Finite Group Databasebeta

← #10684203

Subgroups — #10684203

Order 768.

454 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 79 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#76185C211yesmaximal, characteristic
2C3×#2058C211yesmaximal, characteristic, complemented
3#71793C211yesmaximal, characteristic, complemented
4#71792C211yesmaximal, characteristic
5#67638C211yesmaximal, characteristic, complemented
6#67480C211yesmaximal, characteristic, complemented
7#67479C211yesmaximal, characteristic
8#7785—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—21no—
11C2×C3×#106—21no—
12C3×#389C2211yescharacteristic
13C2×#835—21no—
14#4992—21no—
15#4991—21no—
16#4990—21no—
17#4980—21no—
18C2×#809—21no—
19#4939C2211yescharacteristic
20#4929C2211yescharacteristic
21#4673—21no—
22#4670—21no—
23C3×#340—21no—
24C3×#336C2211yescharacteristic
25#4529—21no—
26#4524C2211yescharacteristic
27#4524C2211yescharacteristic
28#2808—31no—
29#2058D311yescharacteristic, complemented
30#2023—31no—
31#2016—31no—
32#1272—31no—
33#1271—31no—
34#1270—31no—
35C22×C3×Q8—21no—
36C22×D12D411yescharacteristic
37C22×Dic6D411yescharacteristic
38C22×C3×C8—21no—
39C3×C4⋊Q8—42no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C42C2311yescharacteristic
46C2×#216—42no—
47C2×#216—42no—
48C2×#216—42no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4×Dic3—21no—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60C3×#106—42no—
61C3×#95—21no—
62#835—42no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68#809—42no—
69C22×Q16—61no—
70C2×C4⋊Q8C2×D311yescharacteristic
71C2×#106—21no—
72C2×#103—61no—
73C2×#102—61no—
74#397—61no—
75#394—61no—
76#393—61no—
77#391—31no—
78#390—31no—
79#389C2×D311yescharacteristic
80#388—61no—
81#388—61no—
82#387—61no—
83#340—21no—
84#340—61no—
85#336—31no—
86#336—31no—
87#336C2×D311yescharacteristic
88C23×D3—21no—
89C2×C3×Q8—22no—
90C2×C3×Q8—41no—
91C2×C3×Q8—41no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—41no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—41no—
107C2×D12—41no—
108C2×Dic6—21no—
109C2×Dic6—21no—
110C2×Dic6—41no—
111C2×Dic6—41no—
112C2×C3×C8—22no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C42—41no—
125#216—48no—
126(C2×D3)⋊C4—41no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—81no—
131(C2×D3)⋊C4—81no—
132(C2×D3)⋊C4—81no—
133(C2×D3)⋊C4—81no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4×Dic3—81no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C22×Q8—21no—
144C22×Q8—31no—
145C22×D4—31no—
146C2×Q16—122no—
147C2×Q16—122no—
148C2×Q16—122no—
149C22×C8—21no—
150C22×C8—61no—
151C22×C8—61no—
152C4⋊Q8—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4#6711yescharacteristic
159C2×C4⋊C4#6711yescharacteristic
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C42—61no—
167C2×C42C22×D311yescharacteristic
168#106—42no—
169#103—122no—
170#102—122no—
171#95—21no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95—61no—
177C23×C3#12011yescharacteristic
178C22×D3—41no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—41no—
182C22×D3—41no—
183C22×D3—41no—
184C22×D3—41no—
185C3×Q8—42no—
186C3×Q8—42no—
187C3×Q8—81no—
188C3×Q8—81no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4#13511yescharacteristic
205C2×C3×C4#13511yescharacteristic
206C2×C3×C4#13511yescharacteristic
207C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
208C2×C3×C4#13711yescharacteristic
209C2×C3×C4#13711yescharacteristic
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219D12—42no—
220D12—81no—
221Dic6—42no—
222Dic6—81no—
223C3×C8—42no—
224C3⋊C8—42no—
225C3⋊C8—81no—
226C24—61no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—121no—
236C2×Q8—121no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—121no—
240C2×D4—121no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4D3×D411yescharacteristic
248C22×C4D3×D411yescharacteristic
249C22×C4C2×#6711yescharacteristic
250Q16—128no—
251C2×C8—22no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—62no—
255C2×C8—121no—
256C2×C8—121no—
257C2×C8—121no—
258C2×C8—121no—
259C2×C8—121no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—241no—
276C22⋊C4—241no—
277C22⋊C4—241no—
278C22⋊C4—241no—
279C42—41no—
280C42—241no—
281C22×C3#44911yescharacteristic
282C22×C3#45211yescharacteristic
283C22×C3#45211yescharacteristic
284C22×C3#45311yescharacteristic
285C22×C3#45511yescharacteristic
286C22×C3#45611yescharacteristic
287C22×C3B(4,𝔽2)11yescharacteristic
288C2×D3—81no—
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—81no—
303C3×C4—81no—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307C23—121no—
308C23—121no—
309C23—121no—
310C23—121no—
311C23—121no—
312C23—121no—
313C23—121no—
314C23#93711yescharacteristic
315Q8—42no—
316Q8—42no—
317Q8—81no—
318Q8—81no—
319Q8—122no—
320Q8—241no—
321D4—122no—
322D4—241no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4#91411yescharacteristic
348C2×C4#91511yescharacteristic
349C2×C4#91611yescharacteristic
350C2×C4#91611yescharacteristic
351C2×C4#94211yescharacteristic
352C2×C4#95011yescharacteristic
353C8—42no—
354C8—122no—
355C8—241no—
356C2×C3#160011yescharacteristic
357C2×C3#160311yescharacteristic
358C2×C3#160611yescharacteristic
359C2×C3#163411yescharacteristic
360C2×C3#164011yescharacteristic
361C2×C3#165211yescharacteristic
362C2×C3#166111yescharacteristic
363D3—161no—
364C22—241no—
365C22—241no—
366C22—241no—
367C22—241no—
368C22—241no—
369C22—241no—
370C22—241no—
371C22#503211yescharacteristic
372C22#504711yescharacteristic
373C22#509511yescharacteristic
374C22#517211yescharacteristic
375C22#517311yescharacteristic
376C22#519811yescharacteristic
377C22#520211yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—81no—
385C4—82no—
386C4—81no—
387C4—241no—
388C4—241no—
389C4—241no—
390C3#778511yescharacteristic, complemented, Sylow
391C2—481no—
392C2#6970511yescharacteristic
393C2#7097311yescharacteristic
394C2#7097711yescharacteristic
395C2#7112811yescharacteristic
396C2#7114511yescharacteristic
397C2#7134611yescharacteristic
398C2#7139211yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2058C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#2058D3
hypocenterC3#7785
LCSG ▷ C2×C3×C42 ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
2. C3×#2058C2semidirect
3. #71793C2semidirect
5. #67638C2semidirect
6. #67480C2semidirect
29. #2058D3semidirect
390. C3#7785semidirect

Maximal subgroups

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

subgroupindex
#761852
C3×#20582
#717932
#717922
#676382
#674802
#674792
#77853