82no—
215C3×D4—82no—
216C3×D4—82no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—82no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C3×C8—24no—
236C3⋊C8—42no—
237C3⋊C8—81no—
238C24—22no—
239#41—44no—
240C2×Q8—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—22no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—82no—
254C22×C4—41no—
255C22×C4—62no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4C2×#6711yescharacteristic
261SD16—48no—
262D8—48no—
263M16—24no—
264C2×C8—22no—
265C2×C8—62no—
266C2×C8—121no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—244no—
280C22⋊C4—244no—
281C42—122no—
282C42D3×D411yescharacteristic
283C42#23711yescharacteristic
284C42C2×#6711yescharacteristic
285C42C2×#6711yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—88no—
289C22×C3—81no—
290C22×C3—82no—
291C22×C3—81no—
292C22×C3—82no—
293C22×C3—82no—
294C22×C3—81no—
295C22×C3#39211yescharacteristic
296C22×C3#45312yes—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—81no—
303C3×C4#45412yes—
304Dic3—82no—
305Dic3—81no—
306Dic3—81no—
307C23—44no—
308C23—44no—
309C23—41no—
310C23—42no—
311C23—44no—
312C23#93711yescharacteristic
313Q8—42no—
314D4—42no—
315D4—44no—
316D4—42no—
317D4—82no—
318D4—81no—
319D4—82no—
320D4—82no—
321D4—82no—
322D4—82no—
323D4—82no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—82no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4#93711yescharacteristic
337C2×C4#93811yescharacteristic
338C2×C4#93911yescharacteristic
339C2×C4#93911yescharacteristic
340C2×C4#94011yescharacteristic
341C2×C4#95311yescharacteristic
342C8—24no—
343C8—122no—
344C8—241no—
345C2×C3—22no—
346C2×C3—161no—
347C2×C3—162no—
348C2×C3#200211yescharacteristic
349C2×C3#200311yescharacteristic
350C2×C3#200811yescharacteristic
351C22—21no—
352C22—21no—
353C22—88no—
354C22—81no—
355C22—82no—
356C22—81no—
357C22—82no—
358C22—82no—
359C22—81no—
360C22#507712yes—
361C22#522311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—81no—
368C4—242no—
369C4—241no—
370C4—241no—
371C4#520112yes—
372C3#1008911yescharacteristic, complemented, Sylow
373C2—22no—
374C2—162no—
375C2—161no—
376C2#7176911yescharacteristic
377C2#7209611yescharacteristic
378C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3151C2
socleC22×C3#392
hypercenter#3151D3
hypocenterC3#10089
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
2. #83273C2semidirect
3. #83180C2semidirect
4. #71812C2semidirect
5. #71760C2semidirect
6. #69863C2semidirect
7. #67665C2semidirect
372. C3#10089semidirect

Maximal subgroups

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

subgroupindex
C3×#31512
#832732
#831802
#718122
#717602
#698632
#676652
#100893