Finite Group Databasebeta

← #10697214

Subgroups — #10697214

Order 768.

528 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 101 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#83223C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3#79059C211yesmaximal, characteristic
4C3×#1546C211yesmaximal, characteristic, complemented
5#70783C211yesmaximal, characteristic
6#70768C211yesmaximal, characteristic, complemented
7#67714C211yesmaximal, characteristic, complemented
8#10186—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C2×C4×D12C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×#835—21no—
16#5131—21no—
17#5121—21no—
18#5094—22no—
19#5093—21no—
20#5083—22no—
21#5083—22no—
22#5080—21no—
23#4971—22no—
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4847—22no—
27#4846—21no—
28#4842—21no—
29C3×#332—22no—
30#4468—22no—
31#4456—22no—
32#3229—31no—
33#3077—31no—
34C2×#461—31no—
35#1546D311yescharacteristic, complemented
36#1481—31no—
37#1460—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—24no—
51C2×#215—42no—
52C2×#215—42no—
53C2×(C2×D3)⋊C4—21no—
54#926—21no—
55C2×C4⋊Dic3D411yescharacteristic
56#924—21no—
57#924—21no—
58C22×C3⋊C8—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#835—24no—
74#830D412yes—
75#821—22no—
76#821—22no—
77#818—22no—
78#817—22no—
79#810—24no—
80#808—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C22×SD16—61no—
86#533—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×D4—31no—
89#498—61no—
90#487—61no—
91#483—62no—
92#478—62no—
93#476—61no—
94#475—62no—
95#474—61no—
96#461—62no—
97#461—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#428—61no—
101#423C2×D311yescharacteristic
102C2×C4⋊C8—31no—
103#418—61no—
104C2×#103—61no—
105#332—22no—
106Q8⋊C8—62no—
107#323—62no—
108C23×D3—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12SD1612yes—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—82no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143#217—41no—
144#215—48no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—21no—
161C22×D4—31no—
162C23×C4—61no—
163C2×SD16—64no—
164C2×SD16—122no—
165C2×SD16—122no—
166C2×M16—21no—
167C22×C8—61no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173C4⋊Q8D1212yes—
174#125—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183#117—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—31no—
187C2×C22⋊C4—61no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190#106—64no—
191#106—62no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8#6712yes—
198#103—64no—
199#103—64no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C4×C8—21no—
205C4×C8—61no—
206C23×C3#12111yescharacteristic
207C22×D3—22no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—81no—
211C22×D3—81no—
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—81no—
215C3×Q8—81no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4#13711yescharacteristic
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—81no—
235D12—22no—
236D12—22no—
237D12—41no—
238D12—41no—
239C4×D3—82no—
240C4×D3—82no—
241Dic6—42no—
242Dic6—42no—
243C3×C8—42no—
244C3×C8—81no—
245C3⋊C8—42no—
246C3⋊C8—42no—
247C24—61no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×D4—32no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C22×C4—41no—
263C22×C4—61no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—242no—
267C22×C4#23511yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270SD16—128no—
271M16—41no—
272M16—41no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—62no—
276C2×C8—62no—
277C2×C8—121no—
278C2×C8—121no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—121no—
295C42C2×D1211yescharacteristic
296C42D3×D411yescharacteristic
297C42D3×D411yescharacteristic
298C42C2×#6711yescharacteristic
299C22×C3—21no—
300C22×C3—21no—
301C22×C3#39211yescharacteristic
302C22×C3#46511yescharacteristic
303C22×C3#46811yescharacteristic
304C2×D3—42no—
305C2×D3—82no—
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—82no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—82no—
313C3×C4#44811yescharacteristic
314C3×C4#44911yescharacteristic
315C3×C4#45011yescharacteristic
316C3×C4#45211yescharacteristic
317C3×C4#47112yes—
318Dic3—81no—
319Dic3—81no—
320Dic3—81no—
321C23—62no—
322C23—121no—
323C23—121no—
324C23—241no—
325C23—241no—
326C23#88411yescharacteristic
327Q8—42no—
328Q8—42no—
329Q8—81no—
330Q8—81no—
331Q8—122no—
332Q8—122no—
333D4—62no—
334D4—62no—
335D4—121no—
336D4—121no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—242no—
346C2×C4—242no—
347C2×C4—241no—
348C2×C4#88211yescharacteristic
349C2×C4#88211yescharacteristic
350C2×C4#88411yescharacteristic
351C2×C4#91111yescharacteristic
352C2×C4#91211yescharacteristic
353C2×C4D3×SD1611yescharacteristic
354C2×C4#91511yescharacteristic
355C2×C4#93011yescharacteristic
356C2×C4#93711yescharacteristic
357C2×C4#94011yescharacteristic
358C8—42no—
359C8—81no—
360C8—122no—
361C8—122no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3#200011yescharacteristic
365C2×C3#200911yescharacteristic
366C2×C3#201911yescharacteristic
367D3—82no—
368C22—21no—
369C22—21no—
370C22—122no—
371C22—242no—
372C22—242no—
373C22—241no—
374C22—241no—
375C22#477811yescharacteristic
376C22#478011yescharacteristic
377C22#496011yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—241no—
383C4—241no—
384C4—241no—
385C4#475412yes—
386C4#477611yescharacteristic
387C4#478311yescharacteristic
388C4#515811yescharacteristic
389C4#517011yescharacteristic
390C3#1018611yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—242no—
394C2#7011711yescharacteristic
395C2#7020511yescharacteristic
396C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1546D3
hypocenterC3#10186
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
1. #83223C2semidirect
4. C3×#1546C2semidirect
6. #70768C2semidirect
7. #67714C2semidirect
35. #1546D3semidirect
390. C3#10186semidirect

Maximal subgroups

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

subgroupindex
#832232
C2×#50832
#790592
C3×#15462
#707832
#707682
#677142
#101863