Finite Group Databasebeta

← #10698112

Subgroups — #10698112

Order 768.

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

normal subgroups only

This group has 672 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#82178C211yesmaximal, characteristic, complemented
2#82177C211yesmaximal, characteristic
3#78642C211yesmaximal, characteristic
4C3×#1821C211yesmaximal, characteristic, complemented
5#69980C211yesmaximal, characteristic, complemented
6#69977C211yesmaximal, characteristic
7#67683C211yesmaximal, characteristic, complemented
8#10331—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#929—22no—
12C2×#924C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×#407C2211yescharacteristic
15#5210—22no—
16C4×#217C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5134—22no—
19#4993—22no—
20#4988—21no—
21#4983—22no—
22#4975—21no—
23#4975—21no—
24#4973—21no—
25#4738—21no—
26#4736—21no—
27#4731—21no—
28#4722—21no—
29C3×#343—22no—
30#4470—22no—
31#4467—22no—
32#3193—31no—
33#2502—31no—
34#2501—31no—
35#1821D311yescharacteristic, complemented
36#1508—31no—
37#1507—31no—
38#1297—31no—
39C23×C3×C4D412yes—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×D12D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C2×C3×M16D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59#929—48no—
60C2×(C2×D3)⋊C4—24no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64#924—21no—
65#924—21no—
66#924—21no—
67#924—21no—
68C2×Dic3⋊C4—24no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—21no—
73C2×#204—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3⋊#95—24no—
79C3⋊#95D412yes—
80C3⋊#95D412yes—
81#821—22no—
82#818—22no—
83#816—22no—
84C2×#123—62no—
85C2×C22⋊Q8—31no—
86C2×#121—31no—
87C22×C4⋊C4C2×D311yescharacteristic
88#484—61no—
89#480—61no—
90#468—62no—
91#450—61no—
92#448—61no—
93#407C2×D311yescharacteristic
94#407C2×D311yescharacteristic
95#399—62no—
96#398—62no—
97#391—61no—
98#390—61no—
99#385—62no—
100#384—61no—
101#384—61no—
102C4×C22⋊C4—31no—
103C4×C22⋊C4—31no—
104#343—22no—
105#328—62no—
106#327—62no—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—42no—
113C22×C3×C4—44no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—44no—
121C2×#67—41no—
122C2×#67—41no—
123C22×Dic3—22no—
124C22×Dic3—22no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—42no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—42no—
134C3×M16—42no—
135C3×M16—42no—
136C2×C3×C8—22no—
137C2×C3×C8—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142#217—24no—
143#217—24no—
144#217—41no—
145#217—41no—
146(C2×D3)⋊C4—48no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—48no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—84no—
157C4×Dic3—42no—
158C4×Dic3—42no—
159#204—82no—
160C22×Q8—31no—
161C22×D4—31no—
162C22×D4—61no—
163C23×C4D1212yes—
164C23×C4C22×D311yescharacteristic
165C2×SD16—62no—
166C2×M16—21no—
167C2×M16#6712yes—
168#123—128no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173#121—61no—
174#121—61no—
175#121—61no—
176#121—61no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—64no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—64no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—64no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C42—61no—
194C2×C42—61no—
195#107—62no—
196#103—62no—
197#102—62no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202#95—32no—
203#95—32no—
204#95—64no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—41no—
208C23×C3#12011yescharacteristic
209C23×C3#12011yescharacteristic
210C23×C3#12111yescharacteristic
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—41no—
214C22×D3—42no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—44no—
224C2×C3×C4—82no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229#67—82no—
230#67—82no—
231C2×Dic3—24no—
232C2×Dic3—24no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243D12—42no—
244D12—81no—
245Dic6—42no—
246Dic6—81no—
247C3×C8—42no—
248C3×C8—81no—
249C24—61no—
250C24C2×D1211yescharacteristic
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—122no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—122no—
257C2×D4—124no—
258C2×D4—121no—
259C2×D4—121no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4#23511yescharacteristic
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4C2×#6711yescharacteristic
276SD16—242no—
277M16—42no—
278M16—42no—
279C2×C8—22no—
280C2×C8—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—128no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C4⋊C4—244no—
293C22⋊C4—64no—
294C22⋊C4—64no—
295C22⋊C4—128no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—122no—
301C42—122no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—44no—
310C22×C3#39211yescharacteristic
311C22×C3#45312yes—
312C22×C3#45612yes—
313C2×D3—84no—
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—82no—
321C3×C4#45812yes—
322Dic3—44no—
323Dic3—44no—
324Dic3—81no—
325Dic3—81no—
326Dic3—81no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—122no—
331C23—122no—
332C23—121no—
333C23—122no—
334C23#88211yescharacteristic
335C23#88211yescharacteristic
336C23#93011yescharacteristic
337Q8—122no—
338Q8—241no—
339D4—122no—
340D4—241no—
341D4—242no—
342D4—242no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—64no—
353C2×C4—64no—
354C2×C4—82no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4#88411yescharacteristic
366C2×C4#88411yescharacteristic
367C2×C4#93711yescharacteristic
368C2×C4#94011yescharacteristic
369C8—42no—
370C8—81no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—41no—
374C2×C3#200211yescharacteristic
375C2×C3#200511yescharacteristic
376C2×C3#202211yescharacteristic
377D3—161no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—44no—
386C22—244no—
387C22—241no—
388C22—241no—
389C22—241no—
390C22#474712yes—
391C22#475212yes—
392C22#496011yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—82no—
397C4—124no—
398C4—124no—
399C4—241no—
400C4—241no—
401C4—241no—
402C4#505312yes—
403C3#1033111yescharacteristic, complemented, Sylow
404C2—22no—
405C2—22no—
406C2—41no—
407C2—481no—
408C2#6953911yescharacteristic
409C2#7008211yescharacteristic
410C2#7011911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1821C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1821D3
hypocenterC3#10331
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
1. #82178C2semidirect
4. C3×#1821C2semidirect
5. #69980C2semidirect
7. #67683C2semidirect
35. #1821D3semidirect
403. C3#10331semidirect

Maximal subgroups

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

subgroupindex
#821782
#821772
#786422
C3×#18212
#699802
#699772
#676832
#103313