Finite Group Databasebeta

← #10697618

Subgroups — #10697618

Order 768.

604 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 103 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#82379C211yesmaximal, characteristic
2#82296C211yesmaximal, characteristic
3#79313C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic
5#70792C211yesmaximal, characteristic, complemented
6#70779C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10248—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#924—22no—
11#5470C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15#5221—21no—
16#5215—21no—
17#5009—22no—
18#5008—21no—
19#5007—21no—
20#4975—21no—
21#4964—21no—
22#4949—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4924C2211yescharacteristic
25#4849—22no—
26#4739—22no—
27#4699—21no—
28#4681—21no—
29C3×D4⋊C8—22no—
30#4481—22no—
31#4467—22no—
32#3209—31no—
33#3077—31no—
34#2542—31no—
35#1548—31no—
36#1503—31no—
37#1478D311yescharacteristic
38#1303—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×Dic6D412yes—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#217—24no—
50#926—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54#924—21no—
55#924—21no—
56#924—48no—
57C2×Dic3⋊C4—24no—
58C2×C4×Dic3—21no—
59C2×#208—21no—
60C2×Dic12—24no—
61#870—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76#832—22no—
77#832—22no—
78C3⋊#95D412yes—
79#816—24no—
80#808—22no—
81#807—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84#533—31no—
85C2×C4⋊Q8—31no—
86C2×C22⋊Q8—62no—
87C2×C4×D4C2×D311yescharacteristic
88#494—61no—
89#487—61no—
90#478—62no—
91#476—61no—
92#474—61no—
93#462—62no—
94#451—62no—
95#431C2×D311yescharacteristic
96#431C2×D311yescharacteristic
97#428—61no—
98#423—31no—
99#418—61no—
100#399—62no—
101#390—61no—
102#384—61no—
103C4×C4⋊C4—31no—
104#333—62no—
105#328—62no—
106D4⋊C8—22no—
107C24×C3—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4Q1612yescomplemented
114C22×C3×C4—21no—
115C22×C3×C4—44no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—22no—
122C22×Dic3—22no—
123C2×Dic6—22no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—22no—
127C2×Dic6—44no—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—48no—
139#217—41no—
140C4⋊Dic3—24no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149Dic3⋊C4—48no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—22no—
158Dic12—84no—
159C22×Q8—32no—
160C22×D4D1211yescharacteristic
161C23×C4—21no—
162C2×Q16—64no—
163C2×M16—61no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170#125—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173C22⋊Q8—128no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180#117—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4D1211yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—64no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191#106—62no—
192C4⋊C8—21no—
193C4⋊C8—32no—
194C4⋊C8—61no—
195C4⋊C8#6712yes—
196#103—64no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203#95—32no—
204C23×C3—22no—
205C23×C3—42no—
206C23×C3—42no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3#12011yescharacteristic
210C3×D4—22no—
211C3×D4—22no—
212C3×D4—41no—
213C3×D4—41no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—82no—
219C2×C3×C4—82no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12711yescharacteristic
226C2×C3×C4C2×Q1611yescharacteristic
227C2×C3×C4#13511yescharacteristic
228C2×C3×C4#13711yescharacteristic
229C2×C3×C4#13711yescharacteristic
230C2×Dic3—24no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—44no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237Dic6—42no—
238Dic6—44no—
239Dic6—81no—
240Dic6—82no—
241C3×C8—42no—
242C3×C8—81no—
243C3⋊C8—42no—
244C24—21no—
245C2×Q8—61no—
246C2×Q8—62no—
247C2×Q8—62no—
248C2×Q8—61no—
249C2×Q8—124no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4Dic1212yescomplemented
256C22×C4—21no—
257C22×C4—44no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4C2×D1211yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265Q16—244no—
266M16—121no—
267M16—121no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—62no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—64no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—128no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—128no—
288C22⋊C4—121no—
289C42—121no—
290C42—121no—
291C42—121no—
292C42#23511yescharacteristic
293C42D3×D411yescharacteristic
294C42D3×D411yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—82no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3#39211yescharacteristic
304C22×C3#45111yescharacteristic
305C22×C3#45211yescharacteristic
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—81no—
310C3×C4#45111yescharacteristic
311C3×C4#45211yescharacteristic
312C3×C4#45312yes—
313C3×C4#46711yescharacteristic
314C3×C4#47011yescharacteristic
315Dic3—44no—
316Dic3—82no—
317Dic3—81no—
318Dic3—81no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23#88211yescharacteristic
325Q8—122no—
326Q8—124no—
327Q8—242no—
328Q8—241no—
329D4—22no—
330D4—22no—
331D4—41no—
332D4—41no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—64no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—124no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4#88211yescharacteristic
347C2×C4#88411yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#90411yescharacteristic
350C2×C4C2×Dic1211yescharacteristic
351C2×C4#90911yescharacteristic
352C2×C4#90911yescharacteristic
353C2×C4#93011yescharacteristic
354C2×C4#93711yescharacteristic
355C2×C4#94011yescharacteristic
356C8—42no—
357C8—81no—
358C8—122no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—82no—
362C2×C3#200111yescharacteristic
363C2×C3#200211yescharacteristic
364C2×C3#200911yescharacteristic
365C22—21no—
366C22—21no—
367C22—44no—
368C22—42no—
369C22—82no—
370C22—81no—
371C22—81no—
372C22#473311yescharacteristic
373C22#473911yescharacteristic
374C22#496011yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—81no—
379C4—124no—
380C4—241no—
381C4—242no—
382C4—241no—
383C4#473311yescharacteristic
384C4#473911yescharacteristic
385C4#511611yescharacteristic
386C4#513711yescharacteristic
387C4#519812yes—
388C3#1024811yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—82no—
392C2#6996411yescharacteristic
393C2#7032411yescharacteristic
394C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1478D3
hypocenterC3#10248
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
3. #79313C2semidirect
5. #70792C2semidirect
6. #70779C2semidirect
7. #67746C2semidirect
113. C2×C3×D4Q16semidirect
255. C2×D4Dic12semidirect
388. C3#10248semidirect

Maximal subgroups

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

subgroupindex
#823792
#822962
#793132
C3×#14782
#707922
#707792
#677462
#102483