Finite Group Databasebeta

← #10696484

Subgroups — #10696484

Order 768.

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

normal subgroups only

This group has 648 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×#2819C211yesmaximal, characteristic
2C2×#5060C211yesmaximal, characteristic, complemented
3C2×#5055C211yesmaximal, characteristic, complemented
4C2×#4540C211yesmaximal, characteristic, complemented
5C2×#4540C211yesmaximal, characteristic, complemented
6#69555C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#10081—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C22×#216—21no—
12C22×#214—21no—
13C2×#869C2211yescharacteristic
14C3×#394—22no—
15C3×#393—21no—
16C3×#393—21no—
17C3×#392—21no—
18C3×#392—21no—
19C3×#382C2211yescharacteristic
20C2×#835—21no—
21C2×#832—21no—
22#5060—22no—
23#5060—22no—
24#5055—22no—
25#5055—22no—
26C2×#803C2211yescharacteristic
27C2×#803C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4540—22no—
30#4540—22no—
31#4529—22no—
32C2×#497—31no—
33C2×#489—31no—
34#2819D311yescharacteristic
35#1993—31no—
36C2×#329—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4—22no—
42C22×C3×D4D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C22×Dic6—21no—
45C3×#124—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—21no—
52C3×#124—21no—
53C2×C3×C4⋊C4D412yes—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
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#869D412yescomplemented
74C3×#95—22no—
75C3×#95—21no—
76C3×#95—21no—
77#835—24no—
78#832—24no—
79#803—21no—
80#803—21no—
81#803—21no—
82#803—21no—
83#803D412yescomplemented
84#803D412yescomplemented
85C22×Q16—61no—
86C22×SD16—61no—
87C2×C4⋊Q8—31no—
88C2×#124C2×D311yescharacteristic
89C2×#124C2×D311yescharacteristic
90#497—62no—
91#497—62no—
92#489—62no—
93#489—62no—
94C2×#103—61no—
95C2×#102—61no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97—31no—
99#394—22no—
100#393—21no—
101#393—21no—
102#392—21no—
103#392—21no—
104#382C2×D311yescharacteristic
105#340—62no—
106#329—62no—
107#329—62no—
108C24×C3—21no—
109C24×C3—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—42no—
115C2×C3×Q8—42no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—82no—
123C2×C3×D4—82no—
124C2×C3×D4—82no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3—41no—
132C2×Dic6—22no—
133C2×Dic6—42no—
134C2×Dic6—42no—
135C2×Dic6—41no—
136C3×C4⋊C4—42no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—82no—
146C3×C22⋊C4—82no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#216—48no—
152#214—48no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C2×C3⋊C8—24no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—42no—
163C2×C3⋊C8—41no—
164C22×Q8—61no—
165C22×Q8#6711yescharacteristic
166C22×Q8#6711yescharacteristic
167C22×D4—22no—
168C22×D4#6711yescharacteristic
169C22×D4#6711yescharacteristic
170C2×Q16—64no—
171C2×Q16—124no—
172C2×SD16—64no—
173C2×SD16—124no—
174C22×C8—61no—
175C22×C8—61no—
176C22×C8—61no—
177C4⋊Q8—32no—
178C4⋊Q8—61no—
179C4⋊Q8—61no—
180C4⋊Q8—61no—
181#124—21no—
182#124—21no—
183#124—21no—
184#124—21no—
185#124—21no—
186#124—21no—
187#124—21no—
188#124—21no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6712yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42C22×D311yescharacteristic
199#103—64no—
200#102—64no—
201#97—32no—
202#97—32no—
203#97—61no—
204#97—61no—
205#97—61no—
206#97—61no—
207#95—22no—
208#95—21no—
209#95—21no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3#12011yescharacteristic
221C3×Q8—42no—
222C3×Q8—42no—
223C3×Q8—81no—
224C3×Q8—81no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—81no—
228C3×D4—81no—
229C3×D4—88no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12711yescharacteristic
247C2×Dic3—42no—
248C2×Dic3—82no—
249Dic6—42no—
250Dic6—42no—
251Dic6—82no—
252C3⋊C8—24no—
253C3⋊C8—42no—
254C3⋊C8—42no—
255C24—21no—
256C24—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—62no—
264C2×Q8—122no—
265C2×Q8—122no—
266C2×Q8—121no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4—82no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—121no—
280C22×C4C2×#6711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282C22×C4C2×#6711yescharacteristic
283Q16—128no—
284SD16—128no—
285C2×C8—64no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—121no—
290C2×C8—122no—
291C2×C8—121no—
292C4⋊C4—42no—
293C4⋊C4—62no—
294C4⋊C4—62no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C42D3×D411yescharacteristic
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42#23711yescharacteristic
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3#39211yescharacteristic
322C22×C3#45312yes—
323C22×C3#45511yescharacteristic
324C22×C3#45512yes—
325C22×C3#45611yescharacteristic
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333C3×C4—81no—
334C3×C4—81no—
335Dic3—82no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23#95311yescharacteristic
347Q8—42no—
348Q8—42no—
349Q8—81no—
350Q8—81no—
351Q8—122no—
352Q8—122no—
353Q8—242no—
354D4—42no—
355D4—42no—
356D4—81no—
357D4—81no—
358D4—88no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—122no—
371C2×C4—242no—
372C2×C4#93711yescharacteristic
373C2×C4#93711yescharacteristic
374C2×C4#93811yescharacteristic
375C2×C4#93911yescharacteristic
376C2×C4#93911yescharacteristic
377C2×C4#94011yescharacteristic
378C8—64no—
379C8—122no—
380C8—122no—
381C2×C3—161no—
382C2×C3—161no—
383C2×C3#165912yes—
384C2×C3#166112yes—
385C2×C3#200211yescharacteristic
386C2×C3#200411yescharacteristic
387C2×C3#200711yescharacteristic
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22#522311yescharacteristic
399C22#524112yes—
400C22#524211yescharacteristic
401C22#524212yes—
402C22#524611yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4—242no—
413C3#1008111yescharacteristic, complemented, Sylow
414C2—161no—
415C2—161no—
416C2#7185611yescharacteristic
417C2#7185711yescharacteristic
418C2#7188111yescharacteristic
419C2#7194912yes—
420C2#7195312yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2819C2
socleC23×C3#120
hypercenter#2819D3
hypocenterC3#10081
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2819

Complements

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

NH ≅ G/Ntype
2. C2×#5060C2semidirect
3. C2×#5055C2semidirect
4. C2×#4540C2semidirect
5. C2×#4540C2semidirect
6. #69555C2semidirect
7. #67630C2semidirect
73. #869D4semidirect
83. #803D4semidirect
84. #803D4semidirect
413. C3#10081semidirect

Maximal subgroups

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

subgroupindex
C3×#28192
C2×#50602
C2×#50552
C2×#45402
#695552
#676302
#100813