Finite Group Databasebeta

← #10695701

Subgroups — #10695701

Order 768.

568 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 109 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×#4722C211yesmaximal, characteristic
3#73517C211yesmaximal, characteristic
4#69946C211yesmaximal, characteristic
5#69925C211yesmaximal, characteristic
6#69457C211yesmaximal, characteristic
7C3×#1282C211yesmaximal, characteristic
8#9959—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×#818—21no—
18C2×#816—21no—
19#4950—22no—
20#4935—21no—
21#4929—21no—
22#4739—21no—
23#4739—21no—
24#4722—21no—
25#4722—21no—
26#4722—22no—
27#4722—22no—
28#4661—22no—
29#4551—22no—
30#4527—22no—
31#4467—22no—
32#3189—31no—
33C2×#484—31no—
34#2502—31no—
35#1996—31no—
36#1824—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⋊Dic3—24no—
57C2×C4⋊Dic3D412yes—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60#924—21no—
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67#924—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic12—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75C3×#95—21no—
76C3⋊#95—24no—
77C3⋊#95—21no—
78#818—24no—
79#818—22no—
80#816—24no—
81#816—22no—
82#816—22no—
83C2×C22⋊Q8—31no—
84C2×C22⋊Q8—31no—
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⋊C8C2×D311yescharacteristic
96C2×C22⋊C8C2×D311yescharacteristic
97#398—62no—
98#391—61no—
99#385—62no—
100#384—61no—
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—22no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—41no—
121C22×Dic3—41no—
122C22×Dic3—41no—
123C22×Dic3—82no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—42no—
133C2×C3×C8—42no—
134C3×C22⋊C4—22no—
135C3×C22⋊C4Q1614yescomplemented
136C3×C42—42no—
137#217—41no—
138#217—41no—
139#217—41no—
140#217—41no—
141#217—42no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—42no—
151C4⋊Dic3—84no—
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×C4⋊C4—64no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4D1212yes—
188C2×C42—21no—
189#107—64no—
190#107—62no—
191#103—64no—
192#103—62no—
193#103—62no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197C22⋊C8D1212yes—
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—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—24no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
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—44no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—82no—
230C2×Dic3—82no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236Dic6—42no—
237Dic6—42no—
238Dic6—42no—
239Dic6—81no—
240C3×C8—42no—
241C3×C8—42no—
242C24C2×D1211yescharacteristic
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—122no—
248C2×Q8—122no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—61no—
252C22×C4—62no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4—242no—
260C22×C4C2×D1211yescharacteristic
261C22×C4C2×D1211yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4#23911yescharacteristic
266Q16—242no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—62no—
276C4⋊C4—61no—
277C4⋊C4—122no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—244no—
285C22⋊C4—22no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—122no—
291C22⋊C4Dic1214yescomplemented
292C42—42no—
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—44no—
310C3×C4#45812yes—
311Dic3—81no—
312Dic3—82no—
313Dic3—81no—
314Dic3—81no—
315Dic3—81no—
316Dic3—161no—
317C23—41no—
318C23—41no—
319C23#87411yescharacteristic
320C23#88211yescharacteristic
321C23#89511yescharacteristic
322C23C2×Dic1212yes—
323C23#90912yes—
324Q8—122no—
325Q8—122no—
326Q8—122no—
327Q8—241no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—122no—
338C2×C4—124no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—242no—
346C2×C4—242no—
347C2×C4—241no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4#88211yescharacteristic
353C2×C4#88211yescharacteristic
354C2×C4#89511yescharacteristic
355C2×C4#89511yescharacteristic
356C8—42no—
357C8—42no—
358C2×C3—21no—
359C2×C3—22no—
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#467111yescharacteristic
374C22#469911yescharacteristic
375C22#471511yescharacteristic
376C22#473912yes—
377C22#484912yes—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—44no—
382C4—241no—
383C4—241no—
384C4—242no—
385C4—241no—
386C4—241no—
387C4—481no—
388C4#474212yes—
389C3#995911yescharacteristic, complemented, Sylow
390C2—22no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2#6868811yescharacteristic
395C2#6896611yescharacteristic
396C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1282D3
hypocenterC3#9959
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
135. C3×C22⋊C4Q16semidirect
291. C22⋊C4Dic12semidirect
389. C3#9959semidirect

Maximal subgroups

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

subgroupindex
#786172
C2×#47222
#735172
#699462
#699252
#694572
C3×#12822
#99593