Finite Group Databasebeta

← #10695887

Subgroups — #10695887

Order 768.

562 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C3×#3192C211yesmaximal, characteristic
2C2×#5034C211yesmaximal, characteristic, complemented
3C2×#5033C211yesmaximal, characteristic, complemented
4#71676C211yesmaximal, characteristic, complemented
5#71676C211yesmaximal, characteristic, complemented
6#69973C211yesmaximal, characteristic, complemented
7#67688C211yesmaximal, characteristic, complemented
8#9992—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C3×#481—21no—
13C3×#481—21no—
14C3×#469—22no—
15C3×#447—21no—
16C3×#447—21no—
17C3×#407C2211yescharacteristic
18C2×#832—21no—
19C2×#832—21no—
20#5207C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#5034—22no—
24#5034—22no—
25#5033—22no—
26#5033—22no—
27C2×#808—21no—
28C2×#807—21no—
29#4554—22no—
30#4525—22no—
31#4525—22no—
32#3192D311yescharacteristic
33C2×#481—31no—
34C2×#480—31no—
35#1821—31no—
36#1471—31no—
37#1471—31no—
38#1298—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C2×C3×D8—22no—
46C2×C3×M16D412yes—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3D412yes—
65C2×#208—21no—
66C22×C3⋊C8—21no—
67C22×C3⋊C8—21no—
68C3×#106—22no—
69C3×#102—22no—
70C3×#102—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73#832—24no—
74#832—24no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79#830D412yes—
80#830D412yes—
81#808—24no—
82#807—24no—
83C2×#121C2×D311yescharacteristic
84C2×#121C2×D311yescharacteristic
85C22×C4⋊C4—31no—
86#481—21no—
87#481—21no—
88#481—62no—
89#481—62no—
90#480—62no—
91#480—62no—
92#469—22no—
93#447—21no—
94#447—21no—
95C2×#107—61no—
96C2×#106—61no—
97C2×#102—61no—
98C2×#102—61no—
99#407—31no—
100#407C2×D311yescharacteristic
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103#343—62no—
104#323—62no—
105#323—62no—
106C24×C3—21no—
107C24×C3—21no—
108C24×C3#4111yescharacteristic
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—82no—
120C2×C3×D4—82no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—42no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×Dic3—22no—
131C22×Dic3—41no—
132C22×Dic3—41no—
133C22×Dic3—82no—
134C3×D8—82no—
135C3×M16—42no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149#208—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—42no—
153C2×C3⋊C8—42no—
154C2×C3⋊C8—41no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4#6711yescharacteristic
158C22×D4#6711yescharacteristic
159C23×C4—61no—
160C23×C4C22×D311yescharacteristic
161C2×D8—22no—
162C2×M16—61no—
163C2×M16#6712yes—
164C22×C8—61no—
165C22×C8—61no—
166#121—21no—
167#121—21no—
168#121—21no—
169#121—21no—
170#121—21no—
171#121—21no—
172#121—21no—
173#121—21no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4#6711yescharacteristic
181C2×C4⋊C4#6711yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184#107—64no—
185#106—22no—
186#106—64no—
187#102—22no—
188#102—22no—
189#102—64no—
190#102—64no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—32no—
194C22⋊C8—32no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C23×C3—22no—
200C23×C3—22no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—42no—
205C23×C3—42no—
206C23×C3—81no—
207C23×C3—81no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3#12111yescharacteristic
211C23×C3#12111yescharacteristic
212C23×C3#12111yescharacteristic
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—42no—
216C3×D4—42no—
217C3×D4—82no—
218C3×D4—82no—
219C3×D4—82no—
220C3×D4—82no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12711yescharacteristic
235C2×Dic3—42no—
236C2×Dic3—82no—
237C2×Dic3—82no—
238C2×Dic3—82no—
239C3×C8—42no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C3⋊C8—81no—
243C24—21no—
244C24—21no—
245C24#23711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—62no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4—242no—
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4C2×#6711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271D8—82no—
272M16—42no—
273M16—122no—
274C2×C8—22no—
275C2×C8—62no—
276C2×C8—62no—
277C2×C8—122no—
278C2×C8—122no—
279C2×C8—121no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—44no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3—162no—
307C22×C3—162no—
308C22×C3#39211yescharacteristic
309C22×C3#47112yes—
310C22×C3#47212yes—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—81no—
315C3×C4—81no—
316C3×C4#45812yes—
317Dic3—82no—
318C23—22no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23#93811yescharacteristic
330C23#93911yescharacteristic
331C23#93911yescharacteristic
332D4—42no—
333D4—42no—
334D4—42no—
335D4—42no—
336D4—82no—
337D4—82no—
338D4—82no—
339D4—82no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—122no—
351C2×C4—242no—
352C2×C4—242no—
353C2×C4—242no—
354C2×C4#93711yescharacteristic
355C2×C4#93711yescharacteristic
356C2×C4#94011yescharacteristic
357C2×C4#95311yescharacteristic
358C8—42no—
359C8—122no—
360C8—122no—
361C8—241no—
362C2×C3—22no—
363C2×C3—22no—
364C2×C3—41no—
365C2×C3—161no—
366C2×C3—161no—
367C2×C3#201811yescharacteristic
368C2×C3#201911yescharacteristic
369C2×C3#202211yescharacteristic
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—44no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—162no—
385C22—162no—
386C22#519912yes—
387C22#520012yes—
388C22#522311yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—81no—
393C4—81no—
394C4—242no—
395C4#524312yes—
396C3#999211yescharacteristic, complemented, Sylow
397C2—22no—
398C2—22no—
399C2—41no—
400C2—161no—
401C2—161no—
402C2#7184911yescharacteristic
403C2#7209511yescharacteristic
404C2#7209711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3192C2
socleC22×C3#392
hypercenter#3192D3
hypocenterC3#9992
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3192

Complements

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

NH ≅ G/Ntype
2. C2×#5034C2semidirect
3. C2×#5033C2semidirect
4. #71676C2semidirect
5. #71676C2semidirect
6. #69973C2semidirect
7. #67688C2semidirect
396. C3#9992semidirect

Maximal subgroups

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

subgroupindex
C3×#31922
C2×#50342
C2×#50332
#716762
#699732
#676882
#99923