Finite Group Databasebeta

← #10698119

Subgroups — #10698119

Order 768.

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

normal subgroups only

This group has 600 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#82180C211yesmaximal, characteristic
2#82179C211yesmaximal, characteristic, complemented
3#78642C211yesmaximal, characteristic
4C3×#1821C211yesmaximal, characteristic, complemented
5#69980C211yesmaximal, characteristic, complemented
6#69977C211yesmaximal, characteristic
7#67683C211yesmaximal, characteristic, complemented
8#10332—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14#5213—22no—
15#5211—22no—
16C4×#217C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5134—22no—
19#4992—22no—
20#4988—21no—
21#4985—22no—
22#4977—21no—
23#4977—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#2504—31no—
34#2503—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—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—24no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×#204—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95D412yes—
79C3⋊#95D412yes—
80#821—22no—
81#818—22no—
82#816—22no—
83C2×C22⋊Q8—31no—
84C2×#121—31no—
85C22×C4⋊C4C2×D311yescharacteristic
86#484—61no—
87#480—61no—
88#468—62no—
89#450—61no—
90#448—61no—
91#407C2×D311yescharacteristic
92#407C2×D311yescharacteristic
93#400—62no—
94#397—62no—
95#391—61no—
96#390—61no—
97#386—62no—
98#382—61no—
99#382—61no—
100#381—62no—
101C4×C22⋊C4—31no—
102C4×C22⋊C4—31no—
103#343—22no—
104#328—62no—
105#327—62no—
106C24×C3C2×D411yescharacteristic
107C23×D3—21no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—42no—
112C22×C3×C4—44no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—44no—
120C2×#67—41no—
121C2×#67—41no—
122C22×Dic3—22no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—22no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—42no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—42no—
133C3×M16—42no—
134C3×M16—42no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141#217—42no—
142#217—42no—
143#217—41no—
144#217—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—84no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—42no—
155C4×Dic3—42no—
156#204—82no—
157C22×Q8—31no—
158C22×D4—31no—
159C22×D4—61no—
160C23×C4D1212yes—
161C23×C4C22×D311yescharacteristic
162C2×SD16—62no—
163C2×M16—21no—
164C2×M16#6712yes—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169#121—61no—
170#121—61no—
171#121—61no—
172#121—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—64no—
187C2×C42—61no—
188C2×C42—61no—
189#107—62no—
190#103—62no—
191#102—62no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196#95—32no—
197#95—32no—
198#95—64no—
199#95—64no—
200#95—64no—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—41no—
204C23×C3#12011yescharacteristic
205C23×C3#12011yescharacteristic
206C23×C3#12111yescharacteristic
207C22×D3—42no—
208C22×D3—42no—
209C22×D3—41no—
210C22×D3—42no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—44no—
218C2×C3×C4—42no—
219C2×C3×C4—44no—
220C2×C3×C4—82no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225#67—82no—
226#67—82no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239D12—42no—
240D12—81no—
241Dic6—42no—
242Dic6—81no—
243C3×C8—42no—
244C3×C8—81no—
245C24—61no—
246C24C2×D1211yescharacteristic
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—122no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—122no—
253C2×D4—124no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4#23511yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271C22×C4C2×#6711yescharacteristic
272SD16—242no—
273M16—42no—
274M16—42no—
275C2×C8—22no—
276C2×C8—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C22⋊C4—122no—
288C22⋊C4—122no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—244no—
294C42—122no—
295C42—122no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—44no—
304C22×C3#39211yescharacteristic
305C22×C3#45412yes—
306C22×C3#45512yes—
307C2×D3—84no—
308C2×D3—81no—
309C2×D3—81no—
310C2×D3—81no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4#45812yes—
316Dic3—82no—
317Dic3—82no—
318Dic3—81no—
319Dic3—81no—
320Dic3—81no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—122no—
325C23—122no—
326C23—121no—
327C23—122no—
328C23#88211yescharacteristic
329C23#88211yescharacteristic
330C23#93011yescharacteristic
331Q8—122no—
332Q8—241no—
333D4—122no—
334D4—241no—
335D4—242no—
336D4—242no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4#88411yescharacteristic
360C2×C4#88411yescharacteristic
361C2×C4#93711yescharacteristic
362C2×C4#94011yescharacteristic
363C8—42no—
364C8—81no—
365C2×C3—22no—
366C2×C3—22no—
367C2×C3—41no—
368C2×C3#200311yescharacteristic
369C2×C3#200411yescharacteristic
370C2×C3#202211yescharacteristic
371D3—161no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—44no—
380C22—244no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#474612yes—
385C22#475312yes—
386C22#496011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—242no—
392C4—242no—
393C4—241no—
394C4—241no—
395C4—241no—
396C4#505312yes—
397C3#1033211yescharacteristic, complemented, Sylow
398C2—22no—
399C2—22no—
400C2—41no—
401C2—481no—
402C2#6953911yescharacteristic
403C2#7008411yescharacteristic
404C2#7011611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1821C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1821D3
hypocenterC3#10332
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
2. #82179C2semidirect
4. C3×#1821C2semidirect
5. #69980C2semidirect
7. #67683C2semidirect
35. #1821D3semidirect
397. C3#10332semidirect

Maximal subgroups

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

subgroupindex
#821802
#821792
#786422
C3×#18212
#699802
#699772
#676832
#103323