Finite Group Databasebeta

← #10695912

Subgroups — #10695912

Order 768.

510 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 87 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#82742C211yesmaximal, characteristic
2C2×#5044C211yesmaximal, characteristic
3C2×#4721C211yesmaximal, characteristic
4C3×#1472C211yesmaximal, characteristic
5#69973C211yesmaximal, characteristic
6#69972C211yesmaximal, characteristic
7#67680C211yesmaximal, characteristic
8#9995—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14#5228—21no—
15C2×#835—21no—
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#817—21no—
19C2×#816—21no—
20#5052—22no—
21#5050—21no—
22#5044—22no—
23#5044—22no—
24#5043—21no—
25#4971—21no—
26C2×#807—21no—
27#4721—22no—
28#4721—22no—
29C3×#324—22no—
30#4554—22no—
31#4469—22no—
32#3195—31no—
33C2×#484—31no—
34C2×#483—31no—
35#1821—31no—
36#1472—31no—
37#1472D311yescharacteristic
38#1298—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C22×C3×C8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×#217—21no—
53C2×#216—22no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D412yes—
60C2×C4⋊Dic3D411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×#208D412yes—
67C22×C3⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8D412yes—
72#835—24no—
73#835—22no—
74#830—21no—
75#830—21no—
76#830—21no—
77#830D412yes—
78#817—24no—
79#816—24no—
80#810—22no—
81#808—22no—
82#807—24no—
83C2×C22⋊Q8—31no—
84C2×C22⋊Q8C2×D311yescharacteristic
85C22×C4⋊C4—31no—
86#484—62no—
87#484—62no—
88#483—62no—
89#483—61no—
90#483—61no—
91#483—62no—
92#470—62no—
93#451—61no—
94#451—61no—
95C2×#107—61no—
96C2×#106—61no—
97C2×#103—61no—
98C2×#103—61no—
99#407—31no—
100#407C2×D311yescharacteristic
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#343—62no—
104#324—22no—
105#324—62no—
106C24×C3#4111yescharacteristic
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×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—22no—
122C22×Dic3—41no—
123C22×Dic3—41no—
124C22×Dic3—82no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C3×M16—42no—
129C2×C3×C8—22no—
130C2×C3×C8—42no—
131C2×C3×C8—41no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138#217—41no—
139#217—41no—
140#216—82no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149#208—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—42no—
153C22×Q8—31no—
154C22×Q8D1211yescharacteristic
155C23×C4—61no—
156C23×C4C22×D311yescharacteristic
157C2×Q16—62no—
158C2×M16—21no—
159C2×M16—32no—
160C22×C8—21no—
161C22×C8—61no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4D1211yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—61no—
182#107—64no—
183#106—62no—
184#106—64no—
185#103—62no—
186#103—64no—
187#103—64no—
188#103—62no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—32no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196C22⋊C8#6712yes—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3—41no—
200C23×C3#12111yescharacteristic
201C23×C3#12111yescharacteristic
202C23×C3#12111yescharacteristic
203C3×Q8—42no—
204C3×Q8—42no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—81no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12711yescharacteristic
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—82no—
224C2×Dic3—82no—
225C2×Dic3—81no—
226C2×Dic3—82no—
227Dic6—42no—
228Dic6—42no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—42no—
232C3⋊C8—42no—
233C24#23511yescharacteristic
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—42no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—122no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—62no—
246C22×C4—121no—
247C22×C4—121no—
248C22×C4—242no—
249C22×C4C2×D1211yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4C2×#6711yescharacteristic
255Q16—242no—
256M16—42no—
257M16—122no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—122no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—61no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—61no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—44no—
288C22×C3#39211yescharacteristic
289C22×C3#47112yes—
290C22×C3#47312yes—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4#45812yes—
297Dic3—82no—
298Dic3—81no—
299Dic3—81no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23#88411yescharacteristic
304C23#88411yescharacteristic
305C23#93011yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—122no—
309Q8—122no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—242no—
325C2×C4—242no—
326C2×C4—242no—
327C2×C4—241no—
328C2×C4#88211yescharacteristic
329C2×C4#88211yescharacteristic
330C2×C4#93711yescharacteristic
331C2×C4#94011yescharacteristic
332C8—42no—
333C8—81no—
334C8—122no—
335C8—122no—
336C2×C3—22no—
337C2×C3—22no—
338C2×C3—41no—
339C2×C3#201911yescharacteristic
340C2×C3#202011yescharacteristic
341C2×C3#202211yescharacteristic
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—44no—
350C22#475512yes—
351C22#496011yescharacteristic
352C22#514112yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—81no—
357C4—81no—
358C4—242no—
359C4—241no—
360C4—241no—
361C4#474212yes—
362C3#999511yescharacteristic, complemented, Sylow
363C2—22no—
364C2—22no—
365C2—41no—
366C2#7000711yescharacteristic
367C2#7003111yescharacteristic
368C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1472C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenter#1472D3
hypocenterC3#9995
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
362. C3#9995semidirect

Maximal subgroups

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

subgroupindex
#827422
C2×#50442
C2×#47212
C3×#14722
#699732
#699722
#676802
#99953