Finite Group Databasebeta

← #10697404

Subgroups — #10697404

Order 768.

464 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3254C211yesmaximal, characteristic
2#83372C211yesmaximal, characteristic, complemented
3#83319C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic, complemented
5#71812C211yesmaximal, characteristic
6#70515C211yesmaximal, characteristic, complemented
7#67716C211yesmaximal, characteristic, complemented
8#10215—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#477—21no—
13C3×#474—21no—
14C3×#468—22no—
15C3×#461—21no—
16C3×#460—21no—
17C3×#423C2211yescharacteristic
18#5239—22no—
19#5094—21no—
20#5091—21no—
21#5090—21no—
22#5075—21no—
23#5069—21no—
24#5064—21no—
25#5060—21no—
26#5055—21no—
27#4996—22no—
28#4932C2211yescharacteristic
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31#4555—22no—
32#4538—22no—
33#4537—22no—
34#3254D311yescharacteristic
35#3237—31no—
36#3216—31no—
37#1995—31no—
38#1506—31no—
39#1505—31no—
40#1307—31no—
41C2×C3×#41—21no—
42C2×C3×SD16—22no—
43C2×C3×M16D412yes—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#125—21no—
49C3×#124—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×#117—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#216—22no—
61C2×#214—22no—
62#926—21no—
63#924—21no—
64#924—21no—
65C2×C4×Dic3—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#107—22no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#102—22no—
76#835—22no—
77#832—22no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830D412yes—
81#830D412yes—
82#810—22no—
83#810—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90#533—31no—
91#533C2×D311yescharacteristic
92#532C2×D311yescharacteristic
93#501—61no—
94#497—61no—
95#491—61no—
96#489—61no—
97#485—62no—
98#477—21no—
99#477—61no—
100#475—61no—
101#474—21no—
102#468—22no—
103#462—61no—
104#461—21no—
105#460—21no—
106#460—61no—
107#452—62no—
108#432—31no—
109#432—31no—
110#423C2×D311yescharacteristic
111#376—31no—
112#339—62no—
113#328—62no—
114#327—62no—
115C3×#41—82no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×Dic3—41no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C3×SD16—82no—
131C3×M16—42no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—41no—
150#216—82no—
151#214—82no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C4×Dic3—22no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×#41—21no—
164C2×Q16—62no—
165C2×SD16—22no—
166C2×SD16—62no—
167C2×M16#6712yes—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170C4⋊Q8#6711yescharacteristic
171C4⋊Q8#6711yescharacteristic
172C4⋊Q8#6711yescharacteristic
173#127#6711yescharacteristic
174#125—21no—
175#125—61no—
176#124—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181#121—21no—
182#121—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—61no—
186C4×Q8—61no—
187C4×D4—21no—
188C4×D4—21no—
189#117—21no—
190#117—61no—
191C2×C42—61no—
192C2×C42C22×D311yescharacteristic
193#107—22no—
194#107—62no—
195#106—62no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#103—22no—
201#103—62no—
202#103—62no—
203#103—62no—
204#102—22no—
205#102—62no—
206C22⋊C8—32no—
207C22⋊C8—32no—
208#97—61no—
209#97—61no—
210#95—61no—
211#95—61no—
212C23×C3—41no—
213C23×C3#12711yescharacteristic
214C3×Q8—42no—
215C3×Q8—41no—
216C3×Q8—41no—
217C3×Q8—81no—
218C3×D4—42no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—81no—
222C3×D4—82no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—82no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242Dic6—41no—
243Dic6—41no—
244Dic6—42no—
245C3×C8—42no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248#41—82no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C22×C4—41no—
259C22×C4—121no—
260C22×C4D3×D411yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4C2×#6711yescharacteristic
263Q16—242no—
264SD16—82no—
265SD16—242no—
266M16—42no—
267C2×C8—22no—
268C2×C8—62no—
269C2×C8—62no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—121no—
288C42—41no—
289C42—62no—
290C42—121no—
291C42—121no—
292C42D3×D411yescharacteristic
293C42#23711yescharacteristic
294C42C2×#6711yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3#39211yescharacteristic
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4#45511yescharacteristic
312C3×C4#45611yescharacteristic
313Dic3—82no—
314Dic3—81no—
315Dic3—81no—
316C23—41no—
317C23#94011yescharacteristic
318Q8—42no—
319Q8—41no—
320Q8—41no—
321Q8—81no—
322Q8—121no—
323Q8—121no—
324Q8—122no—
325D4—42no—
326D4—41no—
327D4—41no—
328D4—81no—
329D4—82no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—82no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4#93711yescharacteristic
344C2×C4#93711yescharacteristic
345C2×C4#93811yescharacteristic
346C2×C4#93911yescharacteristic
347C2×C4#93911yescharacteristic
348C2×C4#95311yescharacteristic
349C8—42no—
350C8—122no—
351C8—122no—
352C2×C3—41no—
353C2×C3—161no—
354C2×C3#200711yescharacteristic
355C2×C3#200811yescharacteristic
356C2×C3#200911yescharacteristic
357C22—21no—
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#522311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—242no—
373C4—241no—
374C4—241no—
375C4#524211yescharacteristic
376C4#524611yescharacteristic
377C3#1021511yescharacteristic, complemented, Sylow
378C2—41no—
379C2—161no—
380C2#7188111yescharacteristic
381C2#7209911yescharacteristic
382C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3254C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3254D3
hypocenterC3#10215
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3254

Complements

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

NH ≅ G/Ntype
2. #83372C2semidirect
4. #71836C2semidirect
6. #70515C2semidirect
7. #67716C2semidirect
377. C3#10215semidirect

Maximal subgroups

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

subgroupindex
C3×#32542
#833722
#833192
#718362
#718122
#705152
#677162
#102153