Finite Group Databasebeta

← #10693038

Subgroups — #10693038

Order 768.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#82375C211yesmaximal, characteristic
2#79263C211yesmaximal, characteristic, complemented
3C3×#1486C211yesmaximal, characteristic
4#71598C211yesmaximal, characteristic
5#70528C211yesmaximal, characteristic, complemented
6#67738C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9486—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#924—21no—
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5222C412yes—
15#5212—21no—
16C2×#824—21no—
17#4983C412yes—
18#4944—22no—
19#4936C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4830—22no—
23#4718—22no—
24C8×Dic6—22no—
25C3×D4⋊C8—22no—
26#4554—21no—
27D4⋊(C3⋊C8)—22no—
28#2982—31no—
29#2536—31no—
30#1801—31no—
31#1486—31no—
32#1486D311yescharacteristic
33#1286—31no—
34#1286—31no—
35C22×C3×D4C2×C411yescharacteristic
36C23×C3×C4—21no—
37C22×Dic6D411yescharacteristic
38C2×C3×M16—21no—
39C3×C4×D4—21no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#217—22no—
47C2×#217—22no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50#924—42no—
51#924—42no—
52C2×Dic3⋊C4—22no—
53C2×Dic3⋊C4C2×C412yes—
54C2×C4×Dic3D412yes—
55C2×#208—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C4×Dic6—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C4×C8D412yes—
65C3⋊#95—21no—
66C3⋊#95—22no—
67C3⋊#95C2×C412yes—
68#830—21no—
69#824—42no—
70Dic3⋊C8—24no—
71C8×Dic3—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4×C3⋊C8D412yes—
76C2×C22⋊Q8—61no—
77C2×C4×Q8—31no—
78C2×C4×D4C2×D311yescharacteristic
79C8×Q8—62no—
80C8×Q8—62no—
81#431—31no—
82#431—62no—
83#431C2×D311yescharacteristic
84#423—31no—
85#423C2×D311yescharacteristic
86C2×#101—61no—
87#398—32no—
88#394—32no—
89#388—62no—
90#382—31no—
91#380—61no—
92#343—61no—
93D4⋊C8—22no—
94D4⋊C8—62no—
95C24×C3—21no—
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—42no—
99C22×C3×C4—21no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—22no—
110C22×Dic3C22⋊C412yes—
111C22×Dic3C22⋊C412yes—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6D812yes—
117C2×Dic6SD1612yes—
118C3×M16—41no—
119C3×M16—41no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130#217—44no—
131#217—44no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136Dic3⋊C4—22no—
137Dic3⋊C4—22no—
138Dic3⋊C4—44no—
139C4×Dic3—22no—
140C4×Dic3—22no—
141#208—41no—
142#208—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C22×Q8—31no—
146C22×D4C4×D311yescharacteristic
147C23×C4—21no—
148C2×M16—21no—
149C2×M16—61no—
150C22⋊Q8—122no—
151C22⋊Q8—122no—
152C4×Q8—61no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—21no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—62no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4C4×D311yescharacteristic
165C2×C22⋊C4—21no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C42—32no—
169C2×C42C22×D311yescharacteristic
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—64no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177#101—122no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180C4×C8—32no—
181C4×C8—64no—
182C4×C8#6712yes—
183#95—32no—
184#95—62no—
185#95—61no—
186C23×C3—22no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3C2×C22⋊C411yescharacteristic
194C3×D4—22no—
195C3×D4—22no—
196C3×D4—41no—
197C3×D4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×C3×C4C2×D811yescharacteristic
214C2×C3×C4C2×SD1611yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4#13511yescharacteristic
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3#10214yes—
225Dic6—22no—
226Dic6—22no—
227Dic6—22no—
228Dic6—22no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24—21no—
234C2×Q8—32no—
235C2×Q8—32no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—42no—
243C22×C4—21no—
244C22×C4—32no—
245C22×C4—32no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—62no—
253C22×C4C2×C4×D311yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4D3×D411yescharacteristic
256M16—41no—
257M16—41no—
258M16—121no—
259M16—121no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—62no—
267C4⋊C4—61no—
268C4⋊C4—62no—
269C4⋊C4—61no—
270C4⋊C4—124no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—124no—
276C22⋊C4—124no—
277C42—62no—
278C42—62no—
279C42#23511yescharacteristic
280C42D3×D411yescharacteristic
281C42#23711yescharacteristic
282C42C2×#6711yescharacteristic
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—42no—
286C22×C3—42no—
287C22×C3—42no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3#38611yescharacteristic
293C22×C3C2×#10211yescharacteristic
294C22×C3#41611yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—81no—
300C3×C4#46511yescharacteristic
301C3×C4#46511yescharacteristic
302C3×C4#46611yescharacteristic
303C3×C4#46611yescharacteristic
304Dic3—24no—
305Dic3—42no—
306Dic3—82no—
307C23—22no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23D3×C22⋊C411yescharacteristic
315Q8—62no—
316Q8—62no—
317Q8—62no—
318Q8—62no—
319D4—22no—
320D4—22no—
321D4—41no—
322D4—41no—
323C2×C4—34no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4#67⋊C411yescharacteristic
341C2×C4#88311yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4D3×D811yescharacteristic
344C2×C4#91211yescharacteristic
345C2×C4D3×SD1611yescharacteristic
346C2×C4#91611yescharacteristic
347C2×C4C4×#6711yescharacteristic
348C2×C4#93711yescharacteristic
349C2×C4#93811yescharacteristic
350C8—42no—
351C8—81no—
352C8—122no—
353C8—241no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—82no—
357C2×C3#189111yescharacteristic
358C2×C3#189311yescharacteristic
359C2×C3#191211yescharacteristic
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22D3×#10211yescharacteristic
370C22#477211yescharacteristic
371C22#495611yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—64no—
377C4—81no—
378C4—122no—
379C4—242no—
380C4#477511yescharacteristic
381C4#477811yescharacteristic
382C4#515711yescharacteristic
383C4#517111yescharacteristic
384C3#948611yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—82no—
388C2#6999211yescharacteristic
389C2#7020211yescharacteristic
390C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#1486D3
hypocenterC3#9486
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #79263C2semidirect
5. #70528C2semidirect
6. #67738C2semidirect
7. #67713C2semidirect
384. C3#9486semidirect

Maximal subgroups

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

subgroupindex
#823752
#792632
C3×#14862
#715982
#705282
#677382
#677132
#94863