Finite Group Databasebeta

← #10696541

Subgroups — #10696541

Order 768.

598 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3213C211yesmaximal, characteristic
2#83288C211yesmaximal, characteristic, complemented
3#82978C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#70824C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#10090—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#951—22no—
12C2×#869C2211yescharacteristic
13C3×#496—21no—
14C3×#489—21no—
15C3×#463—22no—
16C3×#448—21no—
17C3×#447—21no—
18C3×#432C2211yescharacteristic
19#5229—21no—
20#5221—21no—
21#5092—21no—
22#5076—21no—
23#5060—22no—
24#4953—22no—
25#4947—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4540—22no—
30#4537—22no—
31#4481—22no—
32#3213D311yescharacteristic
33#3152—31no—
34#2640—31no—
35#1558—31no—
36#1527—31no—
37#1505—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C22×Dic6D412yes—
43C2×C3×SD16—22no—
44C2×C3×D8—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—24no—
60#951—48no—
61C2×#216—24no—
62C2×#214—24no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4×Dic3—21no—
66C2×#208—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71C3×C4⋊C8D412yes—
72C3×#103—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3⋊#95—24no—
77C3⋊#95—21no—
78C3⋊#95D412yes—
79#830—21no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803D412yes—
83C4×C3⋊C8—21no—
84C2×#137—62no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87C2×#124C2×D311yescharacteristic
88#497—62no—
89#496—21no—
90#494—61no—
91#492—61no—
92#489—21no—
93#463—22no—
94#448—21no—
95#447—21no—
96#432—31no—
97#432C2×D311yescharacteristic
98C4×M16—31no—
99#397—62no—
100#393—62no—
101#391—61no—
102#390—61no—
103#376—31no—
104#333—62no—
105#329—62no—
106#327—62no—
107C24×C3—21no—
108C3×#41—44no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—42no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C2×C3×D4—41no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×Dic3—22no—
126C22×Dic3—22no—
127C2×Dic6—22no—
128C2×Dic6—22no—
129C2×Dic6—44no—
130C3×SD16—82no—
131C3×D8—82no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#217—84no—
146#216—48no—
147#214—48no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C4×Dic3—42no—
153#208—24no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×#41—21no—
157C22×Q8—32no—
158C22×Q8#6711yescharacteristic
159C22×D4#6711yescharacteristic
160#137—128no—
161C2×Q16—64no—
162C2×SD16—22no—
163C2×SD16—64no—
164C2×D8—22no—
165C2×M16—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8#6711yescharacteristic
171#127#6711yescharacteristic
172#124—21no—
173#124—21no—
174#124—21no—
175#124—21no—
176#124—21no—
177#121—21no—
178#121—21no—
179C4×D4—21no—
180C4×D4—21no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—64no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—61no—
189C4⋊C8#6712yes—
190#103—22no—
191#102—22no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194#97—21no—
195#97—32no—
196#97—61no—
197C4×C8—61no—
198#95—32no—
199#95—64no—
200#95—61no—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—42no—
206C23×C3#12011yescharacteristic
207C3×Q8—42no—
208C3×Q8—42no—
209C3×Q8—81no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—81no—
214C3×D4—81no—
215C3×D4—82no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—82no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—44no—
234Dic6—44no—
235Dic6—82no—
236C3×C8—42no—
237C3⋊C8—24no—
238C3⋊C8—81no—
239C24—21no—
240#41—44no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—42no—
245C2×Q8—62no—
246C2×Q8—62no—
247C2×Q8—124no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—42no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—128no—
263SD16—82no—
264SD16—128no—
265D8—82no—
266M16—64no—
267C2×C8—22no—
268C2×C8—62no—
269C2×C8—121no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—244no—
283C42—122no—
284C42D3×D411yescharacteristic
285C42#23711yescharacteristic
286C42C2×#6711yescharacteristic
287C42C2×#6711yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—84no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#39211yescharacteristic
298C22×C3#45312yes—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—81no—
305C3×C4—81no—
306C3×C4#45512yes—
307Dic3—82no—
308Dic3—82no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23#93711yescharacteristic
315Q8—42no—
316Q8—42no—
317Q8—81no—
318Q8—124no—
319Q8—242no—
320D4—42no—
321D4—42no—
322D4—42no—
323D4—81no—
324D4—81no—
325D4—82no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—82no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—124no—
337C2×C4—122no—
338C2×C4#93711yescharacteristic
339C2×C4#93811yescharacteristic
340C2×C4#93911yescharacteristic
341C2×C4#93911yescharacteristic
342C2×C4#94011yescharacteristic
343C2×C4#95311yescharacteristic
344C8—42no—
345C8—64no—
346C8—241no—
347C2×C3—22no—
348C2×C3—161no—
349C2×C3—161no—
350C2×C3#200211yescharacteristic
351C2×C3#200411yescharacteristic
352C2×C3#200911yescharacteristic
353C22—21no—
354C22—21no—
355C22—84no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#519812yes—
363C22#522311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—81no—
370C4—81no—
371C4—242no—
372C4—242no—
373C4#524212yes—
374C3#1009011yescharacteristic, complemented, Sylow
375C2—22no—
376C2—161no—
377C2—161no—
378C2#7185711yescharacteristic
379C2#7209411yescharacteristic
380C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3213C2
socleC22×C3#392
hypercenter#3213D3
hypocenterC3#10090
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3213

Complements

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

NH ≅ G/Ntype
2. #83288C2semidirect
3. #82978C2semidirect
4. #71812C2semidirect
5. #71718C2semidirect
6. #70824C2semidirect
7. #67748C2semidirect
374. C3#10090semidirect

Maximal subgroups

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

subgroupindex
C3×#32132
#832882
#829782
#718122
#717182
#708242
#677482
#100903