Finite Group Databasebeta

← #10695715

Subgroups — #10695715

Order 768.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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#78617C211yesmaximal, characteristic
2C2×#4721C211yesmaximal, characteristic
3#73518C211yesmaximal, characteristic
4#69946C211yesmaximal, characteristic
5#69924C211yesmaximal, characteristic
6#69457C211yesmaximal, characteristic
7C3×#1282C211yesmaximal, characteristic
8#9961—31nomaximal, Sylow
9C22×C4⋊Dic3C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15#5212C2211yescharacteristic
16#5116—22no—
17C2×#817—21no—
18C2×#816—21no—
19#4943—22no—
20#4936—21no—
21#4929—21no—
22#4739—21no—
23#4739—21no—
24#4722—21no—
25#4722—21no—
26#4721—22no—
27#4721—22no—
28#4659—22no—
29#4551—22no—
30#4527—22no—
31#4467—22no—
32#3189—31no—
33C2×#483—31no—
34#2503—31no—
35#1996—31no—
36#1823—31no—
37#1520—31no—
38#1282D311yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C22×Dic6D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8D412yes—
45C2×C3×C22⋊C4D412yes—
46C2×C3×C42—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D412yes—
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59#924—21no—
60#924—21no—
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic12—22no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8D412yes—
74C3×#95—21no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—21no—
78#818—22no—
79#817—24no—
80#816—24no—
81#816—22no—
82#816—22no—
83C2×C22⋊Q8—31no—
84C2×C22⋊Q8—31no—
85C22×C4⋊C4—31no—
86#484—61no—
87#484—61no—
88#483—62no—
89#483—62no—
90#467—62no—
91#451—61no—
92#451—61no—
93C2×#106—61no—
94C2×#103—61no—
95C2×C22⋊C8C2×D311yescharacteristic
96C2×C22⋊C8C2×D311yescharacteristic
97#400—62no—
98#391—61no—
99#382—61no—
100#381—62no—
101#380—31no—
102C4×C22⋊C4C2×D311yescharacteristic
103#342—62no—
104#340—62no—
105#328—62no—
106C24×C3C2×D411yescharacteristic
107C22×C3×C4—22no—
108C22×C3×C4—42no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—41no—
121C22×Dic3—41no—
122C22×Dic3—42no—
123C22×Dic3—42no—
124C22×Dic3—41no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—42no—
130C2×Dic6—42no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—42no—
134C2×C3×C8—42no—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—22no—
137C3×C42—42no—
138#217—41no—
139#217—41no—
140#217—41no—
141#217—41no—
142#217—42no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—22no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—42no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic12—82no—
157C22×Q8—31no—
158C22×Q8—31no—
159C23×C4—61no—
160C23×C4C22×D311yescharacteristic
161C2×Q16—62no—
162C22×C8—21no—
163C22×C8D1212yes—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4D1212yes—
187C2×C42—21no—
188#107—62no—
189#106—64no—
190#103—64no—
191#103—62no—
192#103—62no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8D1212yes—
197#95—21no—
198#95—64no—
199#95—64no—
200#95—61no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3#12011yescharacteristic
204C23×C3#12111yescharacteristic
205C23×C3#12711yescharacteristic
206C23×C3#13512yes—
207C23×C3#13712yes—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228C2×Dic3—44no—
229C2×Dic3—82no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237Dic6—42no—
238Dic6—42no—
239Dic6—42no—
240Dic6—81no—
241C3×C8—42no—
242C3×C8—42no—
243C24C2×D1211yescharacteristic
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—122no—
249C2×Q8—122no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4—122no—
260C22×C4—122no—
261C22×C4—121no—
262C22×C4C2×D1211yescharacteristic
263C22×C4C2×D1211yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4#23911yescharacteristic
268Q16—242no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—42no—
272C2×C8—42no—
273C4⋊C4—62no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—62no—
278C4⋊C4—61no—
279C4⋊C4—122no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—122no—
293C42—42no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3#39211yescharacteristic
303C22×C3#45212yes—
304C22×C3#46412yes—
305C22×C3#49511yescharacteristic
306C22×C3#49711yescharacteristic
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—82no—
311C3×C4#45812yes—
312Dic3—81no—
313Dic3—81no—
314Dic3—82no—
315Dic3—81no—
316Dic3—81no—
317Dic3—161no—
318C23—41no—
319C23—41no—
320C23#87411yescharacteristic
321C23#88211yescharacteristic
322C23#89511yescharacteristic
323C23#90412yes—
324C23#90912yes—
325Q8—122no—
326Q8—122no—
327Q8—122no—
328Q8—241no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—124no—
346C2×C4—241no—
347C2×C4—241no—
348C2×C4—242no—
349C2×C4—241no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4#88211yescharacteristic
355C2×C4#88211yescharacteristic
356C2×C4#89511yescharacteristic
357C2×C4#89511yescharacteristic
358C8—42no—
359C8—42no—
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—22no—
363C2×C3—21no—
364C2×C3#200911yescharacteristic
365C2×C3#201511yescharacteristic
366C2×C3#202211yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—42no—
374C22—42no—
375C22#467111yescharacteristic
376C22#469611yescharacteristic
377C22#471511yescharacteristic
378C22#473312yes—
379C22#484512yes—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—82no—
384C4—242no—
385C4—241no—
386C4—241no—
387C4—241no—
388C4—241no—
389C4—481no—
390C4#474212yes—
391C3#996111yescharacteristic, complemented, Sylow
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2#6867711yescharacteristic
397C2#6896611yescharacteristic
398C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1282D3
hypocenterC3#9961
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
391. C3#9961semidirect

Maximal subgroups

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

subgroupindex
#786172
C2×#47212
#735182
#699462
#699242
#694572
C3×#12822
#99613