Finite Group Databasebeta

← #10696494

Subgroups — #10696494

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×#4715C211yesmaximal, characteristic
2C2×#4702C211yesmaximal, characteristic, complemented
3#74255C211yesmaximal, characteristic
4C2×#4465C211yesmaximal, characteristic, complemented
5C2×#4464C211yesmaximal, characteristic
6#69555C211yesmaximal, characteristic, complemented
7C3×#1295C211yesmaximal, characteristic, complemented
8#10082—31nomaximal, Sylow
9C22×Dic12—21no—
10C22×#204—21no—
11C2×#875C2211yescharacteristic
12C2×#870C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15C2×C3×#97C2211yescharacteristic
16C3×#382C2211yescharacteristic
17C2×#818—21no—
18C2×#817—21no—
19#4994—22no—
20C2×#803C2211yescharacteristic
21#4715—22no—
22#4715—22no—
23#4702—22no—
24#4702—22no—
25#4677—21no—
26#4674—21no—
27#4672—21no—
28#4656—21no—
29#4529—22no—
30#4465—22no—
31#4464—22no—
32C2×#501—31no—
33C2×#497—31no—
34#2820—31no—
35#1993—31no—
36C2×#330—31no—
37C2×#329—31no—
38#1295D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C2×C3×C4⋊C4D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C22×C3⋊C8—21no—
58C2×Dic12—24no—
59C2×Dic12—44no—
60C2×#204—24no—
61C2×#204—44no—
62#875—21no—
63#875—21no—
64#875—21no—
65#875—21no—
66#870—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70#869—21no—
71#869—21no—
72#869—21no—
73#869D412yes—
74C3×#97—21no—
75C3×#97—21no—
76C3×#97D412yes—
77C3×#97D412yes—
78C3×#95—21no—
79C3×#95—21no—
80C3⋊#95—22no—
81#818—24no—
82#817—24no—
83#803—21no—
84#803—21no—
85C22×Q16—61no—
86C22×SD16—61no—
87C2×C4⋊Q8—31no—
88C2×#125—31no—
89C2×#124—31no—
90#501—62no—
91#501—62no—
92#497—62no—
93#497—62no—
94C2×#107—61no—
95C2×#106—61no—
96C2×#97—31no—
97C2×#97C2×D311yescharacteristic
98C2×#97C2×D311yescharacteristic
99#399—61no—
100#397—61no—
101#396—62no—
102#395—61no—
103#394—61no—
104#382C2×D311yescharacteristic
105#340—62no—
106#330—62no—
107#329—62no—
108C23×D3—21no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—41no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—42no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—42no—
124C2×Dic6—42no—
125C2×Dic6—42no—
126C2×Dic6—41no—
127C2×C3×C8—24no—
128C2×C3×C8—24no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C3×C4⋊C4—42no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—82no—
141(C2×D3)⋊C4—82no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—82no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158Dic12—48no—
159#204—48no—
160C22×Q8—31no—
161C22×Q8—61no—
162C22×D4—31no—
163C2×Q16—64no—
164C2×Q16—124no—
165C2×SD16—64no—
166C2×SD16—124no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—61no—
170C4⋊Q8—32no—
171C4⋊Q8—61no—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174#125—61no—
175#125—61no—
176#125—61no—
177#125—61no—
178#124—61no—
179#124—61no—
180#124—61no—
181#124—61no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4D1212yes—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C42C22×D311yescharacteristic
195#107—64no—
196#106—64no—
197#97—21no—
198#97—21no—
199#97—61no—
200#97—61no—
201#97D1212yes—
202#97D1212yes—
203#95—21no—
204#95—21no—
205#95—62no—
206C23×C3#12011yescharacteristic
207C22×D3—41no—
208C22×D3—42no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—42no—
212C2×C3×C4—22no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—22no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—82no—
234D12—42no—
235D12—81no—
236Dic6—42no—
237Dic6—42no—
238Dic6—42no—
239Dic6—82no—
240Dic6—81no—
241C3×C8—24no—
242C3×C8—24no—
243C3⋊C8—42no—
244C24—61no—
245C2×Q8—61no—
246C2×Q8—62no—
247C2×Q8—61no—
248C2×Q8—122no—
249C2×Q8—122no—
250C2×Q8—122no—
251C2×Q8—121no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—122no—
255C22×C4—22no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—121no—
260C22×C4C2×D1211yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263Q16—128no—
264SD16—128no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—62no—
270C2×C8—121no—
271C2×C8—121no—
272C4⋊C4—42no—
273C4⋊C4—61no—
274C4⋊C4—62no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—62no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—242no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—242no—
291C22⋊C4—242no—
292C42C2×D1211yescharacteristic
293C42C2×D1211yescharacteristic
294C42D3×D411yescharacteristic
295C42#23911yescharacteristic
296C22×C3#39211yescharacteristic
297C22×C3#45312yes—
298C22×C3#45511yescharacteristic
299C22×C3#45611yescharacteristic
300C22×C3#45612yes—
301C2×D3—82no—
302C2×D3—82no—
303C2×D3—81no—
304C2×D3—81no—
305C2×D3—81no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—82no—
313Dic3—82no—
314Dic3—81no—
315Dic3—81no—
316Dic3—81no—
317C23—121no—
318C23—121no—
319C23—121no—
320C23—122no—
321C23—122no—
322C23#88211yescharacteristic
323Q8—122no—
324Q8—122no—
325Q8—122no—
326Q8—241no—
327Q8—242no—
328D4—122no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—122no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—242no—
346C2×C4#87411yescharacteristic
347C2×C4#88211yescharacteristic
348C2×C4#88211yescharacteristic
349C2×C4#89511yescharacteristic
350C2×C4#89511yescharacteristic
351C2×C4#89511yescharacteristic
352C8—24no—
353C8—24no—
354C8—122no—
355C2×C3#166012yes—
356C2×C3#166112yes—
357C2×C3#200211yescharacteristic
358C2×C3#200511yescharacteristic
359C2×C3#200711yescharacteristic
360D3—161no—
361C22—242no—
362C22—242no—
363C22—241no—
364C22—241no—
365C22—241no—
366C22#467111yescharacteristic
367C22#474611yescharacteristic
368C22#475211yescharacteristic
369C22#482212yes—
370C22#482612yes—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—241no—
379C4—241no—
380C4—241no—
381C4—242no—
382C3#1008211yescharacteristic, complemented, Sylow
383C2—481no—
384C2#6835712yes—
385C2#6837212yes—
386C2#6876111yescharacteristic
387C2#6876811yescharacteristic
388C2#6900911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1295D3
hypocenterC3#10082
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. C2×#4702C2semidirect
4. C2×#4465C2semidirect
6. #69555C2semidirect
7. C3×#1295C2semidirect
38. #1295D3semidirect
382. C3#10082semidirect

Maximal subgroups

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

subgroupindex
C2×#47152
C2×#47022
#742552
C2×#44652
C2×#44642
#695552
C3×#12952
#100823