Finite Group Databasebeta

← #10695700

Subgroups — #10695700

Order 768.

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

normal subgroups only

This group has 568 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#82775C211yesmaximal, characteristic
2#82741C211yesmaximal, characteristic
3C2×#4722C211yesmaximal, characteristic
4C3×#1520C211yesmaximal, characteristic
5#69529C211yesmaximal, characteristic
6#69502C211yesmaximal, characteristic
7#67678C211yesmaximal, characteristic
8#9959—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5229—21no—
15#5228—21no—
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#818—21no—
19C2×#816—21no—
20#5051—22no—
21#5050—21no—
22#5044—21no—
23#4981—22no—
24#4975—21no—
25#4970—21no—
26#4946—22no—
27#4722—22no—
28#4722—22no—
29C3×#328—22no—
30#4529—22no—
31#4527—22no—
32#3189—31no—
33C2×#484—31no—
34#2502—31no—
35#1996—31no—
36#1824—31no—
37#1520D311yescharacteristic
38#1282—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—21no—
42C22×Dic6D411yescharacteristic
43C22×C3×C8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54C2×#217D412yes—
55C2×#216—22no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3D412yes—
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—24no—
69C2×C4×Dic3—21no—
70C22×C3⋊C8D412yes—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#95—21no—
74#835—22no—
75C3⋊#95—24no—
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#818—24no—
80#816—24no—
81#810—22no—
82#807—22no—
83C2×C22⋊Q8—31no—
84C2×C22⋊Q8C2×D311yescharacteristic
85C22×C4⋊C4—31no—
86#484—62no—
87#484—61no—
88#484—61no—
89#484—62no—
90#467—62no—
91#451—61no—
92#451—61no—
93C2×#107—61no—
94C2×#103—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97#398—62no—
98#391—61no—
99#385—62no—
100#384—61no—
101#380C2×D311yescharacteristic
102C4×C22⋊C4—31no—
103#342—62no—
104#340—62no—
105#328—22no—
106C24×C3C2×D411yescharacteristic
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—42no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—42no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—22no—
121C22×Dic3—22no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—41no—
125C22×Dic3—41no—
126C22×Dic3—82no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—42no—
130C2×C3×C8—22no—
131C2×C3×C8—42no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—42no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—42no—
140#217—22no—
141#217—41no—
142#217—41no—
143#217Q1614yescomplemented
144#216—82no—
145C4⋊Dic3—21no—
146C4⋊Dic3—22no—
147C4⋊Dic3—21no—
148C4⋊Dic3—22no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—84no—
154C4×Dic3—42no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157C22×Q8—31no—
158C22×Q8D1211yescharacteristic
159C23×C4—61no—
160C23×C4C22×D311yescharacteristic
161C2×Q16—62no—
162C22×C8—21no—
163C22×C8—32no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—31no—
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—64no—
183C2×C4⋊C4D1211yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—32no—
187C2×C22⋊C4—61no—
188C2×C42—61no—
189#107—62no—
190#107—64no—
191#103—62no—
192#103—64no—
193#103—62no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C22⋊C8#6712yes—
198#95—21no—
199#95—64no—
200#95—61no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3#12011yescharacteristic
204C23×C3#12111yescharacteristic
205C23×C3#12711yescharacteristic
206C23×C3C2×Q1612yes—
207C23×C3#13712yes—
208C3×Q8—42no—
209C3×Q8—81no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×Dic3—24no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—44no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—82no—
235C2×Dic3—82no—
236C2×Dic3—81no—
237C2×Dic3—81no—
238Dic6—42no—
239Dic6—42no—
240C3×C8—42no—
241C3⋊C8—42no—
242C24D3×D411yescharacteristic
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—42no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—122no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—62no—
254C22×C4—62no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4—242no—
260C22×C4C2×D1211yescharacteristic
261C22×C4#23511yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4C2×#6711yescharacteristic
266Q16—242no—
267C2×C8—22no—
268C2×C8—42no—
269C2×C8—62no—
270C2×C8—122no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—61no—
277C4⋊C4—62no—
278C4⋊C4—62no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—244no—
285C22⋊C4—34no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—62no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C42—122no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3#39211yescharacteristic
302C22×C3#45112yes—
303C22×C3#46212yes—
304C22×C3#49411yescharacteristic
305C22×C3#49711yescharacteristic
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4—161no—
312C3×C4#45812yes—
313Dic3—44no—
314Dic3—82no—
315Dic3—81no—
316Dic3—81no—
317C23—41no—
318C23—41no—
319C23#88211yescharacteristic
320C23#88411yescharacteristic
321C23#91512yes—
322C23D3×Q1612yes—
323C23#94011yescharacteristic
324Q8—42no—
325Q8—81no—
326Q8—122no—
327Q8—122no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—64no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—124no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—242no—
349C2×C4—242no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4#88211yescharacteristic
353C2×C4#88411yescharacteristic
354C2×C4#93011yescharacteristic
355C2×C4#93711yescharacteristic
356C8—42no—
357C8—122no—
358C2×C3—22no—
359C2×C3—21no—
360C2×C3—22no—
361C2×C3—21no—
362C2×C3#200811yescharacteristic
363C2×C3#201611yescharacteristic
364C2×C3#202211yescharacteristic
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—42no—
372C22—42no—
373C22#479512yes—
374C22#480912yes—
375C22#496011yescharacteristic
376C22#516711yescharacteristic
377C22#518311yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—81no—
382C4—81no—
383C4—124no—
384C4—161no—
385C4—242no—
386C4—241no—
387C4—241no—
388C4#474212yes—
389C3#995911yescharacteristic, complemented, Sylow
390C2—22no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2#7003111yescharacteristic
395C2#7028311yescharacteristic
396C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1520D3
hypocenterC3#9959
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
143. #217Q16semidirect
389. C3#9959semidirect

Maximal subgroups

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

subgroupindex
#827752
#827412
C2×#47222
C3×#15202
#695292
#695022
#676782
#99593