Finite Group Databasebeta

← #10714031

Subgroups — #10714031

Order 768.

594 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#5183C211yesmaximal, characteristic, complemented
2C22×#828C211yesmaximal, characteristic
3C22×#800C211yesmaximal, characteristic
4C2×C3×#358C211yesmaximal, characteristic, complemented
5#70628C211yesmaximal, characteristic
6C2×#4492C211yesmaximal, characteristic, complemented
7#67804C211yesmaximal, characteristic, complemented
8#13327—31nomaximal, Sylow
9C22×C3×Q16C2211yescharacteristic
10C2×#944—21no—
11C22×#216—21no—
12C22×Dic12C2211yescharacteristic
13C22×D24C2211yescharacteristic
14C22×C3×C16C2211yescharacteristic
15C2×C3×#107C2211yescharacteristic
16#5183—22no—
17#5183—22no—
18C2×#828—22no—
19C2×#828—22no—
20C2×#828—22no—
21C2×C8×Dic3C2211yescharacteristic
22C22×C3⋊C16C2211yescharacteristic
23#4988—21no—
24C2×#809—21no—
25C2×#800—22no—
26C2×#800—22no—
27C2×#800—22no—
28C3×#358—22no—
29#4492—22no—
30C22×SD32—31no—
31C22×SD32—31no—
32C2×#494—31no—
33C2×#358—31no—
34C2×#358D311yescharacteristic, complemented
35#1970—31no—
36#1374—31no—
37C22×C3×Q8—21no—
38C22×D12—21no—
39C22×Dic6—21no—
40C2×C3×Q16—21no—
41C2×C3×Q16—21no—
42C2×C3×Q16—21no—
43C2×C3×Q16—21no—
44C2×C3×Q16—21no—
45C2×C3×Q16D412yes—
46C22×C3×C8C2311yescharacteristic
47C2×C3×C4⋊C4—21no—
48#944—24no—
49#944—42no—
50C2×#216—24no—
51C2×#216—42no—
52C2×#216—42no—
53C2×(C2×D3)⋊C4—42no—
54C2×Dic3⋊C4—42no—
55C2×C4×Dic3D411yescharacteristic
56C22×C3⋊C8D411yescharacteristic
57C2×Dic12—21no—
58C2×Dic12—21no—
59C2×Dic12—21no—
60C2×Dic12—21no—
61C2×Dic12—21no—
62C2×Dic12D412yes—
63C2×D24—21no—
64C2×D24—21no—
65C2×D24—21no—
66C2×D24—21no—
67C2×D24—21no—
68C2×D24D412yes—
69C2×C3×C16—21no—
70C2×C3×C16—21no—
71C2×C3×C16D412yes—
72C3×#107—21no—
73C3×#107—21no—
74#828—28no—
75C8×Dic3—21no—
76C8×Dic3—21no—
77C2×C3⋊C16—21no—
78C2×C3⋊C16—21no—
79C2×C3⋊C16D412yes—
80#809—42no—
81#800—28no—
82C22×Q16—31no—
83C22×Q16—61no—
84C22×Q16C2×D311yescharacteristic
85C22×D8—31no—
86C2×C4⋊Q8—61no—
87C2×SD32—62no—
88C2×SD32—62no—
89C2×SD32—62no—
90C2×SD32—62no—
91C2×SD32—62no—
92C2×SD32—62no—
93C22×C16—31no—
94C22×C16C2×D311yescharacteristic
95#494—62no—
96#494—62no—
97C2×#107C2×D311yescharacteristic
98C2×#102—61no—
99C2×C4×C8—31no—
100#390—61no—
101#358—22no—
102#358—62no—
103C23×D3—41no—
104C2×C3×Q8—22no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—41no—
114C22×Dic3D812yes—
115C2×D12—22no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×Dic6—22no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C3×Q16—22no—
128C3×Q16—22no—
129C3×Q16—41no—
130C3×Q16—41no—
131C2×C3×C8C2×D411yescharacteristic
132C2×C3×C8C2×D411yescharacteristic
133C2×C3×C8C2×D411yescharacteristic
134C2×C3×C8C2×D411yescharacteristic
135C2×C3×C8C2×D411yescharacteristic
136C2×C3×C8#4111yescharacteristic
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139#216—48no—
140(C2×D3)⋊C4—84no—
141Dic3⋊C4—48no—
142C4×Dic3—21no—
143C4×Dic3—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8D812yes—
147Dic12—22no—
148Dic12—22no—
149Dic12—41no—
150Dic12—41no—
151D24—22no—
152D24—22no—
153D24—41no—
154D24—41no—
155C3×C16—22no—
156C3⋊C16—22no—
157C22×Q8—21no—
158C22×Q8—61no—
159C22×D4—61no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—32no—
166C2×Q16—64no—
167C2×Q16—61no—
168C2×Q16—61no—
169C2×Q16—61no—
170C2×Q16—61no—
171C2×Q16—61no—
172C2×Q16—122no—
173C2×Q16—122no—
174C2×Q16D1212yes—
175C2×D8—32no—
176C2×D8—61no—
177C2×D8—61no—
178C2×D8—61no—
179C2×D8—61no—
180C2×D8—61no—
181C22×C8—31no—
182C22×C8C22×D311yescharacteristic
183C4⋊Q8—64no—
184C4⋊Q8—122no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—122no—
187C2×C22⋊C4—122no—
188C2×C42—31no—
189SD32—68no—
190SD32—68no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—32no—
194C2×C16—61no—
195C2×C16—61no—
196C2×C16#6712yes—
197#107—21no—
198#107—21no—
199#102—122no—
200C4×C8—61no—
201C4×C8—61no—
202C23×C3C2×D811yescharacteristic
203C22×D3—42no—
204C22×D3—81no—
205C22×D3—81no—
206C22×D3—82no—
207C22×D3—82no—
208C3×Q8—42no—
209C3×Q8—42no—
210C3×Q8—81no—
211C3×Q8—81no—
212C2×C3×C4—42no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×C3×C4—82no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C2×C3×C4C2×D811yescharacteristic
221C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—42no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228C2×Dic3SD3214yescomplemented
229D12—42no—
230D12—42no—
231D12—81no—
232D12—81no—
233Dic6—42no—
234Dic6—42no—
235Dic6—81no—
236Dic6—81no—
237C3×C8#12011yescharacteristic
238C3×C8#12011yescharacteristic
239C3×C8#12111yescharacteristic
240C3×C8#12111yescharacteristic
241C3⋊C8—22no—
242C24—121no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—62no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×D4—62no—
256C2×D4—121no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C22×C4—32no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—121no—
265C22×C4D3×D411yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—41no—
269Q16—41no—
270Q16—62no—
271Q16—62no—
272Q16—128no—
273Q16—121no—
274Q16—121no—
275D8—62no—
276D8—62no—
277D8—121no—
278D8—121no—
279C2×C8—32no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8C2×D1211yescharacteristic
283C2×C8#23511yescharacteristic
284C2×C8D3×D411yescharacteristic
285C2×C8D3×D411yescharacteristic
286C2×C8D3×D411yescharacteristic
287C2×C8C2×#6711yescharacteristic
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—128no—
291C22⋊C4—244no—
292C42—61no—
293C42—61no—
294C16—22no—
295C16—62no—
296C22×C3#44711yescharacteristic
297C22×C3#45911yescharacteristic
298C22×C3#49311yescharacteristic
299C22×C3C2×SD3212yes—
300C22×C3#51012yes—
301C2×D3—84no—
302C2×D3—82no—
303C2×D3—162no—
304C2×D3—161no—
305C2×D3—161no—
306C3×C4—82no—
307C3×C4—161no—
308C3×C4#39211yescharacteristic
309C3×C4#44711yescharacteristic
310C3×C4#45911yescharacteristic
311C3×C4#49611yescharacteristic
312Dic3—24no—
313Dic3—82no—
314C23—122no—
315C23—241no—
316C23—242no—
317C23—242no—
318C23—241no—
319C23D3×D811yescharacteristic
320Q8—42no—
321Q8—42no—
322Q8—81no—
323Q8—81no—
324Q8—122no—
325Q8—122no—
326Q8—241no—
327Q8—241no—
328D4—122no—
329D4—122no—
330D4—241no—
331D4—241no—
332C2×C4—34no—
333C2×C4—42no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—122no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4#88211yescharacteristic
344C2×C4#88411yescharacteristic
345C2×C4D3×D811yescharacteristic
346C2×C4#91411yescharacteristic
347C2×C4#91411yescharacteristic
348C2×C4#94011yescharacteristic
349C8—62no—
350C8#88211yescharacteristic
351C8#88411yescharacteristic
352C8#93011yescharacteristic
353C8#93711yescharacteristic
354C2×C3#201111yescharacteristic
355C2×C3#218512yes—
356C2×C3#220812yes—
357C2×C3#224811yescharacteristic
358C2×C3#225111yescharacteristic
359D3—162no—
360C22—242no—
361C22—244no—
362C22—481no—
363C22—481no—
364C22—482no—
365C22#477311yescharacteristic
366C22#478611yescharacteristic
367C22D3×SD3212yes—
368C22#491812yes—
369C22#515111yescharacteristic
370C4—64no—
371C4—82no—
372C4—161no—
373C4—242no—
374C4#477311yescharacteristic
375C4#478611yescharacteristic
376C4#496011yescharacteristic
377C4#517611yescharacteristic
378C3#1332711yescharacteristic, complemented, Sylow
379C2—482no—
380C2#6922312yes—
381C2#6923412yes—
382C2#7020911yescharacteristic
383C2#7147911yescharacteristic
384C2#7150111yescharacteristic
subgroupG/H
centerC23D3×D8
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC2×C3×#358C2
socleC23×C3C2×D8
prime radicalC22×D24C22
hypercenterC2×#358D3
hypocenterC3#13327
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. C2×#5183C2semidirect
4. C2×C3×#358C2semidirect
6. C2×#4492C2semidirect
7. #67804C2semidirect
34. C2×#358D3semidirect
228. C2×Dic3SD32semidirect
378. C3#13327semidirect

Maximal subgroups

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

subgroupindex
C2×#51832
C22×#8282
C22×#8002
C2×C3×#3582
#706282
C2×#44922
#678042
#133273