Finite Group Databasebeta

← #10697241

Subgroups — #10697241

Order 768.

496 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 105 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×#474C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic, complemented
3#82296C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic, complemented
5#71601C211yesmaximal, characteristic
6#70429C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10190—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#474—22no—
13C3×#474—22no—
14C2×C3×#107—21no—
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×#102—21no—
17#5215—21no—
18#5093—21no—
19#5091—22no—
20#5078—21no—
21#5009—22no—
22#5008—21no—
23#5007—21no—
24#4970—22no—
25#4964—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30#4525—22no—
31#4484—22no—
32#3230—31no—
33#3077—31no—
34C2×#474D311yescharacteristic
35#1548—31no—
36#1481—31no—
37#1460—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56#926—21no—
57#924—21no—
58#924—21no—
59C2×#208—21no—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×#107—24no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×#102—24no—
69#832—22no—
70#832—22no—
71#830—21no—
72#830D412yes—
73#810—24no—
74#808—22no—
75#807—22no—
76#807—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×C4×D4C2×D311yescharacteristic
86#500—61no—
87#487—61no—
88#484—62no—
89#478—62no—
90#477—62no—
91#476—61no—
92#474—22no—
93#474—22no—
94#474—61no—
95#431—31no—
96#431—31no—
97#428—61no—
98C2×#107—21no—
99#423—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101#418—61no—
102C2×#102—21no—
103#333—62no—
104Q8⋊C8—62no—
105#323—62no—
106C24×C3—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—41no—
109C2×C3×Q8—82no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4Q1612yes—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—82no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141#217—41no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—41no—
149#208—41no—
150#208—41no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—41no—
154C22×Q8—21no—
155C22×D4#6711yescharacteristic
156C23×C4—21no—
157C2×M16—61no—
158C22×C8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8#6712yes—
165#125—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174#117—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4#6711yescharacteristic
178C2×C22⋊C4—21no—
179C2×C42C22×D311yescharacteristic
180#107—24no—
181#107—62no—
182#107—64no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—32no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8#6712yes—
190#103—64no—
191#102—24no—
192#102—62no—
193#102—62no—
194C22⋊C8—32no—
195C22⋊C8—61no—
196C4×C8—61no—
197C4×C8—61no—
198C23×C3—22no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—81no—
202C23×C3—81no—
203C23×C3#12111yescharacteristic
204C3×Q8—82no—
205C3×Q8—82no—
206C3×D4—22no—
207C3×D4—22no—
208C3×D4—41no—
209C3×D4—41no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—82no—
215C2×C3×C4—82no—
216C2×C3×C4—82no—
217C2×C3×C4—81no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×C3×C4C2×Q1611yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4#13711yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×Dic3—41no—
229C2×Dic3—41no—
230Dic6—42no—
231Dic6—42no—
232C3×C8—42no—
233C3⋊C8—42no—
234C3⋊C8—42no—
235C3⋊C8—81no—
236C24—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—82no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4#21612yes—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—82no—
253C22×C4D3×D411yescharacteristic
254C22×C4#23711yescharacteristic
255C22×C4C2×#6711yescharacteristic
256M16—121no—
257M16—121no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—121no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C42—121no—
280C42D3×D411yescharacteristic
281C42D3×D411yescharacteristic
282C42C2×#6711yescharacteristic
283C42C2×#6711yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—42no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#39211yescharacteristic
292C22×C3#46711yescharacteristic
293C22×C3#47011yescharacteristic
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—81no—
298C3×C4—82no—
299C3×C4#45111yescharacteristic
300C3×C4#45111yescharacteristic
301C3×C4#45211yescharacteristic
302C3×C4#45211yescharacteristic
303C3×C4#47212yes—
304Dic3—81no—
305Dic3—81no—
306C23—22no—
307C23—41no—
308C23—41no—
309C23—81no—
310C23—81no—
311C23#93911yescharacteristic
312Q8—82no—
313Q8—82no—
314Q8—122no—
315Q8—122no—
316D4—22no—
317D4—22no—
318D4—41no—
319D4—41no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4#93711yescharacteristic
331C2×C4#93711yescharacteristic
332C2×C4#93811yescharacteristic
333C2×C4#93911yescharacteristic
334C2×C4#94011yescharacteristic
335C2×C4#94211yescharacteristic
336C2×C4C2×#21611yescharacteristic
337C2×C4#95011yescharacteristic
338C2×C4#95111yescharacteristic
339C2×C4#95311yescharacteristic
340C8—42no—
341C8—122no—
342C8—122no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—82no—
347C2×C3#200111yescharacteristic
348C2×C3#200911yescharacteristic
349C2×C3#201811yescharacteristic
350C22—21no—
351C22—21no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—81no—
356C22—81no—
357C22#505111yescharacteristic
358C22#505211yescharacteristic
359C22#522311yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—82no—
365C4—241no—
366C4—241no—
367C4#504711yescharacteristic
368C4#505011yescharacteristic
369C4#516012yes—
370C4#522811yescharacteristic
371C4#523911yescharacteristic
372C3#1019011yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—82no—
376C2#7170011yescharacteristic
377C2#7204611yescharacteristic
378C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#474C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenterC2×#474D3
hypocenterC3#10190
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
2. #83224C2semidirect
4. #71774C2semidirect
6. #70429C2semidirect
7. #67707C2semidirect
372. C3#10190semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4742
#832242
#822962
#717742
#716012
#704292
#677072
#101903