Finite Group Databasebeta

← #10697215

Subgroups — #10697215

Order 768.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3077C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic, complemented
3C2×#5019C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic, complemented
5#71601C211yesmaximal, characteristic
6#70506C211yesmaximal, characteristic, complemented
7#67714C211yesmaximal, characteristic, complemented
8#10186—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C22×#214—21no—
12C2×#869C2211yescharacteristic
13C3×#478—22no—
14C3×#476—21no—
15C3×#474—21no—
16C3×#428—21no—
17C3×#423C2211yescharacteristic
18C3×#418—21no—
19#5080—21no—
20#5079—21no—
21#5043—22no—
22#5020—22no—
23#5019—22no—
24#5019—22no—
25C2×#810—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4535—22no—
30#4525—22no—
31#4482—22no—
32#3229—31no—
33#3077D311yescharacteristic
34C2×#461—31no—
35#1546—31no—
36#1481—31no—
37#1460—31no—
38#1301—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×Dic6—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#214—24no—
59C2×#214—42no—
60C2×#214—42no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C22×C3⋊C8—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D412yes—
68C3×#107—22no—
69C3×#106—22no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#102—22no—
73C3×#102—22no—
74#835—24no—
75#830—21no—
76#830D412yes—
77#810—24no—
78#808—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C22×SD16—61no—
86#533C2×D311yescharacteristic
87C2×C4⋊Q8—31no—
88C2×C4×D4C2×D311yescharacteristic
89#498—61no—
90#487—61no—
91#483—62no—
92#478—22no—
93#476—21no—
94#475—62no—
95#474—21no—
96#461—62no—
97#461—62no—
98#431—31no—
99#431—31no—
100#428—21no—
101#423C2×D311yescharacteristic
102C2×C4⋊C8—31no—
103#418—21no—
104C2×#103—61no—
105#332—62no—
106Q8⋊C8—62no—
107#323—62no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4SD1612yes—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—82no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×Dic3—41no—
125C2×Dic6—22no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150#214—48no—
151C4⋊Dic3—22no—
152C4⋊Dic3—22no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—61no—
161C22×D4#6711yescharacteristic
162C23×C4—21no—
163C2×SD16—64no—
164C2×SD16—122no—
165C2×SD16—122no—
166C2×M16—21no—
167C22×C8—61no—
168C4⋊Q8—32no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171C4⋊Q8—61no—
172C4⋊Q8#6711yescharacteristic
173C4⋊Q8#6711yescharacteristic
174#125—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183#117—21no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6711yescharacteristic
187C2×C22⋊C4—21no—
188C2×C42C22×D311yescharacteristic
189#107—22no—
190#106—22no—
191#106—64no—
192C4⋊C8—21no—
193C4⋊C8—32no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8#6712yes—
198#103—64no—
199#103—64no—
200#102—22no—
201#102—22no—
202C22⋊C8—32no—
203C22⋊C8—61no—
204C4×C8—61no—
205C4×C8—61no—
206C23×C3—22no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3#12111yescharacteristic
212C3×Q8—42no—
213C3×Q8—42no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—41no—
217C3×D4—41no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—82no—
224C2×C3×C4—82no—
225C2×C3×C4—81no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×C3×C4C2×SD1611yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×Dic3—42no—
237C2×Dic3—81no—
238C2×Dic3—81no—
239Dic6—42no—
240Dic6—42no—
241Dic6—81no—
242Dic6—81no—
243C3×C8—42no—
244C3⋊C8—42no—
245C3⋊C8—42no—
246C3⋊C8—81no—
247C24—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—62no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4#21512yes—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—121no—
267C22×C4D3×D411yescharacteristic
268C22×C4#23711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270SD16—128no—
271M16—41no—
272M16—41no—
273C2×C8—22no—
274C2×C8—62no—
275C2×C8—62no—
276C2×C8—121no—
277C2×C8—121no—
278C2×C8—121no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42—41no—
295C42D3×D411yescharacteristic
296C42D3×D411yescharacteristic
297C42C2×#6711yescharacteristic
298C42C2×#6711yescharacteristic
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—42no—
302C22×C3—82no—
303C22×C3—82no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3#39211yescharacteristic
307C22×C3#46511yescharacteristic
308C22×C3#46811yescharacteristic
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4#44811yescharacteristic
316C3×C4#44911yescharacteristic
317C3×C4#45011yescharacteristic
318C3×C4#45211yescharacteristic
319C3×C4#47112yes—
320Dic3—82no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—81no—
325C23—81no—
326C23#93911yescharacteristic
327Q8—42no—
328Q8—42no—
329Q8—122no—
330Q8—122no—
331Q8—241no—
332Q8—241no—
333D4—22no—
334D4—22no—
335D4—41no—
336D4—41no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—122no—
346C2×C4—241no—
347C2×C4—241no—
348C2×C4#93711yescharacteristic
349C2×C4#93711yescharacteristic
350C2×C4#93811yescharacteristic
351C2×C4#93911yescharacteristic
352C2×C4#94011yescharacteristic
353C2×C4C2×#21511yescharacteristic
354C2×C4#94211yescharacteristic
355C2×C4#94911yescharacteristic
356C2×C4#95011yescharacteristic
357C2×C4#95311yescharacteristic
358C8—42no—
359C8—122no—
360C8—122no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—82no—
365C2×C3#200011yescharacteristic
366C2×C3#200911yescharacteristic
367C2×C3#201911yescharacteristic
368C22—21no—
369C22—21no—
370C22—42no—
371C22—82no—
372C22—82no—
373C22—81no—
374C22—81no—
375C22#504811yescharacteristic
376C22#504911yescharacteristic
377C22#522311yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—242no—
385C4#504611yescharacteristic
386C4#504711yescharacteristic
387C4#520012yes—
388C4#522711yescharacteristic
389C4#523811yescharacteristic
390C3#1018611yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—82no—
394C2#7169911yescharacteristic
395C2#7209711yescharacteristic
396C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3077C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3077D3
hypocenterC3#10186
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #83323C2semidirect
4. #71824C2semidirect
6. #70506C2semidirect
7. #67714C2semidirect
390. C3#10186semidirect

Maximal subgroups

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

subgroupindex
C3×#30772
#833232
C2×#50192
#718242
#716012
#705062
#677142
#101863