Finite Group Databasebeta

← #10696491

Subgroups — #10696491

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
1C3×#2820C211yesmaximal, characteristic
2C2×#5064C211yesmaximal, characteristic, complemented
3C2×#5060C211yesmaximal, characteristic
4C2×#4543C211yesmaximal, characteristic, complemented
5C2×#4540C211yesmaximal, characteristic
6#69555C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#10082—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C22×#216—21no—
12C22×#214—21no—
13C2×#869C2211yescharacteristic
14C3×#399—21no—
15C3×#397—21no—
16C3×#396—22no—
17C3×#395—21no—
18C3×#394—21no—
19C3×#382C2211yescharacteristic
20#5064—22no—
21#5064—22no—
22#5060—22no—
23#5060—22no—
24C2×#808—21no—
25C2×#807—21no—
26C2×#803C2211yescharacteristic
27C2×#803C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4543—22no—
30#4540—22no—
31#4529—22no—
32C2×#501—31no—
33C2×#497—31no—
34#2820D311yescharacteristic
35#1993—31no—
36C2×#330—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D412yes—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42C2311yescharacteristic
61C2×#216—24no—
62C2×#216—44no—
63C2×#214—24no—
64C2×#214—44no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70#869—21no—
71#869—21no—
72#869—21no—
73#869D412yes—
74C3×#95—22no—
75C3×#95—21no—
76C3×#95—21no—
77#808—24no—
78#807—24no—
79#803—21no—
80#803—21no—
81#803—21no—
82#803—21no—
83#803D412yes—
84#803D412yes—
85C22×Q16—61no—
86C22×SD16—61no—
87C2×C4⋊Q8—31no—
88C2×#125C2×D311yescharacteristic
89C2×#124C2×D311yescharacteristic
90#501—62no—
91#501—62no—
92#497—62no—
93#497—62no—
94C2×#107—61no—
95C2×#106—61no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97—31no—
99#399—21no—
100#397—21no—
101#396—22no—
102#395—21no—
103#394—21no—
104#382C2×D311yescharacteristic
105#340—62no—
106#330—62no—
107#329—62no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—41no—
123C2×Dic6—22no—
124C2×Dic6—42no—
125C2×Dic6—42no—
126C2×Dic6—41no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—82no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—82no—
142C3×C22⋊C4—82no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#216—48no—
148#214—48no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C2×C3⋊C8—24no—
154C2×C3⋊C8—24no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—61no—
161C22×Q8#6711yescharacteristic
162C22×D4#6711yescharacteristic
163C2×Q16—64no—
164C2×Q16—124no—
165C2×SD16—64no—
166C2×SD16—124no—
167C22×C8—61no—
168C22×C8—61no—
169C22×C8—61no—
170C4⋊Q8—32no—
171C4⋊Q8—61no—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174#125—21no—
175#125—21no—
176#125—21no—
177#125—21no—
178#124—21no—
179#124—21no—
180#124—21no—
181#124—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4#6711yescharacteristic
188C2×C4⋊C4#6711yescharacteristic
189C2×C4⋊C4#6712yes—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C42C22×D311yescharacteristic
195#107—64no—
196#106—64no—
197#97—32no—
198#97—32no—
199#97—61no—
200#97—61no—
201#97—61no—
202#97—61no—
203#95—22no—
204#95—21no—
205#95—21no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—42no—
211C23×C3#12011yescharacteristic
212C3×Q8—42no—
213C3×Q8—81no—
214C3×D4—42no—
215C3×D4—81no—
216C2×C3×C4—22no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×Dic3—42no—
237C2×Dic3—82no—
238Dic6—42no—
239Dic6—42no—
240Dic6—82no—
241C3⋊C8—24no—
242C3⋊C8—24no—
243C3⋊C8—42no—
244C24—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—42no—
248C2×Q8—62no—
249C2×Q8—122no—
250C2×Q8—122no—
251C2×Q8—121no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—42no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—121no—
260C22×C4C2×#6711yescharacteristic
261C22×C4C2×#6711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263Q16—128no—
264SD16—128no—
265C2×C8—64no—
266C2×C8—64no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—121no—
271C2×C8—121no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—82no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—82no—
291C22⋊C4—82no—
292C42D3×D411yescharacteristic
293C42D3×D411yescharacteristic
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
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—82no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4—81no—
316Dic3—82no—
317C23—41no—
318C23—42no—
319C23—41no—
320C23—42no—
321C23—41no—
322C23#95311yescharacteristic
323Q8—42no—
324Q8—81no—
325Q8—122no—
326Q8—122no—
327Q8—242no—
328D4—42no—
329D4—81no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—122no—
345C2×C4—242no—
346C2×C4#93711yescharacteristic
347C2×C4#93711yescharacteristic
348C2×C4#93811yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4#93911yescharacteristic
351C2×C4#94011yescharacteristic
352C8—64no—
353C8—64no—
354C8—122no—
355C2×C3—161no—
356C2×C3#166012yes—
357C2×C3#166112yes—
358C2×C3#200211yescharacteristic
359C2×C3#200511yescharacteristic
360C2×C3#200711yescharacteristic
361C22—82no—
362C22—82no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22#522311yescharacteristic
367C22#524112yes—
368C22#524211yescharacteristic
369C22#524612yes—
370C22#524611yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—242no—
382C3#1008211yescharacteristic, complemented, Sylow
383C2—161no—
384C2#7185611yescharacteristic
385C2#7186811yescharacteristic
386C2#7188111yescharacteristic
387C2#7195312yes—
388C2#7195612yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2820C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenter#2820D3
hypocenterC3#10082
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2820

Complements

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

NH ≅ G/Ntype
2. C2×#5064C2semidirect
4. C2×#4543C2semidirect
6. #69555C2semidirect
7. #67630C2semidirect
382. C3#10082semidirect

Maximal subgroups

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

subgroupindex
C3×#28202
C2×#50642
C2×#50602
C2×#45432
C2×#45402
#695552
#676302
#100823