Finite Group Databasebeta

← #10696488

Subgroups — #10696488

Order 768.

590 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×C3×#497C211yesmaximal, characteristic
2C2×#5081C211yesmaximal, characteristic, complemented
3#82992C211yesmaximal, characteristic
4#71788C211yesmaximal, characteristic, complemented
5C2×#4540C211yesmaximal, characteristic
6C2×#4465C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10082—31nomaximal, Sylow
9C22×C3×Q16—21no—
10C22×C3×SD16—21no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#124C2211yescharacteristic
13C2×#870C2211yescharacteristic
14C3×#497—22no—
15C3×#497—22no—
16C2×C3×#97C2211yescharacteristic
17#5230—21no—
18#5222—21no—
19#5081—22no—
20#5081—22no—
21C2×#808—21no—
22C2×#807—21no—
23#4953—21no—
24#4949—21no—
25#4948—22no—
26#4936C2211yescharacteristic
27C2×#803C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4540—22no—
30#4529—22no—
31#4465—22no—
32C2×#501—31no—
33C2×#497D311yescharacteristic
34#2820—31no—
35#1993—31no—
36C2×#330—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C2×C3×Q16—24no—
43C2×C3×Q16—44no—
44C2×C3×SD16—24no—
45C2×C3×SD16—44no—
46C22×C3×C8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8D412yes—
51C3×#124—21no—
52C3×#124—21no—
53C3×#124—21no—
54C3×#124—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61C2×#217—22no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D412yes—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—22no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70#870—21no—
71#870—21no—
72#870—21no—
73#870—21no—
74C3×#97—21no—
75C3×#97D412yes—
76C3⋊#95—22no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#808—24no—
80#807—24no—
81#803—21no—
82#803—21no—
83#803—21no—
84#803D412yes—
85C22×Q16—21no—
86C22×SD16—21no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#125—31no—
89C2×#124C2×D311yescharacteristic
90#501—62no—
91#501—62no—
92#497—22no—
93#497—22no—
94C2×#107—61no—
95C2×#106—61no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97C2×D311yescharacteristic
99#399—61no—
100#397—61no—
101#396—62no—
102#395—61no—
103#394—61no—
104#382—31no—
105#340—62no—
106#330—62no—
107#329—62no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×Q8—42no—
115C2×C3×Q8—41no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C22×C3×C4—21no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C3×Q16—48no—
128C3×SD16—48no—
129C2×C3×C8—24no—
130C2×C3×C8—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—82no—
144#217—82no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—42no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—82no—
155C2×C3⋊C8—24no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—21no—
161C22×Q8#6711yescharacteristic
162C22×D4#6711yescharacteristic
163C2×Q16—24no—
164C2×Q16—44no—
165C2×SD16—24no—
166C2×SD16—44no—
167C22×C8—21no—
168C22×C8—61no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8#6712yes—
174#125—61no—
175#125—61no—
176#125—61no—
177#125—61no—
178#124—21no—
179#124—21no—
180#124—21no—
181#124—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—62no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C42C22×D311yescharacteristic
195#107—64no—
196#106—64no—
197#97—21no—
198#97—32no—
199#97—61no—
200#97—61no—
201#97—61no—
202#97#6712yes—
203#95—62no—
204#95—61no—
205#95—61no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—42no—
211C23×C3#12011yescharacteristic
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—42no—
215C3×Q8—81no—
216C3×Q8—82no—
217C3×D4—42no—
218C3×D4—81no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—82no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C3×C8—24no—
242C3⋊C8—24no—
243C3⋊C8—42no—
244C24—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—42no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—62no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4D3×D411yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4C2×#6711yescharacteristic
263Q16—48no—
264SD16—48no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—64no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—121no—
271C2×C8—121no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—122no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—242no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—242no—
291C22⋊C4—242no—
292C42D3×D411yescharacteristic
293C42#23711yescharacteristic
294C42C2×#6711yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—82no—
297C22×C3—82no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3#39211yescharacteristic
302C22×C3#45312yes—
303C22×C3#45511yescharacteristic
304C22×C3#45611yescharacteristic
305C22×C3#45612yes—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—81no—
313C3×C4—82no—
314Dic3—82no—
315Dic3—81no—
316Dic3—81no—
317C23—41no—
318C23—41no—
319C23—42no—
320C23—41no—
321C23—42no—
322C23#93711yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—82no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4#93711yescharacteristic
347C2×C4#93811yescharacteristic
348C2×C4#93911yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4#94011yescharacteristic
351C2×C4#95311yescharacteristic
352C8—24no—
353C8—64no—
354C8—122no—
355C2×C3—161no—
356C2×C3#166012yes—
357C2×C3#166112yes—
358C2×C3#200211yescharacteristic
359C2×C3#200511yescharacteristic
360C2×C3#200711yescharacteristic
361C22—81no—
362C22—82no—
363C22—82no—
364C22—81no—
365C22—81no—
366C22#507712yes—
367C22#520211yescharacteristic
368C22#520511yescharacteristic
369C22#520512yes—
370C22#522311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—82no—
379C4—241no—
380C4—242no—
381C4—241no—
382C3#1008211yescharacteristic, complemented, Sylow
383C2—161no—
384C2#7133012yes—
385C2#7133112yes—
386C2#7176911yescharacteristic
387C2#7210811yescharacteristic
388C2#7211011yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#497C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenterC2×#497D3
hypocenterC3#10082
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
2. C2×#5081C2semidirect
4. #71788C2semidirect
6. C2×#4465C2semidirect
7. #67659C2semidirect
382. C3#10082semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4972
C2×#50812
#829922
#717882
C2×#45402
C2×#44652
#676592
#100823