Finite Group Databasebeta

← #10697321

Subgroups — #10697321

Order 768.

528 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 107 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#83320C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic, complemented
4C3×#1980C211yesmaximal, characteristic
5#69486C211yesmaximal, characteristic, complemented
6#69471C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#10201—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×#104—22no—
13C3×C4×M16C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5121—22no—
16#5094—21no—
17#5080—21no—
18#5075—21no—
19#5070—21no—
20#5041—22no—
21#5034—22no—
22#5019—21no—
23#5008—22no—
24#5008—21no—
25C2×#810—22no—
26#4925C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4702—22no—
29#4682—21no—
30#4680—21no—
31#4452—22no—
32#3246—31no—
33#3227—31no—
34#3165—31no—
35#1981—31no—
36#1980D311yescharacteristic
37#1503—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#214—22no—
53#926—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56#924—21no—
57#924—21no—
58C22×C3⋊C8D412yes—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D412yes—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#104—44no—
69C3×#97—21no—
70C3×C4×C8—21no—
71C3×#95—21no—
72#832—24no—
73#832—22no—
74#830D412yes—
75#818—24no—
76#817—24no—
77#810—22no—
78#810—44no—
79#808—24no—
80#808—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86#532C2×D311yescharacteristic
87C2×C4⋊Q8—31no—
88#501—62no—
89#500—61no—
90#498—61no—
91#498—61no—
92#486—61no—
93#481—62no—
94#476—62no—
95#476—61no—
96#475—61no—
97#465—62no—
98#461—61no—
99#460—61no—
100#431—31no—
101#428—62no—
102C2×C4⋊C8—31no—
103C2×#104—22no—
104C2×#103—62no—
105C4×M16C2×D311yescharacteristic
106C4×C4⋊C4C2×D311yescharacteristic
107#328—62no—
108C3×#41—42no—
109C3×#41—42no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—41no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—21no—
124C2×Dic6—42no—
125C2×Dic6—42no—
126C2×Dic6—41no—
127C3×M16—22no—
128C3×M16SD1614yescomplemented
129C2×C3×C8—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—42no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141#217—41no—
142#214—82no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—22no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157C2×#41—21no—
158C22×Q8—61no—
159C2×SD16—62no—
160C2×M16#6712yes—
161C22×C8—32no—
162C4⋊Q8—32no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8D1211yescharacteristic
169#127D1211yescharacteristic
170#125—61no—
171#124—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174#121—21no—
175#121—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C42—21no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187#106—62no—
188#106—64no—
189#106—64no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#104—44no—
194#103—62no—
195#103—124no—
196#102—62no—
197#102—64no—
198C22⋊C8—32no—
199#97—21no—
200C4×C8—21no—
201C4×C8—61no—
202#95—21no—
203C23×C3—41no—
204C23×C3#12711yescharacteristic
205C3×Q8—42no—
206C3×D4—42no—
207C3×D4—41no—
208C3×D4—41no—
209C3×D4—42no—
210C3×D4—81no—
211C3×D4—81no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—41no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4—82no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4C2×SD1612yes—
227C2×C3×C4#13712yes—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—82no—
232Dic6—44no—
233Dic6—41no—
234Dic6—41no—
235Dic6—81no—
236Dic6—81no—
237Dic6—81no—
238C3×C8—24no—
239C3⋊C8—42no—
240C3⋊C8—42no—
241#41—42no—
242#41—42no—
243C2×Q8—21no—
244C2×Q8—61no—
245C2×Q8—62no—
246C2×Q8—61no—
247C2×Q8—122no—
248C2×Q8—122no—
249C2×Q8—121no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—121no—
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4C2×#6711yescharacteristic
260SD16—242no—
261M16—22no—
262M16#21414yescomplemented
263C2×C8—22no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—122no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—62no—
276C4⋊C4—61no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—121no—
284C42—42no—
285C42—121no—
286C42C2×D1211yescharacteristic
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3#39211yescharacteristic
296C22×C3#49211yescharacteristic
297C22×C3#49711yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—81no—
303C3×C4#44812yes—
304C3×C4#45311yescharacteristic
305C3×C4#45411yescharacteristic
306C3×C4#46112yes—
307Dic3—82no—
308Dic3—81no—
309Dic3—81no—
310C23—41no—
311C23#94011yescharacteristic
312Q8—42no—
313Q8—124no—
314Q8—121no—
315Q8—121no—
316Q8—241no—
317Q8—241no—
318Q8—241no—
319D4—42no—
320D4—42no—
321D4—41no—
322D4—41no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—122no—
336C2×C4—242no—
337C2×C4#88211yescharacteristic
338C2×C4#88211yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4#88411yescharacteristic
341C2×C4#93011yescharacteristic
342C2×C4C2×#21412yes—
343C2×C4#93711yescharacteristic
344C2×C4#95112yes—
345C8—24no—
346C8—122no—
347C8—122no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—161no—
351C2×C3#200611yescharacteristic
352C2×C3#200911yescharacteristic
353C2×C3#201311yescharacteristic
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22#496011yescharacteristic
360C22#506011yescharacteristic
361C22#507611yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—81no—
367C4—241no—
368C4—242no—
369C4—241no—
370C4#482211yescharacteristic
371C4#482711yescharacteristic
372C4#501912yes—
373C4#504012yes—
374C3#1020111yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—161no—
378C2#6966111yescharacteristic
379C2#7039211yescharacteristic
380C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1980C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1980D3
hypocenterC3#10201
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
3. #78506C2semidirect
5. #69486C2semidirect
6. #69471C2semidirect
7. #67668C2semidirect
128. C3×M16SD16semidirect
262. M16#214semidirect
374. C3#10201semidirect

Maximal subgroups

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

subgroupindex
#833202
#832712
#785062
C3×#19802
#694862
#694712
#676682
#102013